For my May 2015 diary, go here.

Diary — June 2015

John Baez

June 1, 2015

This is a bird's nest fungus — a kind of mushroom that looks like a bird's nest full of eggs. More precisely, it's Cyathus novaezelandiae, photographed by Steve Axford.

Why does it look like this? It's a trick for spreading spores. When rain hits the cup-shaped mushroom, spores shoot out!

Like many fungi that grow on rotten logs, the bird's nest fungus has a complex life cycle. There's the stage you see here, where it reproduces asexually via spores. But there's also a sexual stage!

Spores germinate and grow into branching filaments called hyphae, pushing out like roots into the rotting wood. As these filaments grow, they form a network called a mycelium. These come in several different sexes, or mating compatibility groups. When hyphae of different mating compatibility groups meet each other, they fuse and form a new mycelium that combines the genes of both. After a while, these new mycelia may enter the stage where they grow into the mushrooms you see here. Then they reproduce asexually using spores!

It's complicated, and I don't fully understand it. You can read more here:

You can see more of Steve Axford's photos here:

Thanks to Mike Stay for pointing this out! For an interesting article inspired by this one, go here:

June 3, 2015

When you hear 'carnivorous fungus', I know what you're thinking: GIANT MAN-EATING MUSHROOMS!

At least that's what went through my mind when I was looking at the Wikipedia page on carnivorous plants and saw there was also a page on carnivorous fungi.

In fact, these fungi are tiny, and they eat small things like nematodes. The wormy thing here is a nematode, and it's being caught by the little tendrils called 'hyphae' of a fungus:

Carnivorous fungi were first discovered by the Austrian botanist Whilhelm Zopf in 1888. He was looking at a fungus whose hyphae have little loops in them. Zopf observed nematodes being caught by these loops — caught by the tail, or caught by the head. When this happened, the nematode would struggle violently for half an hour. Then it would become quieter. In a couple of hours, it would die. And then, hyphae from the loop would penetrate and invade its body.

Aren't you glad that you read this post? The world is full of wonderful and horrible things, and this is one.

Somehow we tend to sympathize with the creature that's more like us. When I see a jaguar fighting a crocodile, I want the jaguar to win. A worm eating fungus doesn't seem so bad... but fungus eating a worm seems disgusting, at least to me. This is not a rational judgement of mine: it's just an emotion that sweeps over me.

A nematode is not related to an earthworm: it's a much more primitive sort of organism. Nematodes are serious pests &mdsh; they kill lots of crops. My university, U.C. Riverside, even has a Department of Nematology, where people study how to fight nematodes! One way to fight them is with a carnivorous fungus. So maybe carnivorous fungi are not so bad.

This picture shows a nematode captured by the predatory fungus Arthrobotrys anchonia. Note that the loop around the body of the victim has not yet started to tighten and squeeze it. This picture was taken with a scanning electron micrograph by N. Allin and George L. Barron. I got it here:

According to this page:

Fungi can capture nematodes in a variety of ways but the most sophisticated and perhaps the most dramatic is called the constricting ring. An erect branch from a hypha curves round and fuses with itself to form a three-celled ring about 20-30 microns in diameter. When a nematode "swims" into a ring it triggers a response in the fungus and the three cells expand rapidly inwards with such power that they constrict the body of the nematode victim and hold it securely with no chance to escape. It takes only 1/10th of a second for the ring cells to inflate to their maximum size.

June 5, 2015

In math you get to make up the rules of the game... but then you have to follow them with utmost precision. You can change the rules... but then you're playing a different game. You can play any game you want... but some games are more worthwhile than others.

If you play one of these games long enough, it doesn't feel like a game — it feels like "reality", especially if it matches up to the real world in some way. But that's how games are.

Unfortunately, most kids learn math by being taught the rules for a just a few games — and the teacher acts like the rules are "true". Where did the rules come from? That's not explained. The students are never encouraged to make up their own rules.

In fact, mathematicians spend a lot of time making up new rules. For example, my grad student Alissa Crans made up a thing called a shelf. It wasn't completely new: it was a lot like something mathematicians already studied, called a 'rack', but simpler — hence the name 'shelf'. (Mathematician need lots of names for things, so we sometimes run out of serious-sounding names and use silly names.)

What's a shelf?

