Check out Cayley's nodal cubic surface: http://blogs.ams.org/visualinsight/2016/08/15/cayleys-nodal-cubic-surface/ pic.twitter.com/g6PXhkJR5h

Large countable ordinals are cool - start here and read all 3 parts: https://johncarlosbaez.wordpress.com/2016/06/29/large-countable-ordinals-part-1/ pic.twitter.com/6PNZOtlCAv

Greg Egan has solved a problem connecting quintics and the icosahedron! Check it out: http://tinyurl.com/quintic-icosahedron pic.twitter.com/AObX0TZCkt

In Newtonian gravity, five particles can shoot to infinity in a finite amount of time! http://tinyurl.com/jcb-continuum-1 pic.twitter.com/TSKyyR1gPr

Why don't atoms collapse? With an inverse cube force law, they would! Learn more here: http://tinyurl.com/jcb-continuum-2 pic.twitter.com/jZMNnerA1b

So hot that rocks boil, this planet with a 16-hour year is disintegrating as we watch - http://tinyurl.com/disintegrating-planet pic.twitter.com/5TOwLrNyrf

Orbiting a black hole's "photon sphere", it occupies half your sky like this. Learn more: http://tinyurl.com/photon-sphere pic.twitter.com/JUMqfiHbl2

A charged point particle can suck energy from its field and zip off to infinity! Nasty! http://tinyurl.com/jcb-continuum-3 pic.twitter.com/JK07QCQ70g

I know a theorem whose proof doesn't fit into the observable universe! Learn it here: http://wp.me/pRBZ9-3id pic.twitter.com/aVcwzORym5

Quantum fields saved us from the ultraviolet paradox, but they bring their own trouble! http://tinyurl.com/jcb-continuum-4 pic.twitter.com/L24gNxK1dW

A chimera is an animal with two different sets of genes. Meet Venus, the chimera cat! http://tinyurl.com/gvaa6ld pic.twitter.com/m0M8mVtJJK

Andrew Hamilton shows us what it's like to orbit a black hole at almost the speed of light! http://tinyurl.com/photon-sphere-2 pic.twitter.com/CjyMrYl4DL

Virtual particles create infinities unless we renormalize. Learn how it works! http://tinyurl.com/jcb-continuum-5 pic.twitter.com/RDHAMeE9EC

Top cybersecurity expert: "Someone is learning how to take down the internet". Read this! http://tinyurl.com/jmpzrp3 pic.twitter.com/IJyRRGaypO

Learn about Togliatti's quintic surface: http://blogs.ams.org/visualinsight/2016/09/15/togliatti-quintic-surface/ pic.twitter.com/MQ9b3LXNFY

Leo Stein's webpage lets you play with photons orbiting a rotating black hole! See http://tinyurl.com/kerr-photon pic.twitter.com/A0FNzduIOC

Barth's sextic surface has 65 "double points". Learn more here: http://blogs.ams.org/visualinsight/2016/04/15/barth-sextic/ pic.twitter.com/0QpIl5sBBt

In 1639, at age sixteen, Pascal proved G,H,K always lie on a line - no matter where A,B,C,D,E,F lie on the ellipse. http://tinyurl.com/mystic-hexagram pic.twitter.com/OKpU3gbWHY

Learn how Penrose diagrams work! They're the best way to understand black holes. http://tinyurl.com/schwarzschild-black-hole pic.twitter.com/nsrEHK8vJI

Stuff can fall into a black hole, but - in theory! - it could fall out of a white hole. Learn more: http://tinyurl.com/white-hole-penrose pic.twitter.com/DzTfQZSqZe

The hottest known planet: its atmosphere contains vaporized titanium dioxide, and its "year" is 30 hours long! http://tinyurl.com/WASP-33b pic.twitter.com/lVixkAkJU3

A great free children's book about infinity - "Life on the Infinite Farm", by Richard Evan Schwartz. http://tinyurl.com/infinite-farm pic.twitter.com/lPVczbJ6EK

The sums over Feynman diagrams in QED probably diverge - because electrons with imaginary charge would attract! http://tinyurl.com/jcb-continuum-6 pic.twitter.com/MivMKIbCfA

Two NASA satellites are making movies of the solar wind. But what is the solar wind made of, and what's it like? http://tinyurl.com/jy8nj32 pic.twitter.com/dwv4R9c2g1

The "Labs septic" has the most double points of any surface of degree 7- as far as we know! http://blogs.ams.org/visualinsight/2016/07/15/labs-septic/ pic.twitter.com/5Yfs1LeQhu

A cubical lattice of points, as shown on "Charlie's daily sketches". Learn what's really going on here: http://tinyurl.com/3d-lattice pic.twitter.com/QKdeM972B0

There's an "exotic atom" that's 1/9th as heavy as hydrogen, but otherwise chemically similar. Learn more: http://tinyurl.com/light-hydrogen pic.twitter.com/wlZCAfDUaj

In general relativity the disastrous breakdown of spacetime is inevitable, thanks to the singularity theorems: http://tinyurl.com/jcb-continuum-7 pic.twitter.com/O4SMTKQ31a

Besides light hydrogen, you can also make tiny atoms like hydrogen - but 1/186 times as big across! http://tinyurl.com/tiny-hydrogen pic.twitter.com/HR7TKlYEaR

For the most double points on a surface of degree 8, you want the "Endrass octics". Learn more: http://blogs.ams.org/visualinsight/2016/08/01/endrass-octic/ pic.twitter.com/yv8Rzxkxw4

Gluons hold quarks together in a proton - but you can also make a particle with just gluons: a "glueball". http://tinyurl.com/gwj9jxs pic.twitter.com/q7C4PoF3Cu

Iron pyrite can form a "pseudoicosahedron"- using Fibonacci numbers to approximate the golden ratio! http://math.ucr.edu/home/baez/golden.html pic.twitter.com/91NjK52dfW

You can store about 10^124 bits of information in the observable universe. Alas, it would then be a black hole. http://tinyurl.com/bits-in-universe pic.twitter.com/EfTFoj3WVp

I got a grant from DARPA to use fancy math - categories and "operads" - to design complex adaptive systems. More at http://tinyurl.com/complex-adaptive-1 pic.twitter.com/lHLBkl9fWX

On Mars, an asteroid impact can cause a flood! Learn more about how and where these riverbeds formed: http://tinyurl.com/mars-flood pic.twitter.com/DDikOiRQ6g

The mathematical structure of a diamond is more beautiful than a diamond - and it's even better in 8 dimensions! http://tinyurl.com/diamond-cubic pic.twitter.com/O8hUIbXUjW

Kosterlitz and Thouless won the physics Nobel for figuring out how spins swirl in magnets! I explain it here: http://tinyurl.com/kosterlitz-thouless pic.twitter.com/Rh2OMEySkC

Greg Egan drew gauge transformations of a vortex and antivortex, illustrating this year's physics Nobel prize work: http://tinyurl.com/kosterlitz-thouless pic.twitter.com/OfDDkrVvvf

Someone named მამუკა ჯიბლაძე made a cool movie of the McGee graph, which has 24 nodes and 36 edges. Learn the math: http://tinyurl.com/jakzd65 pic.twitter.com/7RwbHBjcNP

The maximally extended Kerr black hole has an infinite corridor of universes and 'antiverses'. Explore them here: http://tinyurl.com/kerr-hole pic.twitter.com/qfhEmHooiz

I like this photo by David Burdeny - iceberg in the Weddell Sea chops the world in 4 parts! Learn more: http://tinyurl.com/david-burdeny pic.twitter.com/FuC1mfwgE1

Black hole plus white hole equals wormhole! But you have to go a little faster than light to get across: http://tinyurl.com/gsapnz3 pic.twitter.com/OLp5VvhncW

Astronomers found a thing with rings like Saturn, but hundreds of times bigger! It could be a brown dwarf: http://tinyurl.com/J1407b pic.twitter.com/D0p01CajQv

Greg Egan showed how you can flex the McGee graph in the plane while keeping its edge lengths constant. Learn more: http://tinyurl.com/mcgee-embedding pic.twitter.com/wWvLkPuoDS

What's better than a diamond? A triamond! A possible form of carbon, not yet seen: a "topological crystal". More: http://tinyurl.com/triamond pic.twitter.com/vXY4UMsVJa

70,000 years ago, 2 stars shot past the Solar System! A tiny red dwarf and a even tinier brown dwarf. Learn more: http://tinyurl.com/scholzs-star pic.twitter.com/4MncAB3uzu

Moving right or left? Learn the difference between group and phase velocity, and how *both* can exceed c: http://tinyurl.com/group-velocity pic.twitter.com/Hj7rZ3mxpb

Shiny butterfly wings are made of "photonic crystals". In some, electrons act like massless neutrinos! Amazing: http://tinyurl.com/butterfly-gyroid pic.twitter.com/zCkF1XNVz3

"Mini Saturn": astronomers have found an asteroid with rings! Or it is a comet? It's called Chariklo. For more: http://tinyurl.com/chariklo-rings pic.twitter.com/o3ifXWYEys

Went to San Diego and got briefed on Metron's software for designing complex systems. Secretly it's category theory: http://tinyurl.com/complex-adaptive-2 pic.twitter.com/iHsrfvV34d

2 million years ago, when the axis of Mars tilted more, ice at its poles evaporated and refroze elsewhere. More: http://tinyurl.com/white-mars pic.twitter.com/HBzn4jBaCT

8 grad students worldwide can take a free advanced course in category theory! It'll be great! To apply: http://tinyurl.com/kan-two pic.twitter.com/EYzNb8wvzB

This is what Mars may have looked like 3.8 billion years age - the northern plains may be old ocean bed. More: http://tinyurl.com/jsv8eda pic.twitter.com/tnjcwcPa1O

Have three scientists just shown dark energy isn't real? I'm expecting a big argument - and here's why: http://tinyurl.com/darkenergyquestion pic.twitter.com/jgHGO1y0kM

The US Geological Survey has a free huge map of Mars' north pole -see deep furrows carved by katabatic winds: http://tinyurl.com/mars-north-pole pic.twitter.com/srQZHjHqhM

There's new free open-access math journal "Higher Structures", run by top experts on n-categories and such stuff - http://tinyurl.com/higher-structures pic.twitter.com/IkYj5QVZjO

Liquid water on Mars! Salty water flows down Newton Crater in the spring. Learn more here: http://tinyurl.com/newton-crater pic.twitter.com/mr7Exlru3Y

Did an 87-year-old mathematician just crack a famous problem? Are there no complex structures on the 6-sphere? http://tinyurl.com/atiyah-S6 pic.twitter.com/gAXbcuWtBx

Juan Escudero found a degree-9 surface with 220 real double points, the maximum known. It's beautiful! Read more: http://tinyurl.com/escudero-nonic pic.twitter.com/pRq8O1ep1n

Wolf Barth found a degree-10 surface with 345 double points, the most so far - and icosahedral symmetry! More here: http://tinyurl.com/barth-decic pic.twitter.com/sIYoy2v0Zx

Mars' north polar ice cap has 820,000 cubic kilometers of ice! But here's a cute frozen pool in Vastitas Borealis: http://tinyurl.com/vastitas pic.twitter.com/1me65PhnD4

The thorny devil, an Australian lizard, only eats ants. I think it's cute! Learn how it drinks water: http://tinyurl.com/thorny-devil pic.twitter.com/etIdtyHILJ

Closeup photo of a Martian riverbed 1500 kilometers long and billions of years old. Learn more: http://tinyurl.com/reull-vallis pic.twitter.com/H0GQGFWLQm

Nate Silver's blog gives you just 0.06 bits of information about who will be president. Why so little? See: http://tinyurl.com/relative-information pic.twitter.com/vh2hV1De3S

Learn advanced math at a ski resort - for free! Homotopy type theory workshop for students and postdocs: http://tinyurl.com/hott-snow pic.twitter.com/7Z56gmaJ6X

Going to Santa Fe Institute next week for a workshop on Statistical Mechanics, Information Processing and Biology: http://tinyurl.com/SFI-info-bio pic.twitter.com/JzdCuO0zX1

See slides of my Santa Fe Institute talk on the math of networks and open systems! http://tinyurl.com/santa-fe-networks pic.twitter.com/8ROd0n9DXi

Learn how computation is connected to entropy! Shannon meets Kolmogorov, and they're unified by Gibbs: http://tinyurl.com/alg-thermo pic.twitter.com/ydDLnjW3Oz

Jarzynski gave a great one-hour intro to nonequilibrium statistical mechanics at the Santa Fe Institute - http://tinyurl.com/jarzynski pic.twitter.com/x764CAwgUJ

Billiard balls move chaotically in a shape called the "Bunimovich stadium". They're "ergodic" - learn more here: http://tinyurl.com/bunimovich pic.twitter.com/kOfASv3kYI

Learn about hairy, mammal-like reptiles that lived 60 million years before the reign of the dinosaurs: http://tinyurl.com/pisterognathus pic.twitter.com/TA1y17OXmE

Trump's election has gotten me more motivated to fight climate change. There's a lot going on at the local level: http://tinyurl.com/zvad3yr pic.twitter.com/NQJkpsdZLK

Thanksgiving: I was thankful to have grown up before the Arctic melted. Here's what's happening now: http://tinyurl.com/thanks-for-the-old-planet pic.twitter.com/ATC0qQNYqC

Check out my student Blake Pollard's talk on category theory for studying open systems! It was a hit in Santa Fe: http://tinyurl.com/pollard-sfi pic.twitter.com/M6VO6aC3Vs

Workshop on category theory and computer science here in Berkeley this week! Check out my talk on Tuesday! http://tinyurl.com/baez-compositionality pic.twitter.com/eoWx6FhCIw

My student Brendan Fong explains the syntax and semantics of "network languages", like electrical circuit diagrams: https://www.youtube.com/watch?v=o8JDPYO5ZrQ

@drewvolpe - are enough people downloading and backing up the climate databases listed on your Google Docs page to be done by Jan. 20th?

The climate war is heating up! What will Trump do, and what can we do about it? Join our discussion: http://tinyurl.com/j6236nc pic.twitter.com/Lq6fFWMC1a

Scientists are racing to back up climate data in case Trump tries to delete it. Learn how to help here: http://tinyurl.com/saving-climate-data pic.twitter.com/rDjyrnPGbp

Jerry Brown to climate scientists: "If Trump turns off the satellites, California will launch its own damn satellite!" pic.twitter.com/WPCJtPap2I

The latest news on how we're rushing to back up climate data, and how you can help: http://tinyurl.com/saving-climate-data-2 It's wild - but it's working! pic.twitter.com/D0hsobaGAY

Good computer skills? We need your help at the Azimuth Backup Project. We're busy backing up climate data! http://tinyurl.com/azimuth-backup pic.twitter.com/DRl0lftOMJ

If your friend runs a big government climate database, tell 'em to "hash" it so we can tell if backups are accurate! http://tinyurl.com/saving-climate-data-3 pic.twitter.com/2kFqRUUwuM

Read about Dan Romik's ambidextrous sofa - the biggest known shape that can go around two bends like this! http://blogs.ams.org/visualinsight/2016/12/15/romiks-ambidextrous-sofa/ pic.twitter.com/KgzSGowSo2

"Just a farmer's wife" - but her fiber optics deliver data 35 times faster than the UK average! Farmers love it. http://tinyurl.com/z5faaxd pic.twitter.com/SGMGAR4VYq

The scariest insect I've ever seen: the giant toothed longhorn beetle, from the Amazon basin in Ecuador: http://tinyurl.com/macrodontia pic.twitter.com/zKZpW1DWzc

Happy New Year! This surface, the "Chmutov octic", has 144 nodes. Read about it here: http://tinyurl.com/chmutov pic.twitter.com/ApjcnsjWiY

Check out my pal Tom Leinster's new book, Basic Category Theory. It's good and it's free! It's even open-source: http://tinyurl.com/leinster-book pic.twitter.com/NPPw3igscQ

A mysterious player known only as "Master" started beating EVERYONE IN THE WORLD at go. Now we know who it is: http://tinyurl.com/go-master pic.twitter.com/4Pm4gUpXAA

@timteachesmath chops square cake into 5 equal parts - with same amount of icing! Regular hexagon into 7 parts? http://tinyurl.com/chop-cake pic.twitter.com/pBjAxNLgaQ

Astronomers predict that in 2022, two stars will collide and create a nova - the brightest thing in our night sky! http://tinyurl.com/luminous-red-nova pic.twitter.com/zNx8Q1FuDC

The President has an article in the journal Science: "The Unstoppable Momentum of Clean Energy". Read it here: http://tinyurl.com/pres-science pic.twitter.com/5YCi9qgBus

We're rushing to back up US government climate data before Trump takes office! Please contribute here: https://www.kickstarter.com/projects/592742410/azimuth-climate-data-backup-project

At noon today in DC, all mention of "climate change" was deleted from the White House website. But we were ready: http://tinyurl.com/saving-climate-data-4 pic.twitter.com/c3xWvSmH49

Science goes underground! Rogue twitter sites fight the Trump team's censorship - and they back off, for now: http://tinyurl.com/saving-climate-data-5 pic.twitter.com/CS0Fq23ZQM

Workshop here on quantifying biocomplexity. Check out my talk on biology as information dynamics! https://tinyurl.com/j3vn2tc pic.twitter.com/b6MWQc4maq

The Trump gang is trying to gag scientists. We're gagging, all right! Find out how we're fighting back: https://tinyurl.com/zzw3yql pic.twitter.com/Kpe9l5f89e

My "crackpot index" is now needed in politics! The original is here: http://math.ucr.edu/home/baez/crackpot http://www.huffingtonpost.com/bob-seay/the-crackpot-index-for-me_b_5757016.html

Thanks to 627 people who funded the Azimuth Climate Data Backup Project, we're saving 40 terabytes from Trump! https://tinyurl.com/azimuth-backup-4 pic.twitter.com/jTYcSDCKV6

US citizen? In just 2 minutes you can help restore bears to the North Cascades! Only 2 days left! Go here: https://tinyurl.com/grizzzly pic.twitter.com/UIrgwzuoEo

Here Greg Egan shows the "Desargues graph" in a 5d cube. For the whole movie, go here: https://tinyurl.com/desargues-5d pic.twitter.com/D5ki9VlJE8

The clouds of Jupiter, photographed 14,500 kilometers above a dark storm! For more: https://tinyurl.com/jupiter-baez pic.twitter.com/fJRMdzGRQi

Read about "Mock Modular Mathieu Moonshine Modules" - monstrous math made simple! https://tinyurl.com/mathieu-moonshine pic.twitter.com/wiS1xvGdGa

Last night, in a dream, Ramanujan showed me this formula. For the full story, go here: https://tinyurl.com/ramanujan-pi pic.twitter.com/vgCJdptuGC

I'm running a session on Applied Category Theory at the UCR meeting of the AMS Nov. 4-5. Want to give a talk? https://tinyurl.com/kqajmb3 pic.twitter.com/U3ge7LLAr0

That cloud? It's trillions of dying warriors, in a battle visible from space! For details: https://tinyurl.com/baez-cocco pic.twitter.com/17LrMjTLMp

As wolves live in an ever more human-populated world, they're evolving to become more like dogs! https://tinyurl.com/wolf-new-dog pic.twitter.com/WMWypi53W1