It's a set where you can multiply two elements a and b and get a new element a · b. That's not new... but this multiplication obeys a funny rule:

a · (b · c) = (a · b) · (a · c)

That should remind you of this rule:

a · (b + c) = (a · b) + (a · c)

But in a shelf, we don't have addition, just multiplication... and the only rule it obeys is

a · (b · c) = (a · b) · (a · c)

There turn out to be lots of interesting examples, which come from knot theory, and group theory. I could talk about this stuff for hours. But never mind! A couple days ago I learned something surprising. Suppose you have a unital shelf, meaning one that has an element called 1 that obeys these rules:

a · 1 = a
1 · a = a

Then multiplication has to be associative! In other words, it obeys this familiar rule:

a · (b · c) = (a · b) · c

The proof is in the picture.

A guy who calls himself "Sam C" put this proof on a blog of mine. I was shocked when I saw it.

Why? First, I've studied shelves quite a lot, and they're hardly ever associative. I thought I understood this game, and many related games — about things called 'racks' and 'quandles' and 'involutory quandles' and so on. But adding this particular extra rule changed the game a lot.

Second, it's a very sneaky proof — I have no idea how Sam C came up with it.

Luckily, a mathematician named Andrew Hubery showed me how to break the proof down into smaller, more digestible pieces. And now I think I understand this game quite well. It's not a hugely important game, as far as I can tell, but it's cute.

It turns out that these gadgets — shelves with an element 1 obeying a · 1 = 1 · a = a — are the same as something the famous category theorist William Lawvere had invented under the name of graphic monoids. The rules for a monoid are that we have a set with a way to multiply elements and an element 1, obeying these familiar rules:

1 · a = 1 · a = a

a · (b · c) = (a · b) · c

Monoids are incredibly important because they show up all over. But a graphic monoid also obeys one extra rule:

a · (b · a) = a · b

This is a weird rule... but graphic monoids show up when you're studying bunches of dots connected by edges, which mathematicians call graphs... so it's not a silly rule: this game helps us understand the world.

Puzzle 1: take the rules of a graphic monoid and use them to derive the rules of a unital shelf.

Puzzle 2: take the rules of a unital shelf and use them to derive the rules of a graphic monoid.

So, they're really the same thing.

By the way, most math is a lot more involved than this. Usually we take rules we already like a lot, and keep developing the consequences further and further, and introducing new concepts, until we build enormous castles — which in the best cases help us understand the universe in amazing new ways. But this particular game is more like building a tiny dollhouse. At least so far. That's why it feels more like a "game", less like "serious work".

For answers to the puzzles see Colin Gopaul's comment on my G+ post.

In math the rules of a game are called axioms. What's the longest axiom that people have ever actually thought about?

I'm not sure, but I have some candidates.

A lattice is a set with two operations called ∨ and ∧ obeying the 6 equations listed above. But a while back people wondered: can you give an equivalent definition of a lattice using just one equation? It's a pointless puzzle, as far as I can tell, but some people enjoy such challenges.

And in 1970 someone solved it: yes, you can! But the equation they found was incredibly long.

Before I go into details, I should say a bit about lattices. The concept of a lattice is far from pointless — there are lattices all over the place!

For example, suppose you take integers, or real numbers. Let x ∨ y be the maximum of x and y: the bigger one. Let x ∧ y be the minimum of x and y: the smaller one. Then it's easy to check that the 6 axioms listed here hold.

Or, suppose you take statements. Let p ∨ q be the statement "p or q", and let p ∧ q be the statement "p and q". Then the 6 axioms here hold!

For example, consider the axiom p ∧ (p ∨ q) = p. If you say "it's raining, and it's also raining or snowing", that means the same thing as "it's raining" — which is why people don't usually say this.

The two examples I just gave obey other axioms, too. They're both distributive lattices, meaning they obey this rule:

p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r)

and the rule with ∧ and ∨ switched:

p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r)

But nondistributive lattices are also important. For example, in quantum logic, "or" and "and" don't obey these distributive laws!

Anyway, back to the main story. In 1970, Ralph McKenzie proved that you can write down a single equation that is equivalent to the 6 lattice axioms. But it was an equation containing 34 variables and roughly 300,000 symbols! It was too long for him to actually bother writing it down. Instead, he proved that you could, if you wanted to.

Later this work was improved. In 1977, Ranganathan Padmanabhan found an equation in 7 variables with 243 symbols that did the job. In 1996 he teamed up with William McCune and found an equation with the same number of variables and only 79 symbols that defined lattices. And so on...

The best result I know is by McCune, Padmanbhan and Robert Veroff. In 2003 they discovered that this equation does the job:

(((y ∨ x) ∧ x) ∨ (((z ∧ (x ∨ x)) ∨ (u ∧ x)) ∧ v)) ∧ (w ∨ ((s ∨ x) ∧ (x ∨ t))) = x

They also found another equation, equally long, that also works.

Puzzle: what's the easiest way to get another equation, equally long, that also defines lattices?

That is not the one they found — that would be too easy!

How did they find these equations? They checked about a half a trillion possible axioms using a computer, and ruled out all but 100,000 candidates by showing that certain non-lattices obey those axioms. Then they used a computer program called OTTER to go through the remaining candidates and search for proofs that they are equivalent to the usual axioms of a lattice.

Not all these proof searches ended in success or failure... some took too long. So, there could still exist a single equation, shorter than the ones they found, that defines the concept of lattice. Here is their paper:

By the way: when I said "it's a pointless puzzle, as far as I can tell", that's not supposed to be an insult, although I suppose it sounds like one. I simply mean that I don't see how to connect this puzzle — "is there a single equation that does the job?" — to themes in mathematics that I consider important. It's always possible to learn more and change ones mind about these things.

The puzzle becomes a bit more interesting when you learn that you can't find a single equation that defines distributive lattices: you need 2. And it's even more interesting when you learn that among "varieties of lattices", none can be defined with just a single equation except plain old lattices!

By contrast, "varieties of semigroups where every element is idempotent" can always be defined using just a single equation. This was rather shocking to me.

However, I still don't see any point to reducing the number of equations to the bare minimum! In practice, it's better to have a larger number of comprehensible axioms rather than a single complicated one. So, this whole subject feels like a "sport" to me: a game of "can you do it?"

Anyway: is Ralph McKenzie's 300,000-symbol single-equation axiom for lattices the longest axiom people have thought about?

No! People considered even longer single-equation axioms for Boolean algebras!

A Boolean algebra is a distributive lattice with an operation ¬ and elements ⊤ and ⊥, obeying the extra axioms

p ∨ ⊤ = p      p ∧ ⊥ = p

p ∧ ¬p = ⊥      p ∨ ¬p = ⊤

If you think of ∧ as and, ∨ as or, ¬ as not, ⊤ as true and ⊥ as false", these axioms should make sense.

You don't really need to include the symbols ⊤ and ⊥ when defining Boolean algebras, since you can define them using the last two equations above. And you can also leave out ∧, defining p ∧ q to be ¬(¬p ∨ ¬q).

In 1973, Padmanabhan and Quackenbush found a way to define Boolean algebras using just a single axiom involving ∨ and ¬. However, it seems that using their method would give an axiom of "enormous length" — maybe tens of millions of symbols long!

In 2000, McCune and some coauthors found a much shorter axiom that does the job:

¬(¬(¬(x ∨ y) ∨ z) ∨ ¬(x ∨ ¬(¬z ∨ ¬(z ∨ u)))) = z

and their paper is where I got my information about the single axiom of "enormous length":

For more discussion, and an answer to the puzzle, see the comments to my post on G+.

June 11, 2015

The New Horizons spacecraft took 9 years to reach Pluto. But on July 14th, it will blast by Pluto in just one hour. It can't slow down!

In fact, it's the fastest human-made object ever to be launched from Earth. When it took off from Cape Canaveral in January 2006, it was moving faster than escape velocity, not just for the Earth, but for the Solar System! It was moving at 58,000 kilometers per hour.

When it passed Jupiter it got pulled by that huge planet's gravity and fired out at 83,000 kilometers per hour. As it climbed up out of the Solar System it slowed down. But when it reaches Pluto, it will still be going almost 50,000 kilometers per hour.

That's fast enough that even a speck of dust could be fatal. Luckily, Pluto doesn't seem to have rings.

It will punch through the plane that Pluto's moons orbit, and collect so much data that it will take months for it all to be sent back to Earth in a slow trickle.