Thursday April 20th at 4 pm I'm talking on "Biology as Information Dynamics" at the Stanford Complexity Group! https://tinyurl.com/baez-bio-stanford pic.twitter.com/byY85RkM3j

Decompose the proteins in an organism. Plot the number of peptides you get by mass. Amazing oscillations! Why? https://tinyurl.com/baez-peptide pic.twitter.com/NUwFkk0tZ3

This is a "diamondoid" - a molecule whose carbon atoms are arranged just like a tiny piece of diamond! Learn more: https://tinyurl.com/diamondoid pic.twitter.com/zitrjg134N

Check out Greg Egan's movie of the maximal torus in SU(3), the group that describes the strong nuclear force! https://tinyurl.com/egan-SU3 pic.twitter.com/j1PM0aozYD

When phosphorus and sulfur combine, the result is some fascinating geometry... and interesting puzzles! https://tinyurl.com/baez-PS pic.twitter.com/i5mxU07ggO

The most common of the phosphorus sulfides is structured like a tiny piece of a diamond crystal! Learn more: https://tinyurl.com/baez-PS pic.twitter.com/3Dr0v5Xlj8

Watch animations of "juggling roots". As you change the coefficients of a polynomial, its roots move in fun ways: https://tinyurl.com/baez-juggling pic.twitter.com/hXYXN3ePDy

They'll plant a milkweed to help monarch butterflies and send a mother's day card to your mom - it's free! http://on.nrdc.org/2pCNF5u pic.twitter.com/Wu3b8nc4xu

A honeycomb in hyperbolic space, lit from a point beyond infinity, created by Abdelaziz Nait Merzouk - https://tinyurl.com/merzouk-534 pic.twitter.com/FR7Pbtw714

Klein used this "icosahedral function" to solve the quintic equation. Hiding inside is the E8 lattice: https://tinyurl.com/baez-E8 pic.twitter.com/2SMp7nhjeT

The 5 circles theorem, still hard to prove, is now easy to understand, thanks to a Hungarian math grad student: https://tinyurl.com/5-circles-theorem pic.twitter.com/t4dQsZvfUg

Come to my Hong Kong talk tomorrow or look at my slides, and this 4d polytope will be explained: https://tinyurl.com/hongkong-E8 pic.twitter.com/RjyHp1ibLj

Tabby's star dimming again - astronomers rushing to observe! Aliens back to work after a long break? https://tinyurl.com/baez-tabby pic.twitter.com/iSoWvGK4tA

Proving Rubik's cube can always be solved in at most 20 moves took 30 years - and then 35 years of CPU time! https://tinyurl.com/baez-rubik pic.twitter.com/R1FagRWc6b

In "set-theoretic geology", logicians are digging to the core of the mathematical universe. I explain it here: https://tinyurl.com/baez-bedrock pic.twitter.com/2EHYmXYAXZ

Open chemical reaction networks and category theory! My talk is in Luxembourg June 13, but here's a sneak preview: https://tinyurl.com/baez-lux pic.twitter.com/3YldJkdKjV

"Red sprites" over English Channel - actually balls of cold plasma moving at 10% the speed of light! Learn more: https://tinyurl.com/red-sprite pic.twitter.com/ozsjTU4qIb

Intelligent global warming skeptics are starting to reconsider as the Earth warms up - here's what one says: https://tinyurl.com/baez-warm pic.twitter.com/iAfLm66jXm

Stay cool - check out what Derek Wise can do with the Koch snowflake! Fractals within fractals, here: https://tinyurl.com/wise-snowflake pic.twitter.com/LmBCdxhwdO

Someone named Cleo is doing lots of hard integrals. But she won't say how she does them! Who is she? https://tinyurl.com/baez-cleo pic.twitter.com/iySGChmzqJ

What happens when a black hole moves at the speed of light? This is tricky, but there's a nice answer: https://tinyurl.com/aichelburg-sexl pic.twitter.com/kiJyigu6iY

My latest quest: to see how E8 lurks at the singularity of an 'icosahedral big bang'. It starts here: https://tinyurl.com/baez-mckay-1 pic.twitter.com/13svH2JGTl

I went on a hike in the Alps and saw this amazing mountain! Maybe called Nünenenflue. Read the tale here: https://tinyurl.com/ya8nyoys pic.twitter.com/TVD53NmXfd

Greg Egan's new novel about a universe with two dimensions of time is out now! Dichronauts! Read more: https://tinyurl.com/dichronauts pic.twitter.com/YuOcUsq6yf

My student Kenny Courser has brought bicategories into our research on network theory! I explain here: https://tinyurl.com/courser-1 pic.twitter.com/pVEdeq4Ocr

I hear that .png files with transparency don't show up well on mobile Twitter. Here's the picture I was trying to show y'all. pic.twitter.com/xaFJ2CRPVy

To get E8 from the icosahedron I'm gonna use one of Hilbert's crazy schemes - "the Hilbert scheme". Learn about it: https://tinyurl.com/baez-mckay-2 pic.twitter.com/WpZ9mmSgea

Okay, you've all seen some by now, but WOW! These Juno pics are great! https://www.missionjuno.swri.edu/junocam/processing pic.twitter.com/2SG5hnTQnC

A Chinese hero died today. But why is Liu Xiaobo famous? Find out: https://tinyurl.com/baez-liu His wife Liu Xia remains under house arrest. pic.twitter.com/rCOxEnIJEr

Learn how beetles make circularly polarized light - like this, but turning around the other way. https://tinyurl.com/baez-polarized pic.twitter.com/5o97jWEMda

There are 10 kinds of matter. Amazingly, 8 are connected to the real numbers, and 2 to the complex numbers: https://tinyurl.com/baez-ten pic.twitter.com/y6DQOYDmL1

Back in 1974, this man sparked a revolution in game theory. His beard is longer now. Learn more: https://tinyurl.com/baez-game-1/ pic.twitter.com/8BPT3zLd9i

Who should you trust on climate change: the liars or the hypocrites? My answer is here: https://tinyurl.com/baez-hypocrites pic.twitter.com/OeCyCHE3Rs

The geometry of music! Red lines are major thirds; green are minor thirds; blue are perfect fifths. Learn more: https://tinyurl.com/baez-triads pic.twitter.com/c3OYJ5NBi9

Check out this amazing interactive website on the geometry of viruses! It lets you design your own! https://tinyurl.com/baez-virus pic.twitter.com/Ua2vTr3o0p

Check out my new paper on category theory applied to chemistry! Or don't! It's easier to read a short blog post: https://tinyurl.com/baez-reaction pic.twitter.com/JD4Tl399o4

Next week in Japan I'm giving a big overview of algebraic topology, its history, and how it changed our thinking. https://johncarlosbaez.wordpress.com/2017/08/05/the-rise-and-spread-of-algebraic-topology/

Wow! Length of minimal spanning tree for large complete graph with uniform random edge lengths between 0 and 1! https://tinyurl.com/baez-zeta-3 pic.twitter.com/VW2XRhqo6j

I loved Tom Leinster's talk at the Applied Algebraic Topology conference here in Sapporo! Check it out: https://tinyurl.com/leinster-sapporo pic.twitter.com/CimpohGx1U

Norbert Blum claims to have proved P ≠ NP. For some very nonexpert comments on this, visit my blog: https://tinyurl.com/baez-blum pic.twitter.com/LG2nylxYqD

In the late 1500s, before logs, people would multiply big numbers using a table of cosines! I'll show you how: https://tinyurl.com/baez-trig pic.twitter.com/p3v9wV3oRL

I'm using the math of "operads" to help design networks of mobile agents! But what are operads? Read on! https://johncarlosbaez.wordpress.com/2017/08/17/complex-adaptive-system-design-part-3/

"Network operads" let us build big networks from smaller ones. Here I explain the simplest example. More to come! https://johncarlosbaez.wordpress.com/2017/08/22/complex-adaptive-system-design-part-4/

It's the 40th anniversary of the Voyager spacecraft. Read my tale of their plunge into interstellar space! https://tinyurl.com/baez-voyager pic.twitter.com/Co5ElUbSUY

Yamashita's summary of Mochizuki's proof of the "abc conjecture" is 294 pages long - and absolutely terrifying. https://tinyurl.com/baez-abc pic.twitter.com/Ssldoitpgl

Read about "black" open access, the best known way to make scientific papers free to read. Hint: it's illegal. https://tinyurl.com/baez-black pic.twitter.com/8iDgZwGmIm

Watch the grand finale as Cassini crashes into Saturn at 1200 GMT on September 15th! https://tinyurl.com/baez-cassini pic.twitter.com/NcWqAGc1Ug

We can use algebras of operads, and maps between them, to first design a network roughly and later fill in details. https://johncarlosbaez.wordpress.com/2017/09/04/complex-adaptive-systems-part-5/

After Pluto, the New Horizons spacecraft snoozed - but it's heading toward this object, and today it woke up! https://tinyurl.com/baez-dreamer pic.twitter.com/UTx0KFGGEf

Come to Applied Category Theory 2018 in Leiden next April! There's a "school" for grad students, too! https://johncarlosbaez.wordpress.com/2017/09/12/act-2018/

One of the most beautiful animals in nature: the hellbender! Learn why people are hell-bent on helping it: https://tinyurl.com/baez-hell pic.twitter.com/CN7FaGwHuv

Dry ice made this "Swiss cheese terrain" near the south pole of Mars - but what about the big pit? For more: https://tinyurl.com/baez-swiss pic.twitter.com/hI0bx16xQO

Two infinities have been shown equal! I read a bad explanation, improved it, then Timothy Gowers did much better: https://tinyurl.com/baez-p-vs-t pic.twitter.com/eWdSbuWGZd

Two regular pentagons and a decagon fit snugly at a corner. Islamic tile masters exploited that to create this! https://tinyurl.com/baez-tile pic.twitter.com/SaYNhwZ9Yl

Nobel prize for gravitational waves! But read about what the Virgo detector in Italy saw in August. https://tinyurl.com/baez-virgo pic.twitter.com/4cndhGF0BB

Baloney! A better title would be "Physics works just as expected". Unlike the journalists, I can explain it: https://tinyurl.com/baez-dirac pic.twitter.com/ddA80e97NR

To draw this classic "8-fold rosette" tiling pattern, you'll actually use the identity tan(π/8) = √2 - 1. https://tinyurl.com/baez-rosette pic.twitter.com/Nz6t5wUZNX

Read my obituary of the famous mathematician Vladimir Voevodsky, and the long conversation that follows: https://tinyurl.com/baez-voevodsky pic.twitter.com/zFn5feIyoS

Hot gas ball the size of Neptune's orbit expanding at 1/5 the speed of light - yay, we've seen how gold is made! https://tinyurl.com/baez-kilonova pic.twitter.com/wgXs00Eijk

Now that we've actually seen "the chirp of death", check out Greg Egan's online simulation! https://tinyurl.com/egan-chirp pic.twitter.com/PKDhwgkFbR

This spider can walk on water, dive down, and catch fish! In fact, 12 species of spiders eat fish: https://tinyurl.com/baez-spider pic.twitter.com/VSTKXPFqub

Wednesday morning at UCR I'm explaining how my team saved 40 terabytes of climate data. Come by! https://johncarlosbaez.wordpress.com/2017/10/05/azimuth-backup-project-part-5/

The derivative of acceleration is called "jerk". As dark energy takes over, we're in the middle of a colossal jerk: https://tinyurl.com/baez-jerk pic.twitter.com/fHe01Vw2Ca

Attend this school on applied category theory! I'll help teach you! Applications are due Wednesday. For more: https://tinyurl.com/baez-school pic.twitter.com/RLNW5LF8w7

I'm going to Caltech on Monday Nov. 13th. Come to my talk in the biology seminar at 4 pm, and say hi! It's in room 119 of Kerckhoff Hall. Or just watch this video. https://youtu.be/IKetDJof8pk

Germans built a huge machine to weigh a tiny particle: the electron antineutrino. But this machine may not be big enough: if this particle is less than 0.2 eV, it won't do the job. https://tinyurl.com/baez-KATRIN pic.twitter.com/p91pMWROfK

First Wada dug a canal so that every point was at most 1 mile from the red sea. Then he dug one so every point was at most 1/2 mile from the blue lake - shown here. Then he dug one so every point was at most 1/4 mile from the green lake. Then... read: https://tinyurl.com/baez-wada pic.twitter.com/C2amgRgxC7

I'll talk about "Compositionality in network theory" at the Institut de Recherche en Informatique Fondamentale at Paris 7 next Tuesday, Nov. 21, at 10:30 am. But if you can't make it, you can watch this video of a similar talk! https://www.youtube.com/watch?time_continue=7&v=IyJP_7ucwWo

This is not an animated gif. This is @AkiyoshiKitaoka messing with your brain again! He's the psychologist who wrote "The Oxford Compendium of Illusions." His webpage says "Should you feel dizzy, you had better leave this page immediately." pic.twitter.com/U8tBLInQ2c

Any braid can be used to stir fluid, and we can use this to define its "entropy". For a braid with 3 strands, the most entropy we can get each time we switch strands is the logarithm of the golden ratio! An eternal golden braid! Read more here: https://tinyurl.com/baez-braid pic.twitter.com/quXsIo0Nxr

The universal snake-like continuum! It sounds like something from a bad Star Trek episode... but it's a subset of the plane so wiggly it's impossible to draw, the limit of these pictures here. See more here, from profound math to a Disney video clip: https://tinyurl.com/baez-snake pic.twitter.com/6oAQpGih5H

Brouwer discovered an amazing set, the limit of these pictures. It's a "continuum": a nonempty compact connected metric space. And it's "indecomposable": it's not the union of two proper subsets that are themselves continua! For more: https://tinyurl.com/baez-snake pic.twitter.com/UE0qM45D4N

By cutting away more and more dough from a solid doughnut, forever, you can make a zero-calorie doughnut that wraps around infinitely many times! It's called a "solenoid". It's a compact abelian group and an indecomposable continuum! Read more here: https://tinyurl.com/baez-solenoid pic.twitter.com/PWM4cIvugs

Interview with Witten. "Do you have any ideas about the meaning of existence?" "No." But he's puzzled by the nature of quantum spacetime: "I suspect that there’s an extra layer of abstractness compared to what we’re used to." Read the whole thing: https://www.quantamagazine.org/edward-witten-ponders-the-nature-of-reality-20171128/ pic.twitter.com/cha5ORJ7tG

An ocelot kitten! It kills its prey by looking so cute that they just sit there defenseless. Seriously: ocelots are small wild cats that live in Central and South America... and there's even an ocelot reserve in southern Texas! Read about it here: https://tinyurl.com/baez-ocelot pic.twitter.com/tofYs8U064

The limit of these sets is the "Sierpinski carpet". Technically it's a "curve", since you can refine any open cover of it to one where each point lies in at most 2 open sets. And any curve in the plane is homeomorphic to a subset of the Sierpinski carpet! https://tinyurl.com/baez-carpet pic.twitter.com/rq0I4w1PHK

Roice Nelson's new picture of the {7,3,3} honeycomb! Here 3d hyperbolic space is squashed down to a ball, and the "dents" are hyperbolic planes tiled by regular heptagons, each subdivided into 7 red and 7 blue triangles. For more on the math go here: https://tinyurl.com/baez-733 pic.twitter.com/rNmNqkJ1fp

There's a nice article in Quanta about my friend Minhyong Kim's work connecting number theory and physics. As I explain, he's been thinking about this for decades: https://tinyurl.com/baez-minhyong pic.twitter.com/ffG9lSr7v7

This is an "electron crystal", formed when electrons trapped on a small disk of metal try to minimize their energy by moving as far apart as they can. Can you figure out what the blue and red things are? More here: https://tinyurl.com/baez-wigner pic.twitter.com/bKA1q75PCJ

Excitons are in the news! When an electron orbits the *absence* of an electron, that's called an exciton. But excitons can crystallize and form *excitonium* - and that's even more exciting. I explain it all here: https://tinyurl.com/baez-exciton pic.twitter.com/2W3ZzguApi

First AlphaZero learned go, just by playing itself, becoming the world's best player in 40 days. Now it has taught itself to be the world's best chess player - crushing all others. And it did this in just 24 hours! Details: https://tinyurl.com/baez-alphazero pic.twitter.com/lJnEirtofo

The symmetry group of the icosahedron is this "hypericosahedron" in 4 dimensions, with 600 tetrahedral faces. And from here it's just a short hop to the E8 lattice in 8 dimensions! The key is to use the golden ratio. Learn how: https://tinyurl.com/baez-icos-E8 pic.twitter.com/dZ0IDuPPNV

As you move the coefficients of a polynomial around a loop, its roots can trade places. (The roots live up in a branched covering space!) @duetosymmetry made this great animated gif - and now he's made a webpage where you can explore this yourself: https://duetosymmetry.com/tool/polynomial-roots-toy/ pic.twitter.com/77S0d3U56L

What's even cooler than a superfluid? A *supersolid*, where "vacancies", missing atoms in a crystal, form a Bose-Einstein condensate and move around like a superfluid. A ghostly liquid made of absent atoms!!! Details: https://tinyurl.com/baez-solid pic.twitter.com/ReBpGwTFf0

A new paper claims the "string Lie 2-algebra" my grad student Alissa Crans and I discovered underlies the mysterious 6d theory at the top of this chart. Witten calls it the "pinnacle" - one ring to rule them all and in the darkness bind them. https://tinyurl.com/baez-m5 pic.twitter.com/HjmLKVU5ND

Here Greg Egan shows how 5 tetrahedra fit into a dodecahedron! I just realized the implications for 4d geometry: we can fit five 4d Platonic solids with 24 = 1 × 2 × 3 × 4 vertices into a 4d Platonic solid with 120 = 1 × 2 × 3 × 4 × 5 vertices! For more: https://tinyurl.com/baez-24-600 pic.twitter.com/dhS0i9XsoJ

It's not a Christmas tree ornament - it's a wavefunction with dodecahedral symmetry and well-defined total angular momentum, oscillating between two states! Drawn by Greg Egan, it leads to deep math puzzles. Start here: https://tinyurl.com/baez-q-dodec pic.twitter.com/aaoG9xfawq

David Richter and friends spent 6 hours making this model of "the compound of 15 orthoplexes", an amazing 4d shape. The orthoplex is the 4d version of an octahedron; 15 of them can be inscribed in the 4d version of an icosahedron! For more: https://tinyurl.com/baez-16-600 pic.twitter.com/2gqdVLjLUN

Infinite chess is pretty weird! In this position Black is doomed to lose if he's a computer... but he can draw if he can play in uncomputable ways. For more: https://tinyurl.com/baez-chess pic.twitter.com/0bh3kMCWFi

Another gold nugget of math. Each corner of the icosahedron lies on an edge of the octahedron. But here's the cool part: the corners aren't at the centers of the edges! They divide the edges according to the golden ratio! Details here: https://tinyurl.com/baez-golden pic.twitter.com/qQVJ2NRl1q

Happy New Year! Here @gregeganSF shows that if you bisect the edges of an octahedron you get the corners of a cuboctahedron. But if you chop them according to the golden ratio you get the corners of an icosahedron! pic.twitter.com/hrr8cI8Hpf