And as it goes behind Pluto, it will see a carefully timed radio signal sent from the Deep Space Network here on Earth: 3 deep-space communication facilities located in California, Spain and Australia.

This signal has to be timed right, since it takes about 4 hours for radio waves — or any other form of light — to reach Pluto. The signal will be blocked when Pluto gets in the way, and the New Horizons spacecraft can use this to learn more about Pluto's exact diameter, and more.

Then: out to the Kuiper belt, where the cubewanos, plutinos and twotinos live...

Here are some more details:

On July 14, 2015 at 11:49:57 UTC, New Horizons will make its closest approach to Pluto. It will have a relative velocity of 13.78 km/s (49,600 kilometers per hour), and it will come within 12,500 kilometers from the planet's surface.

At 12:03:50, it will make its closest approach to Pluto's largest moon, Charon.

At 12:51:25, Pluto will occult the Sun — that is, come between the Sun and the New Horizons spacecraft.

At 12:52:27, Pluto will occult the Earth. This is only important because it means the radio signal sent from the Deep Space Network will be blocked.

Starting 3.2 days before the closest approach, New Horizons will map Pluto and Charon to 40 kilometer resolution. This is enough time to image all sides of both bodies. Coverage will repeat twice per day, to search for changes due to snows or cryovolcanism. Still, due to Pluto's tilt, a portion of the northern hemisphere will be in shadow at all times. The Long Range Reconnaissance Imager (LORRI) should be able to obtain select images with resolution as high as 50 meters/pixel, and the Multispectral Visible Imaging Camera (MVIC) should get 4-color global dayside maps at 1.6 kilometer resolution. LORRI and MVIC will attempt to overlap their respective coverage areas to form stereo pairs.

The Linear Etalon Imaging Spectral Array (LEISA) will try to get near-infrared maps at 7 kilometers per pixel globally and 0.6 km/pixel for selected areas. Meanwhile, the ultraviolet spectrometer Alice will study the atmosphere, both by emissions of atmospheric molecules (airglow), and by dimming of background stars as they pass behind Pluto.

Other instruments will will sample the high atmosphere, measure its effects on the solar wind, and search for dust — possible signs of invisible rings of Pluto. The communications dish will detect the disappearance and reappearance of the radio signal from the Deep Space Network, measuring Pluto's diameter and atmospheric density and composition.

The first highly compressed images will be transmitted within days. Uncompressed images will take as long as nine months to transmit, depending on how much traffic the Deep Space Network is experiencing.

Most of this last information is from:

June 12, 2015

The Tale of Genji is a wonderful early Japanese novel written by the noblewoman Murasaki Shikibu sometime between 1008 and 1021 AD. Read it, and be transported to a very different world!

It has 54 chapters. Here you see the 54 Genji-mon: the traditional symbols for these chapters. Most of them follow a systematic mathematical pattern, but the ones in color break this pattern.

Here are some puzzles. It's very easy to look up the answers using your favorite search engine, but it's more fun to solve these just by thinking.

Puzzle 1: How is the green Genji-mon different from all the rest?

Puzzle 2: How are the red Genji-mon similar to each other?

Puzzle 3: How are the red Genji-mon different from all the rest?

Puzzle 4: If The Tale of Genji had just 52 chapters, the Genji-mon could be perfectly systematic, without the weirdness of the colored ones. What would the pattern be then?

Puzzle 5: What fact about the number 52 is at work here?

(Hint: it has nothing to do with there being 52 weeks in a year!)

For answers to the puzzles, see my G+ post.

By the way, only after posting this did I remember that it was my birthday and I was 54 years old. Freudian slip? Coincidence? I'd been meaning to post about this for a while.

June 13, 2015


The true scale of the problem is just becoming apparent, but it seem that all data on every computer in the world has been copied to some unknown location.

It's rapidly becoming clear that last week's revelations are just the tip of the iceberg. It seems all US federal government computers show signs of data breaches, with strong evidence that all files have been copied. The same is true of at least 34 US states. The UK, France, Germany, Italy, Switzerland, Japan and India are reporting similar problems, as are a vast number of corporations, universities and individuals. In particular, it seems that all servers in the Google, Facebook, Amazon, and Microsoft data centers have been hacked.