If you're an advanced math or physics student, sign up for my friend Predrag Cvitanovic's free online course on chaos theory! It starts January 9th! For more info, watch this video, then go here to sign up: http://chaosbook.org/course1/about.html https://www.youtube.com/watch?v=JqaM-y2cBkI

Do you ever get the feeling you're being watched? Most spiders have 8 eyes. But they also have very tricky DNA - even with a supercomputer, scientists failed to transcribe the tarantula genome! For more: https://tinyurl.com/baez-jumping-spider pic.twitter.com/vzp07gClJl

Asteroid families were created when larger objects collided billions of years ago. But finding them is harder than this chart makes it seem! You have to 'correct' the observed orbits, which change over time in complex ways. That's where the fun starts: https://tinyurl.com/baez-asteroid pic.twitter.com/m3HQ1Xu7Mc

Here @gregeganSF shows the 24-cell, a 4d regular polytope with 24 vertices and 24 octahedra as faces. It's also a group, the double cover of the rotational symmetry group of the tetrahedron. And it has 24×24 rotational symmetries of its own! More: https://tinyurl.com/egan-24-cell pic.twitter.com/CZDnWxorxD

This is the smallest known number with the digits 1 through 6 arranged in every order. It has one less digit than expected. People predicted you'd need 1!+2!+3!+4!+5!+6! digits. But this has one less! Timothy Gowers has some thoughts here: https://tinyurl.com/baez-digits pic.twitter.com/RVt24CXWbQ

Al Grant has a great interactive page of tilings that move on "hinges". Check it out: http://algrant.ca/projects/hinged-tessellations/ And Al is short for "Albert" not "Artificial intelligence". pic.twitter.com/h79GeheA32

This is a flying fox - a kind of fruit-eating bat. Bats have traditionally been divided into two main kinds: the *microbats*, which mostly eat insects and can echolocate, and the *megabats*, which mostly eat fruits. But now there's been a change: https://tinyurl.com/baez-megabat pic.twitter.com/uDKRpCQieI

You can't divide a square into an odd number of triangles that all have the same area. But you can still try your best! These are the best known attempts for 7 triangles, discovered just last year. There's interesting math lurking behind this: https://tinyurl.com/baez-triangles pic.twitter.com/lG5AoH7TqG

40% of your genes are retrotransposons - they copy themselves from place to place. A retrotransposon that can wrap itself in a protein coat is a virus - a *retrovirus*, like HIV. The news: your brain also uses these protein coats to form memories! https://tinyurl.com/baez-retro pic.twitter.com/NpPLYGG6rj

When Charles Ehresmann combined geometry and category theory in the 1950s, mathematicians thought he'd lost it. Now they understand. And his journal, started in 1957 and run by his wife Andrée since his death in 1989, is now open-access! It's here: http://ehres.pagesperso-orange.fr/Cahiers/Ctgdc.htm pic.twitter.com/DZOULp3Ftb

I'm not sure Gerard Westendorp's folding table is practical, but it's based on his observation that a "hinged tiling" made of squares can be collapsed down to a single square. For more, go here: https://tinyurl.com/westendorp-tile https://tinyurl.com/westendorp-tile-2 pic.twitter.com/S1CSUB5WdF

I love this computer-generated picture of a "Doyle spiral" by my friend Abdelaziz Nait Merzouk! He puts in the extra work to make his mathematical objects look like real-world things. More: https://tinyurl.com/abdelaziz-G pic.twitter.com/ZMbp8Iw7cW

My student Christian Williams went to the first meeting of *Statebox*: an ambitious attempt to combine categories, open games, dependent types, Petri nets, string diagrams, and blockchains into a universal language for distributed systems. Read on: https://tinyurl.com/baez-statebox-1 pic.twitter.com/VMu7BVKI3t

Gödel's incompleteness theorem cuts deep! There's a *universal program*, which can print out any possible finite sequence of numbers, depending on what model of Peano Arithmetic you're in. And you can't tell which one you're in. Details: https://tinyurl.com/baez-universal pic.twitter.com/9Avg9P94nS

I created the Crackpot Index when the rise of the internet exposed a vast army of physics crackpots, who were previously toiling in obscurity. Now the world needs a crackpot index for cryptocurrencies! Luckily, @dhruvbansal has come to our rescue. https://docs.google.com/document/d/1XaHOc3N_KL6i-PAsSlfiV6A-Wf0WBmuVqs9JgzW9ai8/edit

This is the 3d associahedron. Shocking new discovery: if you take any Taylor series F(x) = x + ax^2 + bx^3 + ... and find the series G with F(G(x)) = x, the coefficients of G come from those of F using a formula based on associahedra! Details here: https://tinyurl.com/baez-assoc pic.twitter.com/4ZAzp8jd6D

My student Mike Stay is interested in using higher categories to describe processes of computation, as shown here for the lambda calculus. Now his startup called *Pyrofex* is worth millions... and he's putting these ideas into practice! Read the story: https://tinyurl.com/baez-pyrofex pic.twitter.com/R01tcx8VMO

The Thue–Morse sequence is defined as shown here. And in 2013, Cooper and Dutle proved it's the fairest way for two equally bad shooters to take turns in a duel... in the limit where their chance of hitting the other guy approaches zero! Details: https://tinyurl.com/baez-thue pic.twitter.com/6tGyx7O6Tw

The school for Applied Category Theory 2018 is up and running! Students are blogging about papers! Here's an intro to Aleks Kissinger's diagrammatic method for studying causality - make sure to read the whole conversation, since it helps a lot: https://tinyurl.com/baez-causal pic.twitter.com/gCesfUj44A

What a great idea! A seminar on homotopy type theory, with talks by top experts, available to everyone with internet connection! It starts February 15th: https://tinyurl.com/hott-seminar pic.twitter.com/CKVGDzCzIb

Now students in the Applied Category Theory class are reading about categories applied to linguistics. This lets us take ideas from quantum mechanics and apply them to sentences! Read their blog article: https://tinyurl.com/baez-linguistics pic.twitter.com/qXybYaqrQM

Dunes are born... and dunes die. This is the decaying corpse of a dune on Mars, its sand gradually stolen by the changing winds. For more: https://tinyurl.com/baez-corpse pic.twitter.com/Ujn2fqUaOj

Steve Awodey is big on homotopy type theory... but his student Spencer Breiner works at the National Institute of Standards and Technology, and he's having a workshop on applied category theory! I'm talk about my work on system design using operads: https://johncarlosbaez.wordpress.com/2018/02/17/applied-category-theory-at-nist/

The "{6,3,3} honeycomb", drawn by Roice Nelson, lives in hyperbolic space. 3 hexagonal tilings of the Euclidean plane meet at any edge of this thing! So, instead of filling space with finite-sized polyhedra, we fill it with shapes that go on forever. https://tinyurl.com/baez-633 pic.twitter.com/svJ3dtxYlG

Check out "Toposes in Como" - a conference/school with courses by bigshots like Borceux, Connes, Lafforgue, and Olivia Caramello here, the firebrand who is getting everyone excited about topos theory. Register here: http://tcsc.lakecomoschool.org/registration/ pic.twitter.com/MyoOoDorQE

A cool way to cut carbon emissions - a "double conference" in two distant locations connected by a video feed! And this one is on higher-dimensional algebra and mathematical physics! Lots of great speakers: https://tinyurl.com/baez-HAMP pic.twitter.com/FrPyA4hg8a

Here @roice713 draws "loxodromic" transformations of the plane, spiralling in at one point and out at another. These extend to transformations of the upper half of 3d space - really hyperbolic space - which map the banana-shaped thing to itself! More: https://tinyurl.com/roice-banana pic.twitter.com/uVBykEUccY

Brigham Young University has a math club. Which is good. And they're trying to get women interested in math. Which is good. But actions speak louder than words. And no, this poster was not satirical. More: https://tinyurl.com/byu-women-in-math pic.twitter.com/seHLMqYCjC

Varignon's theorem says: draw any quadrilateral; then the midpoints of the sides are the corners of a parallelogram! It's easy to prove with a little linear algebra. See how? If not, go here: https://en.wikipedia.org/wiki/Varignon%27s_theorem pic.twitter.com/PijEWfkZmj

The "Bricard octahedron* is the simplest flexible polyhedron! It has faces that cross other faces. 81 years after this was discovered, someone discovered a flexible polyhedron without faces that cross each other. More: https://tinyurl.com/baez-bricard pic.twitter.com/UydxbFnPrw

My student @CCategories is studying bicategories where the morphisms are "open graphs" and the 2-morphisms are "rewrites" turning one open graph into another. Some of these are "cartesian", and together with @_julesh_ he explains what that means! https://johncarlosbaez.wordpress.com/2018/03/01/cartesian-bicategories/

Always wondered about ball lightning. It'd be so cool if it were a "topological soliton", unable to fall apart because its magnetic field is tangled up in ways that are hard to break. I'm far from convinced, but that's what some folks claim: https://www.nature.com/articles/383032a0.epdf pic.twitter.com/IJQ8V1CKXH

Cool! Some nonstandard integers can be found inside the ordinary complex numbers! This was noticed by Alfred Dolich at a pub after a logic seminar at the City University of New York. Here's an easy explanation. https://johncarlosbaez.wordpress.com/2018/03/03/nonstandard-integers-as-complex-numbers/

Isaac Kuo suggested that the Bricard octahedron could be used to make a garlic crusher, and Greg Egan designed a model here: https://plus.google.com/113086553300459368002/posts/KmRhPoPGkNz pic.twitter.com/Ijy5LLlAN7

Policy-makers are starting to assume we'll suck massive amounts of CO2 from the air later in this century, to prevent dangerous global warming. But how is that actually gonna work??? This method is gaining favor, but it's not without problems: https://tinyurl.com/baez-BECCS pic.twitter.com/qYfkjBm9D3

What sort of category has morphisms that act like pieces of conductive wire? A *hypergraph category*, where every object is a commutative Frobenius monoid obeying the conditions shown here! Two students at the Applied Category Theory school explain: https://tinyurl.com/ACT2018-hyper pic.twitter.com/pL7u5nsU7d

An "open Markov process" describes how a frog can randomly hop around, but also hop in or out of the whole picture. We can compose these, so they're morphisms in a category. But we can also "coarse-grain" them and get a double category! Here's how: https://tinyurl.com/baez-coarse pic.twitter.com/zIgzfX5nQY

Here's an upper bound on how many steps it takes to get from one picture of a link to another picture of the same link: 2 to the 2 to the 2... where the number of 2's is 10 to the n millionth power, n being the total number of crossings. Details here: https://tinyurl.com/baez-link pic.twitter.com/4JivjwOr0G

Lying under a bush, looking at the evening sky... No! This is the "{5,3,5} honeycomb" drawn by Jos Leys. 5 dodecahedra meet at any edge of this pattern. That's "too many", so it lives in hyperbolic space. And you can build a 3d manifold from it: https://wp.me/p42Vmc-82 pic.twitter.com/MlmcwsVTaL

I remember Hawking in Dublin conceding his famous bet on whether black holes increase the entropy of the universe when they decay. Much earlier, I snuck into the Institute for Advanced Studies for his talk on virtual black holes. More reminiscences: https://tinyurl.com/baez-hawking pic.twitter.com/hHH08oopBc

I'm live-blogging the meeting "Applied Category Theory: Bridging Theory and Practice", which is happening at the National Institute of Standards and Technology. Dusko Pavlovich just blew my mind about categories and computer security. Go here: https://tinyurl.com/baez-nist pic.twitter.com/Xk8Tbvg0KF

David Spivak's talking about his NASA-funded work using topos theory to design a safer air traffic control system. A topos is a "mathematical universe", and he's using one where everything is time-dependent. Sheaf theory meets airplanes! https://tinyurl.com/spivak-NASA

"America will triumph over you." Never before has a former director of the CIA said something like this to the President of the USA. Things are heating up. pic.twitter.com/WfxQt2zRFy

I'm sure it's happened to you. You're flying a fighter jet. You come out of a hard turn, feeling high g forces, when all of sudden YOU'RE ON THE WING OF THE PLANE! You look in the cockpit and you see... YOURSELF. Full story here: https://tinyurl.com/baez-oobe pic.twitter.com/0SPFr30MKN

Okay, time to admit it: I'm running a free online course on applied category theory, with over 250 students, based on Fong and Spivak's "Seven Sketches in Compositionality". Check out the first lecture here: https://tinyurl.com/baez-sketches-1 pic.twitter.com/yb1pvCxKzi

Ever wondered what "adjoint functors" were all about? Basically they provide the "best approximate solution to an unsolvable problem". For example: taking an odd number, dividing it by 2, and getting an integer. I explain this in my course here: https://tinyurl.com/baez-sketches-5 pic.twitter.com/rHbeDcaN0G

Nice picture! How about a slice parallel to one of the 600-cell's tetrahedral facets? I'd naively guess that has maximal symmetry too. Oh no - I guess it just has tetrahedral symmetry. https://twitter.com/gregeganSF/status/984400529458679808

Category theory applied to integer arithmetic! In my course on applied category theory, Matthew Doty proposed the following puzzle. It may be overkill to prove it using adjoint functors, but if you're a category theorist you'll want to do it that way! https://tinyurl.com/ACT-puzzle-1 pic.twitter.com/kxBqjXQJGJ

Check out the video of my talk at the National Institute of Standards and Technology: https://youtu.be/bL3lTMvpKHE Here are the slides: https://tinyurl.com/baez-nist-talk pic.twitter.com/eX9j94vyq1

Today I finished giving 17 lectures on ordered sets, logic, and "generative effects": situations where the whole is more than the sum of its parts. The grand finale: a baby version of the Adjoint Functor Theorem! See them all here: https://tinyurl.com/baez-ACT-chap1 pic.twitter.com/n2Zc2HDlDF

Email of the day: someone doing a podcast on high-IQ crackpots: "I focus on 3 men specifically, each who claim to have the highest IQ in the world--a bar bouncer (Chris Langan), a male stripper (Rick Rosner) and a cult leader (Keith Raniere)." Wants to know if I'm interested. pic.twitter.com/JodVT3zJGU

Engineers have long used analogies between electrical, mechanical and hydraulic networks. But any good analogy wants to become a functor! So, my students and I have formalized all this using category theory. I explain it in this blog article: https://tinyurl.com/baez-prop pic.twitter.com/P5O5NMIu95

Great art at the Stedelijk Museum, Amsterdam. An enormous concrete block hanging from the ceiling - ho hum. But wait: it's slowly rotating. And wait: NO WIRES, IT'S FLOATING IN MIDAIR! Then the fun starts. pic.twitter.com/QVFFguuY1z

Starting right now you can watch the Applied Category Theory 2018 talks here, on live streaming video: https://statebox.org/events/ The videos will also be available permanently, on YouTube.

Right now Kathryn Hess is talking about neurons, the brain, and categorical approaches to neurioscience. Live streaming video: https://statebox.org/events/

Yay! Our new journal is finally here! It's called "Compositionality". It's all about building big things from smaller parts. Another proposed title that didn't make the cut: "Applied Category Theory". For more, go here: https://tinyurl.com/compositionality pic.twitter.com/g5IOXn1Lb0

Applied category theory! "PERT charts" were developed in 1957 by the US Navy to schedule complex tasks. Now I'm talking about them in my online course. They are really enriched categories, but we're just getting started so don't tell the students: https://tinyurl.com/baez-sketches-19 pic.twitter.com/W42r43SavW

Only a mathematician would look at this recipe for lemon meringue pie and think "monoidal category". But this is a delicious way to start learning about monoidal categories and "resource theories". Join my class in the fun: https://tinyurl.com/baez-sketches-18 pic.twitter.com/zTD7WK4lyQ

Mind-blowing science in a mere "remark". "Frobenius manifolds" show up in topological string theory. Now arXiv:1805.02286 shows that at each point of such a manifold there's sitting an automaton - a probabilistic finite-state machine. Zounds! pic.twitter.com/WchPVbjcTN

These are some of my favorite things: the regular icosahedron, the cuboctahedron and the octahedron. They all have some symmetries in common, so we can morph between them in a nice way... but not _all_ the same symmetries, so we must make some arbitrary choices. pic.twitter.com/YLTWOWU7CH

My book with @JacobBiamonte is out! Learn how quantum ideas like Feynman diagrams can be applied to population biology! Learn how Petri nets let you do computation with chemistry! Plus, it's got a cartoon of me pulling a rabbit out of a hat. https://tinyurl.com/baez-qtism pic.twitter.com/T2OKHDkjwn

On my blog Matteo Polettini explains "effective" thermodynamics, where we only know _some_ of what's flowing into or out of an open system. In reality all systems are open, and we never have complete knowledge of inflows our outflows, so this is big: https://tinyurl.com/baez-polettini pic.twitter.com/t6t6kpzbXi

Nice to see my friend @carlorovelli's new book on the nature of time comes in an audiobook version read by... Benedict Cumberbatch! Rovelli is right, the coolest discovery of 20th-century physics is that "now" has no absolute meaning. https://tinyurl.com/rovelli-cumberbatch pic.twitter.com/B9iyWMVSi3

We often ignore the side-effects of what we do, like the "waste" shown in red here. That's one reason we're in trouble. But category theory can make this process of ignoring into a functor! Then, left and right adjoints can help us "un-ignore": https://tinyurl.com/baez-sketches-28 pic.twitter.com/4Zp0B9gsoz

My friend @DrEugeniaCheng is giving public lectures in Australia! The mathematician Ross Street took this photo. Her slide says "Category theory is the logical study of the logical study of how logical things work." Yeah! Don't fear abstraction - love it! pic.twitter.com/YRkvm47Wln

Algebraic geometers like replacing + by minimization and × by addition: curved surfaces defined by polynomials become shapes like this! But it's also good for railway scheduling - minimizing the total time! Now the two worlds finally meet: https://tinyurl.com/baez-tropical pic.twitter.com/7Y3sgAk0Q3

If the graviton had a mass, no matter how tiny, the sun would bend the path of light 25% less than it actually does! So: no graviton mass. And one possible way around this result, the "Vainshtein mechanism", just got killed. Full details: https://tinyurl.com/massive-gravity pic.twitter.com/mf5WlPhlrx

I want to read what the physicist Faraday wrote to Ada Lovelace! I've only seen a couple quotes, like this: "You drive me to desperation by your invitations. I dare not and must not come and yet find it impossible to refuse such a wish as yours." https://tinyurl.com/farad-love pic.twitter.com/C6Nm8O4NRn

Regular heptagons may be the worst possible convex shape for densely packing the plane - the best anyone has done is fill 89.3% of the plane. See https://tinyurl.com/baez-heptagon. But I never knew that including squares helped so much! Alas, I forget who drew this picture. pic.twitter.com/OQXURBUwhh

Dark matter news: the 1-ton liquid xenon detector XENON1T failed to find weakly interacting massive particles! Next: a 7-ton detector. Then a 40-ton model will push down to the background noise of the "neutrino sea", making further improvements pointless. https://tinyurl.com/xenon1t pic.twitter.com/L0PPIDbMZ6

First came linear logic. Then came "differential" linear logic, where you can take the derivative of a proof. Now they're differentiating Turing machines! What will those crazy logicians do next?! Cool stuff: https://arxiv.org/abs/1805.11813 pic.twitter.com/85RfdCphO7

Fermilab seems to be confirming Los Alamos result: muon antineutrinos are turning into electron antineutrinos after traveling just 541 meters! Too fast for the Standard Model: it's six standard deviations off! https://www.quantamagazine.org/evidence-found-for-a-new-fundamental-particle-20180601/

What's this business about physicists maybe discovering a new elementary particle that will change the Standard Model??? Here's my quick explanation - please ask questions on my blog: https://johncarlosbaez.wordpress.com/2018/06/02/miniboone/ pic.twitter.com/Bnu8BpdOm9

A newly discovered solid made of regular pentagons and squares?!? This movie was created by someone called "TED-43", on Wikipedia. It's called the "associahedron": https://commons.wikimedia.org/wiki/File:Associahedron.gif pic.twitter.com/h0QN1PZwTi

Of course the pentagons are *not* regular - you can see they're not. And if its faces were all regular, people would have known about it for a long time. Such polyhedra are called "Johnson solids" and it's believed there are exactly 92 - though nobody has proved it!

Battles with polynomials! In the 1400's Ferrari challenged this famous guy here, Tartaglia, to a math contest - but Tartaglia refused until he was offered a job *if* he accepted the challenge. After one day he was losing so badly he left town. https://tinyurl.com/ybmzeo4m pic.twitter.com/MgyvLmTD06

You can't square the circle with straight-edge and compass. But you can do it approximately by first drawing a regular 12-sided polygon! Zoltán Kovács explains it in his new paper "Another (wrong) construction of π". Check it out: https://arxiv.org/abs/1806.02218 pic.twitter.com/6DWC5IboEh

You can learn category theory through databases! We can get a category from a "database schema" as shown here; then a functor to the category of sets is an actual database. This is just the start of the fun. Try my course for more: https://tinyurl.com/baez-sketches-34 pic.twitter.com/gVhZnNlelb

In a famous incident, Theodore Roosevelt refused to shoot a bear tied to a tree. A cartoon was made of this, and thus the Teddy Bear was born. Then he had a man kill the bear with a knife to "take it out of its misery", and they ate it for dinner. https://www.npr.org/templates/transcript/transcript.php?storyId=338900449 pic.twitter.com/rP0cGo9UBs

I have a question about monads: https://tinyurl.com/baez-PP. If the answer is yes, we can take a set of sets of sets of sets and turn it into a set of sets in such a way that two resulting ways to turn a set of sets of sets of sets of sets of sets into a set of sets agree! pic.twitter.com/se2xdArRo3

"What's one and one and one and one and one and one and one and one and one and one?" "I don't know", said Alice, "I lost count." "She can't do addition." - Lewis Carroll, Through the Looking Glass.

An awkward shape with an unpleasant name: the "disphenocingulum". True, it's one of only 92 strictly convex solids all of whose faces are regular polygons: it's got 20 triangles and 4 squares. But it's 90th on the list. Sounds like an internal organ. Anything nice about it? pic.twitter.com/XdJQCy8gsR

Greg Egan thinks he answered the question in my post with "yes". I need to check his solution: https://golem.ph.utexas.edu/category/2018/06/sets_of_sets_of_sets_of_sets_o.html#c054092

2⁴ is equal to 4². 3³ is obviously equal to 3³. But π to e is less than e to the π, and the proof is easy if you remember calculus! It can't be new, but I saw it here: https://arxiv.org/abs/1806.03163 pic.twitter.com/nk2xeoZUyG

In QED every charged particle is surrounded by a cloud of "soft photons" - low-energy photons - that extends out to infinity. Physicists are starting to believe there's an infinite-dimensional group of symmetries acting on these clouds! New symmetries of QED = big deal! pic.twitter.com/1ks607xJra

It's almost the 100th anniversary of Noether's theorem relating symmetries and conservation laws! Here's a great intro by Emily Conover in Science News: https://tinyurl.com/y7ltyoyx She quotes me... about the dilemma facing fundamental physics today. pic.twitter.com/iU1Be1gKhw

Count the partitions of n, like p(4) = 5 since 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1. Compare the usual smooth approximate formula for p(n) and look at the percentage error. As @JohnDCook shows, it wiggles! Who knows about this? If you do, tell me! https://tinyurl.com/yczsrpkh pic.twitter.com/noYnfpf5kF

Get ready for "Applied Category Theory 2019" next summer in Oxford! Also watch videos and read summaries of discussions at ACT2018, and view glamorous photos of applied category theorists like Spencer Breiner (at right) and me: https://tinyurl.com/ACT2019-plan pic.twitter.com/SzoQAFasJ0

My guess: AIs will get legal rights by becoming corporations. In the US corporations already count as 'persons', with the right to free speech. People are already working to create AI corporations - see below. Then these will hire lobbyists.... https://bitcoinmagazine.com/articles/bootstrapping-a-decentralized-autonomous-corporation-part-i-1379644274/

Short version: a category is a bunch of dots and arrows. A functor, like F or G here, is a picture of one category drawn in another. A natural transformation, like α, slides one such picture over to another. For the full story, check out this: https://tinyurl.com/baez-sketches-43 pic.twitter.com/isyAiyoVHF

When will someone sail the longest "straight line" path on Earth? 32090 kilometers - starting in Pakistan, squeaking past Africa and Madagascar, dodging Antarctica and Tiera del Fuego, ending in Russia! It was mathematically proved to be the best, here: https://arxiv.org/abs/1804.07389 pic.twitter.com/kBQhjoT0Q0

Star math blogger Tae-Danae Bradley (@math3ma) explains the qualities of a banana in terms of convex sets: https://tinyurl.com/ncafe-convex. The color cube, the taste tetrahedron and the gooeyness interval can be tensored to describe all 3 qualities. Part of a new approach to cognition! pic.twitter.com/FLF2D7NprF

Jonathan Lorand drew this magnificent picture of the citric acid cycle, keeping careful track of each type of atom. It's for a paper we're writing on "biochemical coupling through emergent conservation laws". Great to have a collaborator who is even more obsessive than me! pic.twitter.com/DHRpwpWJQm

Ho-ho-ho! Santa Claus brings phosphoric acids to good little girls and boys! Oh-oh - some of these, like phosphoric anhydride, can cause severe burns! But it looks like a tiny tetrahedron, so it can't be all bad. pic.twitter.com/meUIPprbTK

The exceptional Jordan algebra, built from octonions, was discovered in research on foundations of quantum mechanics. I spent a lot of time studying it... but never noticed how the gauge group of the Standard Model is lurking in its symmetries! I gotta think more now. pic.twitter.com/vi53Roidz3

Summary: the automorphism group of the exceptional Jordan algebra has 3 maximal connected compact subgroups. The intersection of two is (SU(3)×SU(2)×U(1))/(Z/6)), the true gauge group of the Standard Model. (The Z/6 makes quarks have charge 2/3 and -1/3.)

My 7-year-old paper with Tom Leinster and Tobias Fritz is finally "trending". We proved that entropy is the unique continuous convex functor from finite probability measure spaces and measure-preserving maps to nonnegative numbers. Entropy is fundamental! https://twitter.com/arxivtrends/status/1012158771706114048

If you have a database listing Germans and their Italian friends, you can forget the Italians and get a list of Germans. But a "left Kan extension" tries to reverse this process. "Every German needs an Italian friend? Okay, make up some Italians!" https://tinyurl.com/baez-sketches-50 pic.twitter.com/dQz18GexZZ

The world's deepest hole! It's 12,226 meters deep and took almost 20 years to dig, but only 23 centimeters across. They dug halfway through the Earth's crust, but quit when it got too hot. They found microscopic fossils 6 kilometers down! https://tinyurl.com/baez-hole pic.twitter.com/0iRb1S5mnU

Atoms can stick together in threes even when two would not. You can use this to make a molecule of 3 cesium atoms. In fact an infinite series of them, each about e^π times larger than the one before! It's called "Efimov scaling" - a crazy quantum effect, now seen in the lab. pic.twitter.com/tdH1lXGgA0

The sum of the heights here doesn't change: it's a "conserved quantity". That's how you can use a heavy mass to pull up a light one! 3 students and I just showed that biology, too, uses "emergent conserved quantities" to control the flow of energy: https://tinyurl.com/baez-law pic.twitter.com/JbjDzL634q

Is it pointless to do highly abstract math like category theory? What purpose does it truly serve for mathematics? by John Baez https://www.quora.com/Is-it-pointless-to-do-highly-abstract-math-like-category-theory-What-purpose-does-it-truly-serve-for-mathematics/answer/John-Baez-1?srid=uQ6H

I'm studying the math of piss! You use the urea cycle to convert poisonous ammonia into urea. This cycle is described by differential equations that turn out to have 7 "emergent" conserved quantities. Some of these prevent the waste of energy: https://tinyurl.com/baez-urea pic.twitter.com/zTGyzxEojr

114 degrees here tomorrow. Global warming keeps cranking up the temperature, and this corrupt asshole never gave a damn. The new interim guy is a coal industry lobbyist. Trump: "the future of the EPA is very bright!" We gotta get rid of him. https://earther.com/so-long-scandal-boy-1827370997

Randomly pick two points on the unit sphere... in n-dimensional space, just to show off. What's their distance, on average? If n = 4, @gregeganSF showed it's 64/(15 pi). I predict that as n gets big, it approaches sqrt(2). What about the humble n = 2, 3 cases? pic.twitter.com/Wkb8ruuaFa

It's almost 47 Celsius. Tons of flies are buzzing around our front porch. Unusual: I think they're trying to keep cool. And as soon as I started watering a plant, a hummingbird landed on it and started sipping water from a leaf, undeterred by the spray! Welcome to the future. pic.twitter.com/EO4oc3Ehy8

Here they are, folks: all the "algebraic numbers". That is, complex numbers that are roots of polynomials with integer coefficients. The big ones are solutions of simpler equations: zero is gigantic. They're color-coded as explained here: https://tinyurl.com/baez-algebraic pic.twitter.com/oJwSlMUPy8

Believe it or not, you can describe this shape using a polynomial equation of degree 8. It's called the "Chmutov octic"- an attempt to get as many pointy things as possible for degree 8. They're called "nodes", and it has 144. Not the winner, though! https://tinyurl.com/baez-chmutov pic.twitter.com/GZWwEB9uiQ

By the way, I can't succeed in clicking any of the links on my blog anymore. It's like they aren't links anymore. Can somebody try clicking some of the stuff in blue here and tell me if the links work? https://tinyurl.com/baez-chmutov

Human eye "could" help test quantum mechanics - mmm-hmm, yeah right, not holding my breath on that - but mainly it's just fun to see how good people are at seeing single photons. Frogs can definitely do it! https://www.scientificamerican.com/article/the-human-eye-could-help-test-quantum-mechanics/

Amazing how much fun can be had with random points on a sphere! A team of mathematicians on Twitter just discovered a cool connection between Catalan numbers and the distance between random points on the sphere in 4d - using the quaternions! https://johncarlosbaez.wordpress.com/2018/07/10/random-p

We've discovered a surprising fact about the probability distribution of distances between random points on spheres! A mysterious mathematician known only as "Lucia" finished the job. It's fun having collaborators you don't even know. https://johncarlosbaez.wordpress.com/2018/07/12/random-points-on-a-sphere-part-2/

Hell yeah! Looks like they finally found high-energy neutrinos shooting out of a "blazar": a supermassive black hole shooting a beam straight at us. They found 'em using a detector made of a cubic kilo of ice. "It won't be business as usual." https://tinyurl.com/blazr pic.twitter.com/nlO4Mko4Xo

In 2016, California cut its carbon emissions to below what they were in 1990. That's 4 years sooner than our law demands! Next step: reduce carbon emissions by 40% percent below 1990 levels. The deadline: 2030. Rooftop solar is mandatory for new homes starting in 2020! pic.twitter.com/Uv2PE9OVnL

Enriched profunctors sound scary, but here's a nice one. Two islands C and D with one-way toll roads between some cities - and one-way plane trips from some cities in C to some in D. What's the cheapest trip from E to c? I explain the theory here: https://tinyurl.com/baez-sketches-59 pic.twitter.com/YtYwsoF9je

This is the free modular lattice on 3 generators, where "modular" means A∨(B∧C)=(A∨B)∧(A∨C) when A<B or A<C. The free modular lattice on 4 generators is infinite! This one is connected to representations of the D4 quiver, but big mysteries remain: https://tinyurl.com/baez-modular pic.twitter.com/lRCQuyhLDp

The animated gif was created by Jesse McKeown. Surprisingly, you can get it to switch directions and turn the other way using just the power of your own mind. It takes practice though.

Antimatter is amazing: it means negative numbers are real. Yes, you can have -3 particles: it's called having 3 antiparticles! Just as 3 - 3 = 0, three antiparticles can annihilate three particles of the same kind. But the discovery of antimatter is a funny story (cont.) pic.twitter.com/XWGQFLUDTe

Dirac invented a great equation describing electrons but it predicted negative-energy electrons as well as positive-energy ones, because numbers have two square roots. A lesser theorist would have ignored the negative-energy solutions, but he followed where the math led him.

Dirac imagined that the world was packed with a "sea" of negative-energy electrons and noticed that a "hole" in this sea would act like a positive-energy particle of the opposite charge - just like a bubble in the water acts like a thing of its own.

Problem: nobody had seen positively charged "anti-electrons". So Dirac desperately suggested that these were protons. But the mathematician Hermann Weyl said this was ridiculous: a proton is nothing like an electron, it's 1836 times heavier.

So Dirac resigned himself and admitted that his theory predicted a positively charged particle with the same mass as an electron. This would leave tracks bending around in the opposite direction in a magnetic field in a "bubble chamber" (see the picture).

Then physicists looked and YES, THEY FOUND THESE TRACKS! But the funny part is that they'd already been seeing these tracks. Since they "knew" there was no positively charged particle with the same mass as an electron, they assumed someone had flipped the picture over!

One moral is that you should take your own theories seriously enough to admit when they predict weird stuff. But another is that sometimes you need to be "authorized" to see what's right in front of you - if you don't believe it's there, you ignore it.

Sorry, at this time they were using "cloud chambers", not "bubble chambers" - see the nice history here: https://physics.aps.org/story/v17/st5

I really like my cloud at https://scimeter.org/clouds/. I made up the terms "2-group" and "spin foam", and there's a lot of my favorite math jargon, but I also study "electrical circuits", "Markov processes", and even "real numbers", "symbols", "axiom"... and "every". pic.twitter.com/hFGC1nNSn4

The equations of general relativity have solutions that describe weird things that probably don't exist but *might*. Einstein didn't believe in the Big Bang and black holes at first. They exist. His theory also allows black holes that move at light speed. Do those exist? pic.twitter.com/5hz2Yt2kvJ

Probably not, but they're allowed by Einstein's theory. Let the mass of a black hole approach 0 as its speed approaches that of light. In the limit you get a flattened-out gravitational shock wave that's strongest in the middle. Details: https://tinyurl.com/baez-sexl

In 1977, Kip Thorne and Anna Żytkow figured out what happens when a neutron star - a ball of neutronium 25 kilometers across - hits a normal star. It's called a "Thorne-Żytkow object" or TŻO. Now astronomers may have found one in the Small Magellanic Cloud! Read on.... pic.twitter.com/dr7WK0y6OQ

The neutron star slowly eats its host from within. Gas gets sucked into the neutron star and gets very hot: over a billion Celsius! The heat comes from two things: energy released when infalling gas hits the neutron star, and nuclear fusion after the gas hits. (Read on....) pic.twitter.com/pDWYfoKbdH

How can you tell if a star is a Thorne-Żytkow object? If it's a red giant, the neutron star will make its core a lot hotter than usual. So, the "rapid proton process" should create elements that you don't usually see in such a star! (One more....) pic.twitter.com/drXtLmy9O2

And now astronomers have found a red supergiant with a lot more rubidium, strontium, yttrium, zirconium, molybdenum and lithium than usual! Read "Discovery of a Thorne-Zytkow object candidate in the Small Magellanic Cloud": https://arxiv.org/abs/1406.0001

Nobody knows the planar shape of maximum area that can take right-angled turns both to the right and left in a hallway of width 1! But Dan Romik's "ambidextrous sofa" is the current champion, with area ~1.64495. Read more here: https://tinyurl.com/baez-romik pic.twitter.com/BNwLKyyego

My favorite 8-dimensional number system has an energetic new advocate! It's a long shot, but octonions *might* clarify the Standard Model. See also my Scientific American article with John Huerta, on their role in string theory: https://tinyurl.com/baez-strangest https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/

Look! If you graph all stars within 300 light years by mass and temperature, there's a line of "missing" stars with mass about 1/3 our Sun. Maybe it's a phase transition: the onset of "full convection". It's probably not just a glitch in the data: https://aasnova.org/2018/07/11/3796/ pic.twitter.com/MMPSNkQSHz

Let me tell you a theorem whose proof is so long you couldn't fit it in the observable Universe if you wrote one symbol on every electron, proton and neutron available. In fact.... (to be continued) pic.twitter.com/Uq93Exq27f

Let me tell you a theorem whose shortest proof has so many symbols you can't even write down the NUMBER OF SYMBOLS if you write one digit on each electron, proton and neutron in the observable Universe! (to be continued) https://johncarlosbaez.wordpress.com/2012/10/19/insanely-long-proofs/

To prove theorems you need some axioms so let's pick some famous axioms for math, say "Peano Arithmetic" or "PA". The details don't matter but we need to choose *some* axioms: otherwise we can't talk about the length of the shortest proof of a theorem!

There's a slight catch: if our PA axioms are inconsistent there's a short proof of *everything*, even 0=1. So, I'll assume PA is consistent. (Most mathematicians think it is.)

Now, Gödel showed that you could encode the concept of ‘statement’ and ‘proof’ into arithmetic, by coding everything as numbers. You can even use this to create statements that refer to themselves - that is, their own number.

Using this Gödel showed you could make a purely mathematical statement that secretly means "This statement has no proof in Peano Arithmetic." If PA is consistent, this statement can't possibly have a proof in PA - so it's true, but not provable in PA.

In 1936 Gödel noticed that you can also write a mathematical statement that says: "This statement has no proof in Peano arithmetic that contains fewer than 10^{1000} symbols." If PA is consistent, this has no proof in PA with fewer than 10^{1000} symbols!

But because it has no proof with shorter than 10^{1000} symbols, you can PROVE this in Peano Arithmetic just by going through all possible proofs with up to 10^{1000} symbols, and showing that none of them works! (Whew.)

We can easily replace 10^{1000} by 10^{10^{1000}}} and get a theorem of Peano Arithmetic whose shortest proof has so many symbols you can't even write down the number of symbols in the observable Universe.

But the interesting thing is that we can become convinced that this theorem is provable in Peano Arithmetic without actually giving the proof. We're using an argument that cannot be formalized in Peano Arithmetic!

So, by stepping out of an axiom system, assuming it's consistent and using that to draw conclusions, we can shorten proofs of some results by an arbitrary amount! For some more about the world's longest proofs, try this: https://johncarlosbaez.wordpress.com/2012/10/19/insanely-long-proofs/

To understand how Hamilton discovered the quaternions, you have to know his breakthrough in understanding complex numbers. In this interview I explain: https://plus.maths.org/content/os/issue32/features/baez/index Multiplying by a complex number can not only stretch or squash the plane, but rotate it! pic.twitter.com/83W82iwln4

Skimming over the photon sphere, you're orbiting a black hole at almost the speed of light! Above you see starlight, bent and discolored by gravity and your motion. The red below is not real: in reality that would be black. More great images here: http://jila.colorado.edu/~ajsh/insidebh/schw.html pic.twitter.com/uyYbBzhMik

Did you know this? There's an exotic element 1/9th as heavy as hydrogen. It's almost the same size as hydrogen. Its chemical properties are also almost the same. There's just one catch: it's unstable. On average, it decays in just 2.2 microseconds! (continued) pic.twitter.com/lzCzKZ62nV

2.2 microseconds sounds quick, but in chemistry it's not. Many chemical reactions happen much faster. So, there's plenty of time for this light version of hydrogen to form molecules. The picture shows it trapped in a crystal of silicon. pic.twitter.com/3zRFBW2Axw

Here's how you make it. First, shoot a beam of protons at a chunk of beryllium. If the protons have enough energy, you get short-lived particles called pions. Pions come in three kinds: positively charged, negatively charged and neutral.

A positively charged pion quickly decays into another positively charged particle called an antimuon. This lasts much longer: it has a half-life of 2.2 microseconds. It's 1/9 the mass of a proton. As it zips through matter, it can grab an electron.

An antimuon and an electron make an exotic atom called "muonium". It's very much like hydrogen, but about 1/9 as heavy. It's just a bit larger than ordinary hydrogen. You can do fun things with it. In 2002, people made muonium chloride! https://www.theguardian.com/science/life-and-physics/2016/jan/31/muonium-the-most-exotic-atom

Antimuons decay into positrons via the weak force in 2.2 microseconds on average. If the weak force were weaker, antimuons would last longer and we could do more with muonium. Too bad we can't! It would be so cool to work with "light hydrogen". https://www.decodedscience.org/standard-atom-muonium/49333

Miniature atoms! Yesterday I showed you how to make atoms 1/9 as heavy as hydrogen but the same size. Today: how to make atoms slightly heavier than hydrogen... but 1/200 as big. (continued) pic.twitter.com/iYDtseNJaU

Again the trick is to use muons. A muon is just like an electron except 207 times heavier. Unfortunately they are short-lived, decaying into electrons in just 2.2 microseconds on average. But that's enough time to orbit one around a proton! (continued)

The result is called "muonic hydrogen". Thanks to the uncertainty principle, because the muon is much heavier than an electron, muonic hydrogen is much smaller than hydrogen. Using muonic hydrogen to measure the size of the proton gave weird results! https://www.aps.org/publications/apsnews/201806/proton.cfm

Smaller atoms also makes nuclear fusion easier! So people have studied "muon-catalyzed fusion". It works, but it's not economically viable - not yet, anyway - because it takes so much energy to make muons: https://en.wikipedia.org/wiki/Muon-catalyzed_fusion (continued)

Even stranger is "muonic helium", where one of the two electrons in a helium atom is replaced with a muon. The muon orbits much closer to the nucleus than the electron. So chemically, this exotic atom behaves more like hydrogen! (the end) https://www.newscientist.com/article/dn20049-atomic-disguise-makes-helium-look-like-hydrogen/

It's not "bad" that quaternions don't commute. They describe rotations in 3 dimensions, which don't commute! You can do an experiment to test this. If you don't have a book, you can do it with a cell phone. I explain how in this interview: https://plus.maths.org/content/os/issue33/features/baez/index pic.twitter.com/6TdVsJ4V8Y

A billiard ball in a rectangle with two circular ends: slight changes in initial conditions make such a big difference that eventually it could be anywhere. To be precise, we say its motion is "ergodic". This animation was made by @roice713. (to be continued) pic.twitter.com/rQPS0OOdXH

For the precise definition of "ergodic" read my diary: https://tinyurl.com/baez-bunimovich A billiard ball moving in a convex region with smooth boundary can't be ergodic. In fact, in 1973 Lazutkin showed this for a convex table whose boundary has 553 continuous derivatives!

This result was improved by Douady in 1982: to prevent ergodic motion in your convex billiard table, it's enough for its boundary to have 6 continuous derivatives. And he conjectured that 4 is enough! (the end)

Imagine stars where it rains! Luhman 16 is a pair of brown dwarfs 7 light years from us. Luhman 16B is half covered by huge clouds. These clouds are hot — 1200 °C — so they're probably made of molten sand, iron or salts. Sometimes they disappear! Why? Probably rain. (cont.) pic.twitter.com/cDIs1QYlQc

Brown dwarfs are stars less than 80 times the mass of Jupiter — too small to power themselves by fusing hydrogen. They're very common! They start by fusing the isotope of hydrogen called deuterium, which people use in H-bombs. Then this runs out and they cool down. (cont.)

Type L brown dwarfs have clouds! But the cool type T brown dwarfs do not. Why not? This is the mystery we may be starting to understand: the clouds may rain down, with material moving deeper into the star! Luhman 16B is right near this "L/T transition". (cont.)

For more, read Caroline Morley's blog: ttps://tinyurl.com/ya8vkwgu She doesn't like how people call brown dwarfs "failed stars". Neither do I! It's like calling a horse a "failed giraffe". They are, in fact, fascinating worlds of their own. (the end)

My students @JadeMasterMath and @CCategories are going to the 2nd Statebox Summit, with sessions on Compositional Models for Petri Nets Consensus and Sheaves Logic Programming and Hardware Open Games and Cryptoeconomics More info here: https://summit.statebox.org/static.html Really cool! pic.twitter.com/fhhjQNAsrg

Math is such a big subject that someone can win a Fields Medal and I'll read a nice news article about it and be thinking "wow, who is this- and what's this weird stuff they did?" So I read a bit about Birkar's work.... (continued) https://www.quantamagazine.org/caucher-birkar-who-fled-war-and-found-asylum-wins-fields-medal-20180801/

He's studying "algebraic varieties", which are roughly curves, surfaces and higher-dimensional shapes described by polynomial equations. It's an old subject. In the 1800s people realized it's easier to study solutions with *complex numbers*, not just real numbers. (cont) pic.twitter.com/PUtu1v2R2z

It's also easier to study them if you add "points at infinity". For example, then *all* pairs of lines on the plane intersect - perhaps at infinity, like railroad tracks in perspective. This idea is called "projective geometry". https://pointatinfinityblog.wordpress.com/tag/projective-geometry/ pic.twitter.com/pUqbpxiLSz

Combining these ideas, "complex projective space" is a nice framework for studying algebraic varieties. In this framework a circle, ellipse, parabola and hyperbola are all just different views of the same curve! Nice and simple. So we can become more ambitious.... (continued) pic.twitter.com/1jWIzG455L

We can try the ambitious task of classifying *all* algebraic varieties. This sounds insane: classifying all possible curves, surfaces, and higher-dimensional things that you can describe with polynomials? Weird things like the shape here??? So one has to clever... (continued) pic.twitter.com/LJfUJ5paoz

First, we gotta view algebraic varieties through very blurry glasses, so tons of them count as "the same". Not just circles and hyperbolas! We gotta use "birational equivalence", so the red and black curves here count as the same: https://en.wikipedia.org/wiki/Birational_geometry (continued) pic.twitter.com/jeDCYDFQLZ

Classifying algebraic varieties "up to birational equivalence" gets harder as their dimension goes up. Curves are *all* birationally equivalent. Surfaces come in 10 families - I don't know this stuff, but I can look it up on Wikipedia: https://en.wikipedia.org/wiki/Enriques-Kodaira_classification (continued) pic.twitter.com/MOvxG2jiTm

But in higher dimensions things get harder. Starting around 1980, the "minimal model program" seeks to find the "simplest possible" algebraic variety that's birationally equivalent to whichever one you name. I think it succeeded in 3 dimensions.... (continued) pic.twitter.com/QMadeY3g0r

Caucher Birkar is pursuing the minimal model program in higher dimensions. The Quanta magazine explains what he's done about as well as possible w/o getting ultra-technical - so read that! He made progress on "Fano varieties". But what are those? https://www.quantamagazine.org/caucher-birkar-who-fled-war-and-found-asylum-wins-fields-medal-20180801/

Now for some real math. 😈 A smooth complex variety is Fano iff it admits a Kähler metric of positive Ricci curvature. Calabi-Yau manifolds are "Ricci flat", and most varieties are negatively curved. This rough 3-fold classification shows up all over math! (continued) pic.twitter.com/CH2SmHJFHO

So very roughly, Caucher Birkar is contributing to a grand program to classify all shapes described by polynomials... with a focus on "positively curved" ones. But the technical details are way over my head. Check out this video of a Fano variety! https://www.youtube.com/watch?v=y8o7fQw9wGU

Scott Wagner calls a woman "young and naive" when she asks if his wacky views are connected to the $200,000 he got from fossil fuel companies. Better young and naive than old, corrupt and stupid. https://tinyurl.com/baez-naive pic.twitter.com/QwWFoMSAYi

What would a ball inside a slightly stretched sphere with perfectly mirrored walls look like, if you were in there... and somehow there was light? Infinite reflections of reflections of reflections! This picture by John Valentine shows what you'll never see in real life. pic.twitter.com/4hiYMFZtp0

And here's a closeup. By the way, the "slightly stretched sphere" is really a prolate spheroid: that is, an ellipsoid with an axis of symmetry, longer than the other two axes. There should be many nice math puzzles lurking in these infinite reflections. pic.twitter.com/bJ6e8bkjcy

@j_bertolotti shows us chaos. Starting with very similar positions and velocities, two copies of a frictionless double pendulum drift apart... and soon they do completely different things! The "Lyapunov exponent" measures how fast this happens: https://tinyurl.com/yccfok6z pic.twitter.com/WBf7ROgb0Q

Want to learn some cool physics? Follow Jacopo Bertolotti's "Physics Factlets" at @j_bertolotti For example, learn how light can move slower than the speed of light... even in a vacuum!

Basic laws of arithmetic, like a×(b+c) = a×b+a×c, are secretly laws of set theory. But they apply not only to sets, but to many other structures! @emilyriehl explained this in @DrEugeniaCheng's "Categories for All" session. Check out her slides: http://www.math.jhu.edu/~eriehl/arithmetic.pdf pic.twitter.com/2cUEWW342W

Her proof of a×(b+c) = a×b+a×c relies on 3 facts about the category of sets: it has coproducts (the disjoint union A+B of two sets), it has products (the Cartesian product A×B of two sets), and it's cartesian closed (the set Fun(A,B) of functions from A to B).

The defining property of the coproduct A+B is that a function from A+B to any set X is "the same as" a function from A to X and a function from B to X. (Here "the same as" means there's a natural one-to-one correspondence.)

The defining property of the product A×B is that a function from any set X to A×B is the same as a function from X to A and a function from X to B. (If nobody ever tweeted this, the world would be sadly incomplete.)

The defining property of Fun(A,B) is that a function from X to Fun(A,B) is the same as a function from A×X to B. This change of viewpoint is called "currying", and @emilyriehl uses it twice in her proof. The defining properties of + and × are called "pairing". pic.twitter.com/x4C9JFqRxU

Every ingredient is needed for the proof to work! There are categories with coproducts + and products × that are not cartesian closed where A×(B+C) is not isomorphic to A×B+A×C. What's the most famous? Can you find one where A+(B×C) is isomorphic to (A+B)×(A+C)? pic.twitter.com/z1JliBqBBO

To hear @emilyriehl discuss this argument, and math in general, listen to her episode of "My Favorite Theorem": https://blogs.scientificamerican.com/roots-of-unity/emily-riehls-favorite-theorem/ pic.twitter.com/AoxveEEltP

Nobody knows if there are infinitely many "twin primes": primes that are 2 apart. But Viggo Brun proved the sum of the reciprocals of the twin primes converges. Now someone has conjectured a formula for this sum, which is called Brun's constant. (continued) pic.twitter.com/Kr8HFtVVag

The formula involves Catalan's constant G: the alternating sum of reciprocals of odd squares. Nobody knows if this constant is irrational, though I bet it is. (continued) pic.twitter.com/4xh3c6j1cN

The formula also involves the Ramanujan-Soldner constant μ. If you integrate the reciprocal of ln(x) from 0 to this constant, you get zero. Now let me show you the conjectured formula for Brun's constant! (continued) pic.twitter.com/H4s7A9hyHf

Marcin Lesniak recently conjectured this formula relating Brun's constant - the sum of reciprocals of twin primes - to Catalan's constant G and the Ramanujan-Soldner constant μ. My guess: it's false. First: it's "too good to be true". Second: read my tweets carefully. pic.twitter.com/qC1CKIk14u

I'm feeling sick so this elementary puzzle is enough to amuse me. An "isolated" prime is one that's not a twin prime. So, it's a prime p such that neither p-2 nor p+2 is prime. What are the first three isolated primes, and which one doesn't seem very "isolated"?

I find it amusing that you have to go so far up to get the first two *real* isolated primes. A naive experimentalist might conjecture that all odd primes are twin primes.

Is it is possible to untie this knot - that is, wiggle it around without cutting it to turn it into a loop of string? The answer is "yes". But can you write a program that decides, in a finite amount of time, the answer to all questions like this? (continued) pic.twitter.com/O0Y88AP9bb

As you wiggle around a possibly knotted loop of string, and someone watches from above, they'll see you carrying out a sequence of "Reidemeister moves". There are 3 Reidemeister moves, shown here. (continued) pic.twitter.com/G1ErcThA1I

Is there an upper bound on how many Reidemeister moves are required to untie a knot with n crossings, if it's possible at all? Yes, there has to be. But it could be big, since sometimes you have to make a knot more complicated before you can untie it. (continued) pic.twitter.com/KLCFU60P1b

In 1998, Joel Hass and Jeffrey Lagarias found an upper bound on how many Reidemeister moves it takes to untie a knot with n crossings, if it can be done at all: 2^(cn), where c = 10^11. Yikes! That's 2 to the n hundred billionth power! https://arxiv.org/abs/math/9807012 (continued)

So, you can write a computer program that decides whether any knot can be untied (that is, reduced to a simple circle). If your knot has n crossings, just try all possible sequences of 2^(cn} Reidemeister moves, where c = 10^11. Not practical, of course.. (continued)

In 2013 Marc Lackenby vastly improved the upper bound! He proved it takes at most (236 n)^11 Reidemeister moves to untie a knot with n crossings, if it can be done at all. Alas, still not good enough for a polynomial time algorithm. https://arxiv.org/abs/1302.0180 (continued)

What about telling whether any two knots are the same? In 2014, Coward and Lackenby found an upper bound for this. It's *really* absurd: 2 to the 2 to the 2 to the 2… where we go on for 10^1,000,000 times... to the number of crossings in both knots! (continued) pic.twitter.com/kutOPnqalR

This will doubtless be improved. Once you show something is computable, you can start trying to get good at computing it. Many problems in topology have answers that are *not* computable. But now we know: knot theory is computable. For more, my blog: https://johncarlosbaez.wordpress.com/2018/03/09/an-upper-bound-on-reidemeister-moves/

If you're in London October 5-6, you can hear me and other boffins talk about Noether's theorems relating symmetries and conservation laws! Or maybe you can give a talk. But you gotta register: https://philosophy.nd.edu/news/events/noether/ I hope they will make videos of the talks; I don't know. pic.twitter.com/HL5BNnfZfZ

Suppose I have a box of jewels. The average value of a jewel in the box is $10. I randomly pull one out of the box. What’s the probability that its value is at least $100? You can't tell me the exact probability... but you can tell me something about it. (continued) pic.twitter.com/v65zodLNDy

The probability can't be more than 1/10. If the probability of picking out a jewel worth at least $100 were over 1/10, the average value of the jewels in the box would be over $100 × 1/10 = $10. This is an example of Markov's Inequality. (continued)

Markov's Inequality says that if a nonnegative random variable has mean M, the probability of it taking a value ≥ X must be ≤ M/X. The proof is just like the one I gave. Chebyshev's Inequality goes one step further! (continued) https://en.wikipedia.org/wiki/Markov%27s_inequality …

Chebyshev's Inequality: if a random variable has standard deviation D, the probability that its distance from its mean is ≥ X must be ≤ D/X². You can prove this using Markov's Inequality: https://en.wikipedia.org/wiki/Chebyshev's_inequality (continued)

But beware: not every random variable has a well-defined standard deviation or mean! For example, the Cauchy distribution, proportional to 1/(x² + 1) has ill-defined mean and standard deviation. It's too "long-tailed". And there's a moral lesson here... (continued)

In real life unusually severe events are more likely than naive guesses might suggest. Lots of probability distributions are "long-tailed". So there may be no well-defined mean value for the size of a disaster... and results like Markov's inequality may not apply!

The number 5 is more exotic than all the natural numbers that come before. So, take the hyperbolic plane and tile it with pentagons, 5 meeting at each vertex. This is the "{5,5} tiling". Very fivey! We should be able to have fun with this. (continued) pic.twitter.com/ZlwuYKOb8h

Note: the {3,3} tiling of the sphere gives the tetrahedron, with 3 triangles meeting at each vertex. The {4,4} tiling of the plane has 4 squares meeting at each vertex. I could talk all day about these, but we're going straight for the jugular and doing {5,5}. (continued) pic.twitter.com/ljv0VBZpR1

The {5,5} tiling has a big symmetry group. If we mod out by some of these symmetries we get a surface tiled by just 12 pentagons, with 5 meeting at each vertex. Amazingly, we can almost fit this in 3d space. It's called the "great dodecahedron". (continued) pic.twitter.com/fMEivmq82f

The great dodecahedron has 12 vertices, 30 edges and 12 faces, which are pentagons that cross through each other. Now 12 - 30 + 12 = -6 = 2 - 2g with g = 4, so it's really a surface of genus 4. That is, a 4-holed torus! This picture is by "Brokk". (continued) pic.twitter.com/BiQ9DDmBir

The great Felix Klein showed this surface - the great dodecahedron - can also be described by equations in 5 complex variables: a+b+c+d+e = 0 a²+b²+c²+d²+e² = 0 a³+b³+c³+d³+e³ = 0 They're homogeneous so they describe a complex curve in CP⁴. It's a 4-holed torus. (continued) pic.twitter.com/fVpIAYa1Nt

There are many possible conformal structures on a 4-holed torus, but the one we get from the great dodecahedron is the most symmetrical of all! Its symmetry group is S₅: the group of permutations of the 5 variables I showed you. See how fivey it all is? (continued) pic.twitter.com/tWg7SrZKQ8

This amazing complex curve of genus 4 is also the set of ordered 5-tuples of complex numbers, modulo rescaling, that are roots of some quintic of the form z⁵+pz+q=0. You can see a sketch of the proof in my blog article on this stuff: https://blogs.ams.org/visualinsight/2016/06/15/small-stellated-dodecahedron/

What's this weird tree of fractions in that strange book Kepler wrote in 1619, "The Harmony of the World"? It may be a rather deep piece of mathematics! (continued) pic.twitter.com/gsfKEEftLR

It could be the "Calkin-Wilf tree". This tree contains all the positive rational numbers - and each shows up just once! Can you spot the Fibonacci numbers? Can you see the pattern that's being used to get all the numbers in this tree? I'll tell you... (continued) pic.twitter.com/MP5MRfA4tg

The Calkin-Wilf tree starts with 1/1. If a/b is any fraction in the tree, then a/(a + b) is below and to the left. (a + b)/b is below and to the right. Amazingly, each positive rational shows up just once, in lowest terms! (continued) pic.twitter.com/wB0rusSY67

So here's the rule for getting new numbers in the Calkin-Wilf tree. You can see why Fibonacci numbers show up! But the Calkin-Wilf tree also gives a nice way to *list* all positive rational numbers.... (continued) pic.twitter.com/EZWqslofpj

If you follow this spiral through the Calkin-Wilf tree, you'll meet every positive rational number, so you can list them all: 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, …. Amazingly, there's also recursive formula for this sequence! (continued) pic.twitter.com/mCv9ScNzGp

If you can't guess that recursive formula, you can look it up on Wikipedia! That's where I got all this great information... and all these nice pictures: https://en.wikipedia.org/wiki/Calkin-Wilf_tree Here is Calkin and Wilf's paper: http://www.math.upenn.edu/~wilf/website/recounting.pdf Also read about the Stern-Brocot tree!

This image by Greg Egan illustrates the amazing "Cayley–Salmon theorem": any smooth complex cubic surface contains 27 straight lines! In this example all the lines lie in real 3-dimensional space, not just complex 3d space. (continued) pic.twitter.com/vMv2pYx60H

This is the "Clebsch surface", defined by equations a+b+c+d+e = 0 a³+b³+c³+d³+e³ = 0 These pick out a 3-dimensional variety in C⁵ but since they're homogeneous we can count two solutions as "the same" if one is a multiple of another, and get a 2d variety in CP⁴. (continued) pic.twitter.com/8a4qClgRES

We can also eliminate one variable and write the equations as a³+b³+c³+d³ = (a+b+c+d)³ This picks out a 3d variety in C⁴ or, counting two solutions as the same if one is a multiple of another, a 2d variety in CP³. Less symmetrical, but closer to real 3d space! (continued) pic.twitter.com/pOa28L0GkT

If we pick nice coordinates we can see a lot of the Clebsch surface as a *real* surface in RP³... and most of it fits into R³, good old 3d space. Enough to see the 27 lines! That's what @gregeganSF has drawn here. pic.twitter.com/86nOmkoVrP

It takes real work to show every smooth complex cubic surface contains 27 lines! Here's a friendly sketch of two proofs: https://www.quora.com/Algebraic-Geometry/In-Algebraic-Geometry-why-are-there-exactly-27-straight-lines-on-a-smooth-cubic-surface/answer/Jack-Huizenga# You'll see how much algebraic geometry you know... or need to learn. (continued) pic.twitter.com/u0XZwhvwAh

27 is one of my favorite numbers. The 27 lines on the cubic are connected to the 27-dimensional "exceptional Jordan algebra": 3×3 self-adjoint matrices of octonions! (continued) pic.twitter.com/BHiXYbmoXL

For more on the Clebsch cubic, including formulas for the 27 lines and an explanation of the mysterious black dots in these pictures, try my blog: https://tinyurl.com/baez-clebsch Algebraic geometry is not just sheaves on schemes. It's also *geometry*. pic.twitter.com/awV7dKAbQi

Here's what the folks at Elsevier say when we complain about their profit-gouging, monopolistic behavior. They spend millions on lobbyists and lawyers to fight *against* the efficient distribution of knowledge. Then they mock us for trying to create a better system! https://twitter.com/mrgunn/status/1028812448063664129

You can use quaternions to understand electrons! Or: you can use electrons to understand quaternions! Hamilton invented the formula for quaternion multiplication long before people knew quantum mechanics. But it describes what happens when you rotate an electron. (continued) pic.twitter.com/dxophdwUx8

Usually people describe the spin of an electron using a pair of complex numbers... or in other words, 4 real numbers. But we can package the same information in a single quaternion: a + bi + cj + dk where a,b,c,d are real numbers and i,j,k are square roots of -1. (continued)

When you rotate the electron 180 degrees around the x axis, you multiply its quaternion by i. But if you rotate it like this *twice*, it doesn't come back to where it was! Instead, since i² = -1, it gets multiplied by -1. 🤔 But wait - what does that even mean??? (continued)

Shoot an electron through a double slit. It goes through *both* slits, then forms an interference pattern. Make a machine that rotates the electron 360 degrees when it goes through the left slit. This changes the pattern! (continued) pic.twitter.com/Ru2qgmwVf4

With work you can show experimentally that rotating an electron 360° around any axis multiplies it by -1. So, if we say rotating it 180° around the x,y, or z axes multiplies it by i, j, or k respectively, then we conclude i² = j² = k² = -1. (continued) pic.twitter.com/u54RDHcdJT

It's also true that rotating an electron 180° around the x axis, then the y axis, and then the z axis multiplies it by -1. This gives ijk = -1 In fact we get the whole quaternion multiplication table this way! So: quaternions are built into physics. https://en.wikipedia.org/wiki/Quaternion pic.twitter.com/tLFytkes61

Here's how to make a zero-calorie doughnut. Start with a doughnut and keep removing dough. In the limit, you get a space called the "solenoid", first invented by the topologist Vietoris in 1927. It has zero volume. It's connected but not path-connected! (continued) pic.twitter.com/At0VUeHI5N

The cool part: the solenoid is also a compact topological group, the limit of S¹ ⟵ S¹ ⟵ S¹ ⟵ S¹ ⟵ ⋯ where each map is multiplication by 2 (doubling angles). Tychonoff's theorem implies the category of compact topological groups has all limits! There are some weird ones.

@JadeMasterMath and I finished a paper on open Petri nets! Here is a Petri net. Things of various "species" (yellow circles) turn into things of other species via "transitions" (blue squares). For example, a susceptible meets an infected and we get 2 infected. (continued) pic.twitter.com/uYI5CE2gin

Petri nets are used in computer science, biology, chemistry and more. To build big Petri nets from smaller pieces we need "open" Petri nets like this. An open Petri net has a set of "inputs", namely X in this example X, and a set of "outputs", namely Y. (continued) pic.twitter.com/K64aQAIdU8

We can then feed the tokens into the Petri net and move them around using the transitions... perhaps getting all of them to points accessible from the outputs: (continued) pic.twitter.com/BirfTvWBR8

Then we can pull our tokens out, so they label the outputs. This is a very general, very simple model for many kinds of processes. It's called the "reachability semantics" for open Petri nets. (continued) pic.twitter.com/lqOnq77mby

In fact open Petri nets are morphisms of a category, since we can compose them! Here I'm composing an open Petri net from X to Y and one from Y to Z to get one from X to Z. Note that the species C, D and E are all getting "glued together" to give the new C. (continued) pic.twitter.com/7nZ7JG3cvr

But there are also "2-morphisms" going between open Petri nets! For example, we can map the open Petri net on top to the simpler one below by mapping A and A' to A, α and α' to α, and B to B. So, there's actually a "double category" with Petri nets as morphisms! (continued) pic.twitter.com/SHt2IcF2NW

Double categories were invented by Charles Ehresmann in 1963. They're great for studying open systems (which are morphisms) and maps between them (which are 2-morphisms). For the double category of open Petri nets, try my blog article: https://tinyurl.com/baez-petri-1 (continued) pic.twitter.com/BE7jY64Hvk

The "reachability semantics" for open Petri nets is a "double functor": a map between double categories. It sends open Petri nets to relations between sets! For more, try part 2 of my blog series! And then... (continued) https://johncarlosbaez.wordpress.com/2018/08/18/open-petri-nets-part-2/

And then see what's underlying all this math. In fact a Petri net is itself a way of specifying a certain kind of category: the "species" give objects, and the "transitions" give morphisms! So it's turtles, or categories, all the way down. For more: https://johncarlosbaez.wordpress.com/2018/08/19/open-petri-nets-part-3/

We're getting good at making strange forms of matter - like CRYSTALS MADE OF ELECTRONS. Wigner predicted these could exist in 1934. They were only made in the 1980s. Now we can make even cooler stuff, like 2d electron crystals. (continued) pic.twitter.com/o5qGYYUpuK

Most electrons have 6 neighbors. But there are also some red "defects", which are electrons with 5 neighbors. And blue defects: electrons with 7 neighbors. There are always 6 more red defects than blue ones. See why? Yup, it's topology... read this: https://tinyurl.com/y8smpxwo pic.twitter.com/4xwf1L04z7

A "hole" is a missing electron in a crystal. It can move around like a particle in its own right! An "exciton" is an electron and a hole orbiting each other. Last year people made a Bose-Einstein condensate of excitons. Matter is getting damned weird. pic.twitter.com/YlG1nX3ihP

Number theory is a great source of problems that are easy to state, hard to solve. For experts, though, the fun comes from big ideas - often connected to other branches of math! So it ranges from the cute: https://en.wikipedia.org/wiki/Full_reptend_prime to the deep: https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis pic.twitter.com/NdZv8QsjH8

Assuming the Generalized Riemann Hypothesis or P/=NP is a bit bolder, because you *can* do good math in these subjects *without* those assumptions, while trying to avoid ZFC in most math is like being a vegetarian in rural Texas.

One great thing about assuming these hypotheses and doing good stuff with them is that it increases the pressure to prove them - and also exhibits connections that might be useful. Cool applications of GRH: https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequences_of_the_generalized_Riemann_hypothesis

It's open for submissions! Our new journal! Free to read, free to publish in. With a great team of editors. For details, go here: http://www.compositionality-journal.org/ pic.twitter.com/HMKaAtyPh0

Those he commands move only in command, Nothing in love. Now does he feel his title Hang loose about him, like a giant’s robe Upon a dwarfish thief. - William Shakespeare pic.twitter.com/BeQ3PhqWKk

Almost all physics crackpots are men. Men standing atop Mt. Stupid are blessed with supreme confidence. Crackpots reside there permanently. pic.twitter.com/siOFKFCzHi

What's cooler than a superfluid? A supersolid! That's a solid crystal with vacancies - *missing* atoms - that form a Bose-Einstein condensate. In the same quantum state, they flow through the solid with zero viscosity - like a liquid made of ghosts! https://en.wikipedia.org/wiki/Supersolid pic.twitter.com/McwLRncrHt

It would be great if we could make sense of the Standard Model: the 3 generations of quarks and leptons, the 3 colors of quarks vs. colorless leptons, the SU(3)×SU(2)×U(1) symmetry group, etc. This paper may not be on the right track, but I feel a duty to explain it. pic.twitter.com/I5759sifNG

After all, it uses the exceptional Jordan algebra (3×3 self-adjoint octonion matrices), which I've wasted a lot of time thinking about. And the math is probably right. Just don't get me wrong: I'm not claiming this paper is important. I really have no idea.

Summary: if you pick one unit imaginary octonion and call it i, you can identify O (the octonions) with C+C³, nice for one colorless lepton and one colored quark. The subgroup of octonion automorphisms fixing i is SU(3), nice for the strong force. It acts the right way.

The Jordan algebra J3 of 3×3 self-adjoint octonion matrices contains J2, of 2×2 self-adjoint octonions matrices, in various ways. J2 can be identified with 10d Minkowski spacetime. Picking a unit imaginary octonion i chooses a copy of 4d Minkowski spacetime inside J2.

The automorphism group of J3 is the exceptional group F4. All this so far is well-known. New: the subgroup of F4 commuting with the action of "i" is (SU(3)×SU(3))/(Z/3). And the subgroup of *that* group preserving a copy of J2 in J3 is (SU(3)×SU(2)×U(1))/(Z/6).

(SU(3)×SU(2)×U(1))/(Z/6) is the true gauge group of the Standard Model: people often say it's SU(3)×SU(2)×U(1), but there's a Z/6 that acts trivially on all particles, thanks to quarks having exactly the right charges.

So, Todorov and Dubois-Violette are getting the true gauge group of the Standard Model from the symmetry group of J3 by choosing a copy of J2 inside it to serve as 10d Minkowski spacetime, and then choosing a unit imaginary octonion to pick out 4d Minkowski spacetime in that!

They do more, but I think this is the main result. Here is their paper: Michel Dubois-Violette and Ivan Todorov Exceptional quantum geometry and particle physics II https://arxiv.org/abs/1808.08110 pic.twitter.com/fgrFvTcwhq

You can also try my more detailed summary of their paper on the n-Category Cafe: https://golem.ph.utexas.edu/category/2018/08/exceptional_quantum_geometry_a.html

What's the fewest flips of a coin you need, on average, to randomly choose a number from 1 to N? Let's assume the coin is fair and - the hard part - each number 1, 2, ..., N has an equal chance of being chosen. The answer is the purple curve here! Complicated! (continued) pic.twitter.com/PZyjIOeThC

This is from a paper by Matthew Brand, https://arxiv.org/abs/1808.07994, pointed out by @sigfpe. It's easy to pick a random number between 1 and N = 2ⁿ with n coin flips. But when N is not a power of 2, it gets tricky: fractal patterns start to show up! Here number theory reigns. pic.twitter.com/0TZLh0Z390

The geometry of music revealed! Red lines connect notes that are a major third apart. Green lines connect notes a minor third apart. Blue lines connect notes that are a perfect fifth apart. Triangles are "triads", and they come in two mirror-image kinds: major and minor. pic.twitter.com/GTV5qlEYub

This pattern is called a tone net - and while it goes back to Euler, it's studied in "neo-Riemannian" music theory: https://en.wikipedia.org/wiki/Tonnetz The symmetry group of this tone net has 24 elements, and it's important in music theory: https://tinyurl.com/green-tonnetz pic.twitter.com/LKijgG2ooO

After the octonions comes a 16-dimensional number system called the "sedenions". They have some nice features, which I describe in this Quora post. But while the octonions are connected to many amazing mathematical ideas, nobody knows what the sedenions are good for. https://twitter.com/Quora/status/1034078220608581639

This is "right circularly polarized light". Take your right hand, make a fist, and point your thumb in the direction the light is moving. The electric field rotates in the direction your fingers are curling. Want to make "left circularly polarized light"? (continued) pic.twitter.com/8zyRehZZ3C

Get ahold of a golden scarab beetle! The physicist Michelson, who with Morley found that light moves past you at the same speed no matter how fast you go - so, no aether - also discovered that light reflected off this shiny beetle is left circularly polarized! (continued) pic.twitter.com/ozU985J8HR

We now know that the shell of the golden scarab is made of spiral-shaped molecules! These interact with light to make it left circularly polarized. But the math of circularly polarized light involves complex numbers - since these are good for plane geometry. (continued) pic.twitter.com/4B0X1i3AaJ

To describe a photon with a certain energy moving in a certain direction, we need 2 complex numbers! One is the amplitude for it to be right circularly polarized; its electric field rotates as shown here. The other is the amplitude for it to rotate the other way! (continued) pic.twitter.com/kwMoBeybeX

The role of complex numbers is easy to see in the quantum description of a photon, but it's already lurking in the classical Maxwell equations. The space of solutions of the vacuum Maxwell equations is a *complex* Hilbert space! To learn more, try https://tinyurl.com/torre-photon

In this video, Maryam Mirzakhani posed a puzzle like this: in a perfectly mirrored square room, how many bodyguards does it take to block every attempt of a photographer to take a picture of a celebrity? (Assume they're all points.) (continued) https://www.youtube.com/watch?time_continue=1551&v=tprlQMClSYQ

@emilyriehl was there. She solved the puzzle and talked about it to @math3ma, who made a great video about it. You might think infinitely many bodyguards would be required, but no! 16 is enough. (continued) https://www.youtube.com/watch?v=a7gp9c2p0UQ

@math3ma also blogged about it: https://www.math3ma.com/blog/is-the-square-a-secure-polygon Now @gregeganSF has studied this problem for a room shaped like an equilateral triangle: https://plus.google.com/113086553300459368002/posts/gFGPmCXHKUs The photographer takes shots in different directions, but they're all blocked. (continued) pic.twitter.com/aVk9jF3LVd

@gregeganSF also solved the problem for a room shaped like a regular hexagon. Finitely many bodyguards is enough! The trick in all 3 cases is to use a lattice in the plane. Here are all your mirror images as you pace in your mirrored square cell, drawn by Greg Egan. pic.twitter.com/j0JAYfiLV0

Big news! European organizations spending €7.6 billion on research annually will require every paper they fund to be freely available from the moment of publication, starting in 2020! Crush the rip-off journals! http://www.sciencemag.org/news/2018/09/european-science-funders-ban-grantees-publishing-paywalled-journals

After @QuantaMagazine wrote about octonions they decided to write about on quaternions - and they interviewed me! https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/ Did you know the "dot product" and "cross product" were originally two parts of a single thing, the quaternion product? pic.twitter.com/knqJQdaq13

Quaternions were invented in 1843. They became a mandatory examination topic in Dublin, and in some US universities they were the only advanced math taught! But then came Gibbs, the first US math Ph.D., who chopped the quaternion into its scalar and vector parts. pic.twitter.com/vCD4eNL9H7

"Rise for Climate" - world-wide rallies yesterday before the Global Climate Action Summit in San Francisco. But none so cool as the nomads of Kyrgyzstan. They look like a band of superheroes. https://www.flickr.com/photos/350org/43754930564/

But then, the World Nomad Games makes the Olympics look like croquet. Nomads rock! https://www.sbnation.com/2016/9/12/12888720/world-nomad-games-burning-horseriders-dead-goat-basketball-eagle-hunting-wow pic.twitter.com/eUAdplpuT2

This amazing gif by Gábor Damásdi illustrates a result of Jakob Steiner. If you can snugly fit some circles inside one circle and outside another, you can move them around while they stay touching! They may need to change size, though. The gif does this recursively. (cont.) pic.twitter.com/RR7G29EG2u

This result is usually called 'Steiner's porism'. What the heck is a 'porism'? Wikipedia writes: "The subject of porisms is perplexed by the multitude of different views which have been held by geometers as to what a porism really was and is." Okay... (continued) pic.twitter.com/tlmsDf64bs

Here's a "closed Steiner chain" of 12 black circles. Steiner's Porism: If one closed Steiner chain of n circles exists between two circles α and β, then *any* circle touching α and β belongs to such a chain. A more precise statement is here: https://en.wikipedia.org/wiki/Steiner_chain (cont.) pic.twitter.com/xc1VggX1V5

As you move the circles in a closed Steiner chain around, their centers move around an ellipse! It's shown in green here. Their points of tangency move around a circle! It's shown in brown. You can prove some of these things using conformal transformations. pic.twitter.com/p8ZrW9Kicw

"Fast radio bursts" or "FRBs" are one of the biggest mysteries in astronomy! Now Garry Zheng, a grad student at U.C. Berkeley, has used machine learning to detect 72 more FRBs from a galaxy 3 billion light years away - the only known source of *repeated* FRBs. (continued) pic.twitter.com/pnRO5lITAN

FRBs last a few milliseconds and occur in distant galaxies. Nobody knows what causes them! Hyperflares from magnetars? Blitzars? Superradiance from active galactic nuclei? Dark-matter induced pulsar collapses? Or, maybe... umm... extraterrestrials? (continued) pic.twitter.com/aqATASZj3N

I doubt it's aliens trying to fry themselves! But the Breakthrough Listen project paid for 5 hours of observatory time to search for FRBs from the one galaxy that repeatedly puts them out. They found 21. Then Gerry Zhang used machine learning to find 72 more. (continued) pic.twitter.com/diJFPUzoSt

So what's the payoff? So far, they've just ruled out some kinds of periodicity in the FRBs coming from this source (called FRB 121102). All the big mysteries still remain! But machine learning can help solve them! For more, read this older story: https://www.quantamagazine.org/astronomers-trace-radio-burst-to-extreme-cosmic-neighborhood-20180110/

I'm visiting the Centre for Quantum Technologies (in Singapore). Tomorrow I'm giving a talk on Noether's theorem relating symmetries and conserved quantities. It's the 100th anniversary of her paper on this! But we still haven't gotten to the bottom of it. (continued) pic.twitter.com/b2eQKocfvT

Noether showed that in a theory of physics obeying the "principle of least action", any 1-parameter family of transformations preserving the action gives a conserved quantity. This video is a good easy intro. (continued) https://www.youtube.com/watch?reload=9&v=04ERSb06dOg

But Noether's theorem takes different guises in other approaches to physics, and my talk focuses on the *algebraic* approach using Poisson brackets or commutators. I argue that this explains the role of complex numbers in quantum theory! Slides here: https://johncarlosbaez.wordpress.com/2018/09/12/noethers-theorem/

The weather disasters will keep getting worse unless we cut carbon emissions until they're *negative*. That's hard. But how much is the Paris Agreement helping? (continued) pic.twitter.com/wA8tYvWMHg

The EU committed to cut carbon emissions by 40% from 1990 to 2030. They'd gone down 23% by 2016 - but last year they went *up*. The EU's energy chief wants to do more: cut by 45%. But Merkel is pushing back. Germany is already going to miss its 2020 target. (continued) pic.twitter.com/YJs8AQnq2l

The US pledged to cut carbon emissions 17% from 2005 to 2020, and 26% by 2025. Trump pulled us out of the agreement. Many in the US are soldiering on, but right now it looks like we'll miss these goals. (continued) https://www.voanews.com/a/report-us-unlikely-to-meet-paris-climate-pledge/4569150.html pic.twitter.com/kx6G4ChvRc

China's carbon emissions shot up starting in 2000. It leveled off after 2010, even going down a bit - but it went up 3.5% last year. India and "ROW" (rest of world) are also going up. Read this: https://arstechnica.com/science/2017/11/2017-co%E2%82%82-emissions-to-increase-after-several-years-of-stability/ (continued) pic.twitter.com/TH8UdqtQfK

China pledged only to reach peak carbon emissions by 2030. See what other countries pledged - and how they're doing: https://climateactiontracker.org/countries/ Short version: right now we're heading for 2.5-4.7°C warming by 2100. Between bad and disaster. But some good news... (continued) pic.twitter.com/uQsF7IMiJy

California just passed a law saying we'll switch to completely carbon-free electricity production by 2045! Very hard, but look at the chart. And today the gov announced: "with climate science still under attack... we'll launch our own damn satellite" to track carbon emissions. pic.twitter.com/qhnghjvqtB

The vertices of this 4-dimensional shape are the quaternions 1, -1, i, -i, j, -j, k, -k It's a 4d analogue of the regular octahedron! These 8 points also form a group: the "quaternion group". And it's one of the most commutative of noncommutative groups! (continued) pic.twitter.com/rbq6b7h73U

The quaternion group has 2 elements, ±1, that commute with everything. The rest commute with 4 elements each: e.g. i with ±1,±i. 1/4 of the elements commute with all. 3/4 commute with half. Thus the chance that two elements commute is 1/4 × 1 + 3/4 × 1/2 = 5/8 (continued) pic.twitter.com/FL8xLWXKWm

If you randomly choose two elements of a finite group, what's the probability that they commute? For the quaternion group it's 5/8. But what's the *biggest* this probability can be, for *any* noncommutative group? It's 5/8. And I explain why here: https://tinyurl.com/y85bwu8d

Check out Tai-Danae Bradley's new short book "What is Applied Category Theory?" It's free, it's friendly, it's fun. It explains the big ideas, then applies them to chemistry and linguistics. You can follow her here on Twitter: she's @math3ma. https://www.math3ma.com/blog/notes-on-act

Sylow's theorems are key to understanding the structure of finite groups. Take a finite group G, a prime p, and let p^k be the biggest power of p that divides the size of G. Then G has a subgroup of size p^k, called a "Sylow p-subgroup". And more good stuff... (continued) pic.twitter.com/AsapfpE9NP

But what the hell are Sylow's theorems good for? As an undergrad in love with physics, I never really got it. Now on YouTube you can see examples that show what you can do with them - nice! Okay, but how do you prove them? (continued) https://www.youtube.com/watch?v=tx8gYKRc1iU

You can also see proofs of Sylow's theorems on YouTube, like this 3-part series. But I'm often too impatient for videos. I'd rather see really short proofs, then think about them in my spare time: washing the dishes, lying in bed, etc. (continued) https://www.youtube.com/watch?v=w96XSsEbdRU

So here's a really short proof of all 3 Sylow theorems from Robert A. Wilson's book "Finite Simple Groups". That's where I got the statement of the theorems in my first tweet, so compare that. This proof takes *lots of work* to unravel. But I love it: all the ideas are here. pic.twitter.com/gBUBdlg0Qy

But I'll only *really* understand Sylow's theorems if I understand how they're connected to other stuff where you focus on one prime at a time, like "p-completion" in homotopy theory. For my attempts to do that, try this blog post: https://golem.ph.utexas.edu/category/2018/09/plocal_group_theory.html

Sometimes you check just a few examples and decide something is always true. But sometimes even 9.8 × 10⁴² examples is not enough!!! @gregeganSF and I came up with this shocker here on Twitter. To see what's really going on, visit my blog: https://tinyurl.com/baez-fail pic.twitter.com/1FThY9skeZ

Tomorrow Atiyah will talk about his claimed proof of the Riemann Hypothesis. It's all about "the music of the primes". Here the function that counts primes < n is being approximated by waves whose frequencies come from zeroes of the Riemann zeta function. (continued) pic.twitter.com/lzNDyXo3Im

The Riemann zeta function is given by this simple formula when the complex number s has Re(s) > 1. Then the sum converges! But we can "analytically continue" the Riemann zeta function to define it for other values of s, and that's where the fun starts. (continued) pic.twitter.com/E6fUwdfNJF

The Riemann zeta function is zero for some numbers with 0 < Re(s) < 1. These are called the "nontrivial zeros" of his zeta function. Riemann computed a few and hypothesized they all have Re(s) = 1/2. https://www.youtube.com/watch?v=sD0NjbwqlYw

Riemann found a formula for the number of primes < n as a sum over the nontrivial zeros of the zeta function. My first tweet shows the sum over the first k nontrivial zeros. So, if the Riemann Hypothesis is true, we'll get a better understanding of primes! (continued) pic.twitter.com/CbTvsCV1VE

So far people have checked, using a computer, that the first 10,000,000,000,000 nontrivial zeros of the Riemann zeta function have Re(s) = 1/2. This might seem like damned good evidence for the Riemann Hypothesis. But maybe not! (continued) pic.twitter.com/QOlcuXmr2s

For me, the best evidence for the Riemann Hypothesis is that it's part of a much bigger story! Mathematicians like Weil, Grothendieck and Deligne proved similar results for related functions. Much remains mysterious, though. (continued) https://www.youtube.com/watch?v=sD0NjbwqlYw

I bet that Atiyah's claimed proof, if and when he writes it up, will not convince experts. In 2017 he claimed to have a 12-page proof of the Feit-Thompson theorem, which usually takes 255 pages: https://www.maths.ed.ac.uk/~v1ranick/atiyahtimes2017.pdf He showed it to experts, and... silence. (continued)

In 2016 Atiyah put a paper on the arXiv claiming to have solved a famous problem in differential geometry. The argument was full of big holes: https://mathoverflow.net/questions/263301/what-is-the-current-understanding-regarding-complex-structures-on-the-6-sphere So, I'm not holding my breath this time. But of course I'd be happy to be wrong. (the end) pic.twitter.com/YPsR2FSd3M

Here is Atiyah's lecture: on the Riemann Hypothesis https://www.youtube.com/watch?v=UBVy0oOYczQ Here, apparently, is his paper: https://www.dropbox.com/s/pydoj0a8hguebc6/2018-The_Riemann_Hypothesis.pdf?dl=0 It refers extensively to this much longer paper, where he attempts to compute the fine structure constant: https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view

Here's what Science says about Atiyah. Only a few mathematicians were willing to be quoted, and I drew the short straw. By the way, I have huge respect for Atiyah, whose earlier work revolutionized geometry and physics. http://www.sciencemag.org/news/2018/09/skepticism-surrounds-renowned-mathematician-s-attempted-proof-160-year-old-hypothesis

What are the big unsolved mysteries of physics... that we have a decent chance of solving? I'll talk about this on Wednesday Oct. 3 for the Cambridge University Physics Society. 8-9 pm, followed by a reception. Costs £2 - I'll make it worthwhile! http://talks.cam.ac.uk/talk/index/110938 pic.twitter.com/I51HDvO7Os

Here's the smallest known number with the digits 1,2,3,4,5,6 appearing consecutively in every possible order. This was discovered by Robin Houston, a category theorist! https://oeis.org/A180632 pic.twitter.com/22Alo2Elbz

I'm talking about Noether's theorem connecting symmetries and conservation laws twice in England next week: in Cambridge and London! First at Hawking's old hangout, DAMTP, on Thursday Oct. 4 at 1 pm. (continued) pic.twitter.com/bG1JftWg7Y

Then I'll give almost the same talk on Saturday Oct. 5 at 5:30 pm at Fischer Hall, The University of Notre Dame, 1-4 Suffolk Street, London. This is in a cool workshop with lots of great talks - you can read about it here! https://philosophy.nd.edu/news/events/noether/

Fog in a Martian canyon on a chilly morning! This is a high-res image of Valles Marineris: a huge canyon on Mars, 4000 kilometers long and up to 7 kilometers deep. Some believe this fog indicates a source of water, because the canyon is *warmer* than the uplands. (cont) pic.twitter.com/M6IkBBsoN1

Unless you click on that last image you'll just see part. Click and download! Here's another view. This fog is made of water ice, not liquid water, so it's a Martian relative of "pogonip": a dense winter frozen fog in mountain valleys. (continued) pic.twitter.com/XUC1bJNKRA

And here's a view of fog in Noctis Labyrinthus - a maze-like system of deep, steep-walled valleys between Valles Marineris and the nearby Tharsis upland. Fog in the Labyrinth of the Night! How poetic! I want to go see this someday. pic.twitter.com/b78KwqYOm5

ANITA, the Antarctic Impulse Transient Antenna, detects ultra-high-energy cosmic neutrinos. Here it is in front of Mount Erebus, It's seen some amazing things... possibly signs of physics beyond the Standard Model! But first let's talk about how it works. (continued) pic.twitter.com/UBiCsgIRBr

A neutrino shooting through Antarctic ice hits an atom and creates a shower of charged particles moving faster than the speed of light in ice. They create radio waves - Cherenkov radiation! As they leave the ice, the waves refract. It's called the "Askaryan effect". (continued) pic.twitter.com/XnSl93Tf2U

ANITA, hanging from a weather balloon, has seen many events like this. But *two* of these pulses came almost *straight up*, which is not how it's supposed to work! An ultra-high-energy neutrino shouldn't be able to go through the Earth. (continued) https://www.sciencenews.org/article/hints-weird-particles-space-may-defy-physics-standard-model

ANITA saw one ultra-high-energy pulse coming straight up from the Earth in 2006, and one in 2014. Another Antarctic neutrino detector has seen similar strange events. This could be great! But don't believe the theories yet - it's way too early. https://gizmodo.com/new-particle-could-explain-unusual-antarctic-weather-ba-1829372246

For more read: Observation of an inusual upward-going cosmic-ray-like event in the third flight of ANITA, https://arxiv.org/abs/1803.05088 The ANITA anomalous events as signatures of a beyond Standard Model particle, and supporting observations from IceCube, https://arxiv.org/abs/1809.09615

Take n equally spaced points on the unit circle. Draw lines from one point to all the rest. The product of their lengths is exactly n. Cool! But what if we stretch the circle by a factor of √5 at right angles to your chosen point? Then it gets 𝘳𝘦𝘢𝘭𝘭𝘺 cool! (cont.) pic.twitter.com/TRe6P5GuOq

Now the product of the lengths is n times the nth Fibonacci number! Proved by Thomas Price, this result is explained in "Chords of an ellipse, Fibonacci numbers, and cubic equations" by Ben Blum-Smith, Japheth Wood: https://arxiv.org/abs/1810.00492 Fun stuff! pic.twitter.com/0PmqJ9OIQ4

A bunch of us are organizing "Applied Category Theory 2019", a conference followed by a week-long school for grad students, in Oxford, June 15-26 next year. In a while you'll be able to submit papers or apply for the school. Here's some info about it! https://johncarlosbaez.wordpress.com/2018/10/02/applied-category-theory-2019/

If you're near Riverside and like math, come to this workshop that I'm running Oct. 19-20. Lots of good talks on pure and applied math! Please register on this page: http://math.ucr.acsitefactory.com/event-list/2018/10/19/riverside-mathematics-workshop-excellence-and-diversity pic.twitter.com/SOk4sLFo1V

I hear that if you take n equally spaced points on the circle and sum one over the distance squared from one point to all the rest, you get (n^2 - 1)/12. Someone told me this is called "Verlinde's equation", but I can't find it anywhere online.

More progress on a famous problem! A subset of the plane has "diameter 1" if the distance between any two points in this set is ≤ 1. An equilateral triangle with edges of length 1 has diameter 1... and it doesn't fit in the circle with diameter 1. (continued) pic.twitter.com/1NFqUEiotn

In 1914 the famous mathematician Lebesgue sent a letter to his pal Pál. He challenged Pál to find the convex set with least possible area such that every set of diameter 1 fits inside... at least after you rotate, translate and/or reflect it. (continued) pic.twitter.com/En5aYjAA23

Pál showed the smallest regular hexagon containing a disk of diameter 1 would work. Its area is 0.86602540… But he also showed you could cut off two corners, and get a solution of area just 0.84529946… Nice! (continued) pic.twitter.com/LoErDOaW0i

In 1936, Sprague showed another piece of the hexagon could be removed and still leave a universal covering - that is, a shape that can cover any set of diameter 1. This piece has area about 0.02. Cutting it out, we get a universal covering of area 0.84413770843... (continued) pic.twitter.com/m531VEmXOm

In 1992, Hansen removed two microscopic slivers from Sprague's universal covering and got an even better one! They were absurdly small, with areas of roughly 4 × 10^(-11) and 8 × 10^(-21). He got a universal covering of area 0.844137708398... (continued) pic.twitter.com/9vklkADNmS

At this point two mathematicians joked: "...it does seem safe to guess that progress on this problem, which has been painfully slow in the past, may be even more painfully slow in the future." But that's when Philip Gibbs came in! (continued) https://johncarlosbaez.wordpress.com/2013/12/08/lebesgues-universal-covering-problem/

He found a way to remove a piece about a million times bigger than Hansen’s larger sliver. Its area was a whopping 2 · 10^(-5), leaving a universal covering of area 0.8441153769… A student and I helped him publish this result. (continued) https://johncarlosbaez.wordpress.com/2015/02/03/lebesgues-universal-covering-problem-part-2/

I met Philip Gibbs in London last week, and it turns out he's done even better! He has chopped off another enormous chunk, with area 2 · 10^(-5), leaving a universal covering of area just 0.8440935944… Huge progress! For details, read his paper. https://johncarlosbaez.wordpress.com/2018/10/07/lebesgue-universal-covering-problem-part-3/

The moral is that even plane geometry holds deep problems connected to optimization. Lebesgue's universal covering problem, the sofa problem, the Moser worm problem, Bellman's "lost in a forest problem" - we just don't know powerful techniques to tackle these... yet. pic.twitter.com/U0xnATbGTW

The one you're not looking at turns clockwise. I believe this illusion was invented by Arthur Shapiro and collaborators: http://www.illusionsciences.com/2008/12/rotating-reversals.html pic.twitter.com/LXXspHOn2m

Here's the "{7,3,3} honeycomb", drawn by Danny Calegari. It's built of regular heptagons in hyperbolic space. These heptagons lie on infinite sheets, with 3 meeting at each vertex. And 3 sheets meet along each heptagon edge! That's why it's called {7,3,3}. (continued) pic.twitter.com/LXwqPG8QvR

Each sheet of heptagons in the {7,3,3} honeycomb is a "{7,3} tiling". Here's a {7,3} tiling drawn by Anton Sherwood. It lives in the hyperbolic plane. The {6,3} tiling lives on the flat plane, and the {5,3} tiling lives on a sphere: it's a dodecahedron! (continued) pic.twitter.com/8cUNP0SPyg

The symmetry group of the {7,3,3} honeycomb is called o—7—o—3—o—3—o This means it has generators V,E,F,S, each squaring to 1, with (VE)⁷ = (EF)³ = (FS)³ = 1. The symmetries can carry any triangle here to any other triangle. @roice713 drew this picture! (continued) pic.twitter.com/gd8Bmle37E

Here's the 'boundary' of the {7,3,3} honeycomb, where it meets the 'plane at infinity' of hyperbolic space - the surface of the ball in the previous picture. The black circles are the holes in the previous picture. Again this is by @roice713. (continued) pic.twitter.com/dDnBKz4Bft

For more about the math, go here: https://blogs.ams.org/visualinsight/2014/08/14/733-honeycomb-meets-plane-at-infinity/ and here: https://blogs.ams.org/visualinsight/2014/08/14/733-honeycomb-meets-plane-at-infinity/ Have a great weekend and learn lots of cool stuff! Here's yet another picture of the {7,3,3} honeycomb by @roice713... master of honeycombs. pic.twitter.com/6ZZQ20r2Pw

A polygon can be dissected into straight-edged pieces which you can rearrange to get any other polygon of the same area! Sometimes you can even do a "hinged" dissection, as shown here by Rodrigo Camaro. But only if you're lucky. What about higher dimensions? (continued) pic.twitter.com/zJHLV4k5a0

Two polyhedra are "scissors-congruent" if you can cut one into finitely many polyhedral pieces, then reassemble them to get the other. In 1900 Hilbert asked: are any two polyhedra with the same volume scissors-congruent? His student Max Dehn answered this! (continued) pic.twitter.com/ZGzq8oaW2l

The cube, and a regular tetahedron with the same volume, are not scissors-congruent. To prove this Dehn found another invariant of scissors-congruence beside the volume! Sum the edge lengths times their dihedral angles mod rational multiples of π. (continued) pic.twitter.com/XnzjdzeB9P

The "Dehn invariant" is an element of this tensor product of vector spaces over the rational numbers: R tensored with R/πQ. In 1965 Sydler proved that two polyhedra are scissors-congruent if and *only if* they have the same volume and Dehn invariant. (continued)

But this is just the beginning of the story, which now involves hyperbolic space, algebraic K-theory, and even ideas connected to quantum gravity, like the famous "pentagon identity" arising from two ways to chop this shape into tetrahedra. For more: https://tinyurl.com/dehn-scissors pic.twitter.com/wpGI2xqyTk

The population of insects has been crashing worldwide. Butterflies and moths are down 35% in the last 40 years. Other invertebrates are down even more: http://science.sciencemag.org/content/345/6195/401.full (continued) pic.twitter.com/UW8lbWnZDp

A new study from Puerto Rico is even more disturbing. Between January 1977 and January 2013, the catch rate in sticky ground traps fell 60-fold. Says one expert: "Holy crap". https://www.washingtonpost.com/science/2018/10/15/hyperalarming-study-shows-massive-insect-loss/

Here we see the milligrams of arthropods (like insects) caught in sticky traps on the ground (A) and forest canopy (B) in a rain forest in Puerto Rico. Dramatic collapse! And it hurts the whole ecosystem: lizard populations are halved or worse. http://www.pnas.org/content/early/2018/10/09/1722477115 pic.twitter.com/SxteqrWYtZ

String theorists like anti-de Sitter spacetime... but it's very weird. In 2d it's the hyperboloid shown here! Time loops around in a circle, while space extends infinitely far left and right. (The blue cone is just for fun, not part of the spacetime.) (continued) pic.twitter.com/hxTWwjiH6a

Since "closed timelike loops" are disturbing - can you kill your grandfather? - physicists usually unroll anti-de Sitter spacetime so the circle of time becomes a line. This trick is called taking a "universal covering space". (continued) pic.twitter.com/cCvc5BXD1X

Even after we get rid of timelike loops, this spacetime is strange. If you move without accelerating you trace out a curve gotten by intersecting the hyperboloid with a plane through the origin. All such curves go through another "antipodal" point. (continued) pic.twitter.com/g1oip0ECZk

Our universe is a bit more like de Sitter spacetime. In 2d de Sitter spacetime, *space* is a circle, while time extends infinitely far up and down. The universe expands as time passes... at least in the top half of the picture! (continued) pic.twitter.com/COi1AzjZdp

All these pictures and more come from Ugo Moscella's tour of de Sitter and anti-de Sitter spacetime. Check it out: http://www.bourbaphy.fr/moschella.pdf pic.twitter.com/EFEdT7JCGr

As algebraists and topologists soar upward to the paradise of ∞-categories, will they leave their colleagues behind in a mathematical version of the Rapture? I hope not. So I'm glad analysts are starting to study higher gauge theory: https://golem.ph.utexas.edu/category/2018/10/analysis_in_higher_gauge_theor.html pic.twitter.com/dxVNmFhpA8

New progress on a hard problem! @gregeganSF broke the record for the shortest number with the digits 1 through 7 arranged in all possible orders. He found one with just 5908 digits. (continued) pic.twitter.com/lqzSZtuQG7

Here is Egan's solution. It's 4 digits shorter than the previous best, and he got it using ideas from Aaron Williams' work on Hamiltonian cycles on Cayley graphs: https://arxiv.org/abs/1307.2549. But what's the pattern? What's really going on? (continued) pic.twitter.com/aR3ufllvvl

@robinhouston noticed that the new winner containing all permutations of n=7 digits has n! + (n-1)! + (n-2)! + (n-3)! + (n-3) = 5908 digits. And Egan also found new winners for n=8,9, which also fit this pattern! (continued) https://www.youtube.com/watch?v=wJGE4aEWc28

However, this formula does *not* correctly predict the length of the winners for n=3 and n=4... so the true pattern is more complicated. Ask @robinhouston or @gregeganSF for an explanation: they have at least a partial understanding, though much remains mysterious.

25 minutes of photos taken by the Rosetta mission on comet Churyumov-Gerasimenko, nicely assembled into an animated gif by @landru79. This place is called The Cliffs of Hathor. The "snow" is ice and dust moving slowly; you can also see some stars in the background. pic.twitter.com/4I8lSGiR5s

"Super Saturn" is an object 400 light years away with enormous rings 2/3 the size of Earth's orbit. This object could be a big planet or a very small star. It's orbiting another star. The big puzzle is: why don't these huge rings get torn apart? (continued) pic.twitter.com/zXUGK2APmv

If Super Saturn had a moon, the night on that moon might look like this. Super Saturn's real name is J1407b. Learn more about it here: https://tinyurl.com/super-saturn But back to my puzzle! Why don't the huge rings get torn apart by the gravity of the star it orbits? (continued) pic.twitter.com/ZcjHIjUgfn

Simulations show that if the rings turn the opposite way from how Super Saturn orbits its host star, they are more stable! This is called "retrograde" motion. But now the question is: why would the rings turn the other way? No one knows. Details here: https://arxiv.org/abs/1609.08485 pic.twitter.com/rGz31QtLAZ

Mini-Saturn! Chariklo orbits the Sun between Saturn and Uranus. It's a "centaur" — half asteroid, half comet. And it has two tiny rings, just 850 kilometers across! One is 6 kilometers wide and the other — barely visible in this artist's picture — just 3 kilometers wide. pic.twitter.com/HFB17Y2Nbd

A "black Saturn" is a black hole with a black ring - a ring-shaped singularity. We've never seen one. There's a good reason: stationary solutions of the vacuum Einstein equations have to be ordinary black holes. But what if space were 4-dimensional? (continued) pic.twitter.com/l2heCYbptC

In 2001, two physicists proved that general relativity allows rotating but otherwise stationary ring-shaped singularities in 4-dimensional space! (Not 4d spacetime.) And in 2007, Henriette Elvanga and Pau Figueras found that such a ring can orbit a black hole! (continued)

Check 'em out: R. Emparan and H. S. Reall, A rotating black ring in five dimensions, https://arxiv.org/abs/hep-th/0110260 (that's 5d spacetime, 4d space) Henriette Elvang and Pau Figueras, Black Saturns, https://arxiv.org/abs/hep-th/0701035 So "Black Saturns" is not just a good name for a rock band!

And thus ends my mini-series on Super-Saturn, Mini-Saturn and the Black Saturns. A downright Saturnalia! But don't forget the original Saturn. pic.twitter.com/dVBGn1Wjua

Next year is the 150th anniversary of the Periodic Table! I'll give a public lecture about it at Georgia Tech. Here's a boring title: "The math and physics of the Periodic Table". Can you suggest a more intriguing alternative? It'll be a romp through QM and relativity... pic.twitter.com/QVbChpHsqQ

Okay, it doesn't have the quirky charm of many suggestions here, but the organizers of this event took my suggested title: Why These Atoms? Mysteries of the Periodic Table and convinced me to use this title instead: Mathematical Mysteries of the Periodic Table

The axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite. It has shocking consequences... but so does its negation! (continued) pic.twitter.com/QUKFjMo5Az

The axiom of choice implies that we can't define lengths, areas, and volumes for all sets in a way that obeys a short list of reasonable-sounding rules. Indeed, we can chop a ball into 5 subsets and rearrange them to get 2 balls: the "Banach-Tarski paradox". (continued) pic.twitter.com/UE5aP23TEk

If we assume the *negation* of the axiom of choice, we can make this problem go away. Technically: Solovay found models of ZF¬C where all subsets of Euclidean space are measurable! But that doesn't mean life is trouble-free! (continued) pic.twitter.com/hJreKsFIlG

Indeed, in all known models of ZF¬C (set theory and the negation of axiom of choice), there is a set that can be partitioned into strictly more equivalence classes than the original set has elements! It's hard to find axioms that give just what we want, and nothing freaky. pic.twitter.com/MqtCc4KP25

In modern math we often replace equations with arrows, also called "morphisms". When we do this, the boring old associative law (ab)c = a(bc) gives birth to the "associahedra". Here is the 3d associahedron, whose vertices are all ways of bracketing 5 letters. (continued) pic.twitter.com/UEta1AgBLh

This trick of replacing a bunch of equations with a polyhedron is very powerful! Samuel Vidal just typed up Todd Trimble's notes on it: http://math.ucr.edu/home/baez/trimble/polyhedra.html and I'm giving a gentle intro here: http://math.ucr.edu/home/baez/qg-fall2018/ The associahedron pics are by @NilesTopologist! pic.twitter.com/UUNY5tFXuA

Today I learned: unlike in our universe, in 2d space solids melt in two separate stages! Solid, hexatic, liquid - shown below. People have shown this mathematically but confirmed it experimentally using a layer of small magnetized beads. (continued) pic.twitter.com/djVgmFyFL7

The solid phase has long-range correlations between the *orientation* of rows of beads. In the hexatic phase, these correlations decay with distance following a power law. In the liquid phase, they decay exponentially! For details: https://grasp-lab.org/2013/06/18/hexatic/ pic.twitter.com/h4smXH3eW4

We should all be crying now, but if you can't do that at least laugh. The whole cartoon is here: https://www.theguardian.com/commentisfree/2018/nov/02/why-didnt-humanity-save-the-planet-perhaps-they-were-busy (continued) pic.twitter.com/GQvMBYACmS

Please read this report - it's important: https://tinyurl.com/WWF-2018 Starting on the 47th page of the pdf, you can see the crash of mammal, bird, reptile, amphibian and fish populations worldwide. (continued) pic.twitter.com/RtMLOoZHo2

On land, the worst declines are occurring in the "Neotropical" region: Central and South America. Mammals, birds, reptiles and amphibian population have dropped by about 89% since 1970. (continued) pic.twitter.com/yFwBQkZWSv

There's no excuse for inaction. Even if a disaster of some sort is certain, there are different degrees of disaster - and it’s our responsibility to minimize the disaster. Glib optimism, denial and despair are all just different ways to avoid facing up to the situation.

The four elements of Antarctica: air, ice, water and darkness. Canadian photographer David Burdeny captured this in the Weddell Sea off, which has been called "the most treacherous and dismal region on earth." The Ross Sea is apparently more peaceful. pic.twitter.com/iKtvgQIxDp

Slowly lower yourself toward the event horizon of black hole. As you do, look up. Your view of the outside universe will shrink to a point - and become brighter and brighter, tending to infinite brightness! Andrew Hamilton made this: http://jila.colorado.edu/~ajsh/insidebh/schw.html pic.twitter.com/9mGPnnvNq9

These effects don't happen if you simply let yourself fall in to the black hole. If you do that, your view of the outside world will not shrink to a point, and the light you see will not be intensified by blueshifting - because you'll be falling along with it!

Okay, besides going to jury duty tomorrow I also have to do this. Civic duty. https://twitter.com/Public_Citizen/status/1060297213052641280

The deadly needle of Kakeya! "Kakeya's needle problem" ask what's the least area you need to turn around an infinitely thin needle. As @gregeganSF's great animation shows, not much! In fact, you can get the area as close to zero as you want! More: https://en.wikipedia.org/wiki/Kakeya_set pic.twitter.com/deRo4HVACk

Good news: first African-American woman to lead the House Committee on Science, Space, and Technology. First Democrat to lead it since 2011. First who isn't a climate science denier since 2011. First with training in SCIENCE since the 1990s!! https://www.motherjones.com/environment/2018/11/eddie-bernice-johnson-science-returns-to-the-house/

This number has 317 digits, all ones. It's prime. 317 is also prime! That's not a coincidence. A number whose digits are all 1 can only be prime if the number of digits is prime! This works in any base, not just base ten. Can you see the quick proof? (continued) pic.twitter.com/TVtHfvA0ma

A prime whose decimal digits are all ones is called a "repunit prime". The largest known repunit prime has 1031 digits. Mathematicians believe there are infinitely many repunit primes, but nobody can prove it yet. Why does this matter? (continued) pic.twitter.com/naSOQNunpq

The density of primes decreases slowly, like 1/log(N). So if numbers whose digits have "no good reason not to be prime", there should be infinitely many of them. This idea gives a probabilistic argument that there should be infinitely many repunit primes. (continued)

But what does probability really mean when it comes to prime numbers? God didn't choose them by rolling dice! This is why silly-sounding puzzles about primes can actually be important: they challenge our understanding of randomness and determinism. (continued)

There might be infinitely many true facts about primes that are true just because it's overwhelmingly "probable" that they're true... but not for any reason we can convert into a proof. However, even this has not yet been proved. Clouds of mystery surround us. pic.twitter.com/P57XEb5AGn

There might be infinitely many true facts about primes that are true just because it's overwhelmingly "probable" that they're true... but not for any reason we can convert into a proof. However, even this has not yet been proved. Clouds of mystery surround us. pic.twitter.com/HPHCzpAYwH

Philip Gibbs, a master of plane geometry, finally gets his due! Read about his progress on Lebesgue's old, hard problem of finding the least-area shape that covers all shapes of diameter one. Pretty sure the bit about a "drowning insect" wasn't my idea. https://www.quantamagazine.org/amateur-mathematician-finds-smallest-universal-cover-20181115/

Yay! Starting May 20th in 2019, the kilogram will be defined using Planck's constant, the speed of light, and the frequency of a very specific kind of light emitted by a cesium atom! Until now it was defined as the mass of this chunk of metal in Paris. (continued) pic.twitter.com/T8ibiMsQNQ

It was really bad. By definition the official kilogram in Paris has mass exactly 1 kilogram, yet we *know* its mass drifted ~50 micrograms away from other replica kilograms since 1900. Are the others gaining mass, or is it losing mass, or both? AARGH! (continued) pic.twitter.com/TDW89khqlx

The prototype kilogram is stored in a vault. But it slowly gains mass from the air! So people have to take it out and *clean it*, firmly rubbing it with a chamois soaked in ether and ethanol, and then steam cleaning it. 😟 (continued) pic.twitter.com/bI0R21Posy

After the kilogram is cleaned, it "exhibits a short-term instability of about 30 micrograms over a period of about a month". Nobody is sure why. Barbaric! Thankfully, all SI units will soon be defined in terms of physical laws, not chunks of stuff that need cleaning. 👍 pic.twitter.com/sZ36CBwP8D

You can take 8 perfectly rigid regular tetrahedra and connect them along their edges to form a ring that you can turn inside-out! It's called a 'kaleidocycle', and you can make one with any even number of tetrahedra, as long as you have at least 8. (continued) pic.twitter.com/DEoJyOTujP

A ring of 6 tetrahedra forms a 'collidocycle' - they hit each other, as shown here by @gregeganSF. The one with 8 is from here, along with others: http://intothecontinuum.tumblr.com/post/50873970770/an-even-number-of-at-least-8-regular-tetrahedra But is there any deep math nearby? Yes! The Rigidity Theorem and Bellows Theorem! (continued) pic.twitter.com/AJCuHzYstF

The Rigidity Theorem says if the faces of a *convex* polyhedron are made of a rigid material and the polyhedron edges are hinges, the polyhedron can't change shape at all: it's rigid. Cauchy claimed to prove this in 1813. Steinitz fixed a mistake much later. (continued) pic.twitter.com/XZkImMASG5

There are nonconvex polyhedra without a hole that aren't rigid! The first was found by Connelly in 1977. It has 18 triangular faces. Later Steffen found the one here, with 14 triangles. Read about it on @gregeganSF's page: https://tinyurl.com/egan-steffen (continued) pic.twitter.com/6fWRvo4ClR

The Bellows Theorem says that even if a polyhedron with rigid faces and hinged edges is flexible, its volume doesn't change as it flexes. This was proved by Connelly, Sabitov, and Walz in 1997. There are still open questions in this area. Fun stuff! https://en.wikipedia.org/wiki/Flexible_polyhedron

I am thankful for the beauty of mathematics and physics, which always go deeper than I expect. For example, Hamilton's equations describe the motion of a particle if you know its energy. But they turn out to look a lot like Maxwell's relations in thermodynamics! (continued) pic.twitter.com/Lc4OXaXC2s

Maxwell's relations connect the temperature, pressure, volume and entropy of a box of gas - or indeed, a box of *anything* in equilibrium. Nobody told me they're just Hamilton's equations with different letters and vertical lines thrown in. (continued) pic.twitter.com/fPqqucgmks

So I decided to see what happens if I wrote Hamilton's equations in the same style as the Maxwell relations. It freaked me out at first. What does it mean to take the partial derivative of q in the t direction while holding p constant? But it's okay. (continued) pic.twitter.com/cdhLuI17wO

I thought about it longer and realized what was going on. You get equations like Hamilton's whenever a system *extremizes something subject to constraints*. A moving particle minimizes action; a box of gas maximizes entropy. Read how it works: https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/

So: whenever you see unexplained patterns in math or physics, write them down in your notebook. Think about them from time to time. Simplify. Clarify. Soon you'll never be bored. And if you get stuck and frustrated, just ask people. True seekers will be happy to help. pic.twitter.com/WfazUKxP6h

The game of "58 holes", or "hounds and jackals", is very ancient. This copy comes from Thebes, Egypt. It was made around 1810 BC - the twelfth dynasty! Two players took turns rolling dice to move their pieces forward. But why 58? That's a strange number! (continued) pic.twitter.com/jATDSQrfCm

The holes come in two groups of 29. Nobody knows the rules for sure! But the Russian game expert Dmitriy Skiryuk argued that the players move their pieces from holes A to 29 and then the large shared hole H, where they exit the board: http://www.ancientgames.org/hounds-and-jackals/ (continued) pic.twitter.com/f8iOKBD88X

If so, each player really has 30 holes! That makes more sense: the number 60 was very important in Egypt and the Middle East. So "58 holes" is a red herring. The game was really widespread: here's one from a pillaged Iron Age tomb in Necropolis B at Tepe Sialk, Iran. (cont) pic.twitter.com/tPI1KffGuU

But this is even cooler! The game was just found 2000 kilometers from the Middle East - chiseled into a rock by Bronze Age herders! I guess nobody can resist a good game. Someone should popularize this one again. https://www.sciencenews.org/article/bronze-age-game-found-chiseled-stone-azerbaijan

"I am not a scientist. I am just an idiot who expects you to listen while I spout my opinions about science." https://twitter.com/AOC/status/1066833125156753408

Wow! Is this some sneaky trick? No, it's an actual object designed by Kokichi Sugihara, an engineer at Meiji University. Can you figure out how it works? Yes you can. Think about it. (continued) pic.twitter.com/S3JC639Gu2

Suppose you're videotaping an object, looking down at an angle of 45 degrees. Think about one pixel of the object's image. Its height on your viewscreen depends on how far *up* that piece of the object actually is. But it also depends on how far *back* it is.

Namely: pixel height = actual object height + actual distance back But a mirror reverses front and back, so mirror image pixel height = actual object height - actual distance back This is only true if you're looking down at 45 degrees and the object is not too close.

So, create an object where actual object height + actual distance back and actual object height - actual distance back give two different curves: one round and one a diamond! Just take formulas for circle and diamond and solve for object height and distance back!

Here are the details. I'm afraid this makes it look harder than it is, because they consider the case where the object is close to the camera. Then the formulas get more complicated! At the end, though, they simplify it down to what I just said: https://divisbyzero.com/2016/07/05/sugiharas-circlesquare-optical-illusion/

Scientists at Harvard plan to drop 100 grams of calcium carbonate into the atmosphere and see how it disperses. A small, harmless experiment... but it's a big deal because it's one of the first experiments to test "solar geoengineering". For more: https://johncarlosbaez.wordpress.com/2018/11/28/stratospheric-controlled-perturbation-experiment/