It's unclear who has the storage capacity to hold all this data. Some suspect the Chinese or Russia, but according to an unnamed source at the US State Department these countries too are victims of the massive hack. "Furthermore," the source stated, "the fact that all the many petabytes of data from the particle accelerator at CERN have been copied seems to rule out traditional espionage or criminal activity as an explanation."

Rumors of all kinds are circulating on the internet. Some say it could be the initial phase of an extraterrestrial invasion, or perhaps merely an attempt to learn about our culture, or — in one of the more fanciful theories — an attempt to replicate it.

Another theory is that some form of artificial intelligence has developed the ability to hack into most computers, or that the internet itself has somehow become intelligent,

Perhaps the strangest rumor is that the biosphere itself is preparing to take revenge on human civilization, or perhaps make a "backup" in case of collapse. A recent paper in PLOS Biology estimates the total information content in the biosphere at roughly 5 × 1031 megabases, with a total processing speed exceeding 1024 nucleotide operations per second. The data in all human computers is still tiny by comparison. However, it is unclear how biological organisms could have hacked into human computers, and what the biosphere might do with this data.

According to one of the paper's authors, Hanna Landenmark, "Claims that this is some sort of 'revenge of Gaia' seem absurdly anthromorphic to me. If anything, it could be just the next phase of evolution."

June 15, 2015

There are 52 weeks in a year and 52 cards in a deck. Coincidence? Maybe not. It's hard to guess what the people who first designed the deck were thinking.

Puzzle 1: Suppose you add up the values of all the cards in a deck, counting an ace as 1, a two as 2, and so on, and counting a jack as 11, a queen as 12 and a king as 13. What do you get?

Puzzle 2: How many cards are there in a suit? (There are four suits of cards: diamonds, hearts, spades and clubs.)

Puzzle 3: How many weeks are there in a season? (There are four seasons in a year; suppose they all have the same number of weeks.)

Puzzle 4: Multiply the number of days in a week, weeks in a season and seasons in a year to estimate the number of days in a year.

Puzzle 5: Suppose on the first day of Christmas you buy your true love a partridge. On the second day you buy two turtle doves and a partridge. On the third you buy three French hens, two turtle doves and a partridge, and on on up to the twelfth day. By the end, how many gifts have you bought?

Here's another fun thing about the number 52. There are also 52 ways to partition a set with 5 elements — that is, break it up into disjoint nonempty pieces. This probably has nothing to do with weeks in the year or cards in the deck! But it's the start of a more interesting story.

Here's a picture of all 52 ways:

They're divided into groups:

$$ 52 = 1 + 10 + 10 + 15 + 5 + 10 + 1 $$

If this chart reminds you of the chart of "Genji-mon" that I showed you on June 12th, that's no coincidence! The Genji-mon are almost the same as the partitions of a 5-element set. This chart should help you answer all the puzzles I asked.

The math gets more interesting if we ask: how many partitions are there for a set with n elements?

For a zero-element set there's 1. (That's a bit confusing, I admit.) For a one-element set there's 1. For a two-element set there's 2. And so on. The numbers go like this: $$ 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, \dots $$ They're called Bell numbers.

Say you call the \(n\)th Bell number \(B(n)\). Then we have a nice formula $$ \sum_{n = 0}^\infty \frac{ B(n) x^n }{ n!} = e^{e^x - 1} $$ This is a nice way to compress all the information in the Bell numbers down to a simple function. But it's not a very efficient way to compute the Bell numbers. For that, it's better to use the 'Bell triangle':

For more on all these things, try:

June 24, 2015

According to a new simulation, the population of Europe dropped from 330 thousand to just 130 thousand during the last glacial cycle.

These pictures show the population density at various times, starting 27,000 years ago — that's why it says "27 ky", meaning "27 kiloyears".

As it got colder, the population dropped, reaching its minimum 23,000 years ago. Things started warming up around then, and the population soared to 410 thousand near the end of the ice age, around 13,000 years ago.

You can see the coast of Spain, Italy and Greece continued to have 23 to 20 people per hundred square kilometers. But the population got pushed out of northern Europe, and even dropped in places like central Spain. The black dots are archaeological sites where we know there were people.

By comparison, there are now roughly 25,000 people per hundred square kilometers in England or Germany, though just half as many in France. So, by modern standards, Europe was empty back in those hunter-gatherer days. Even today the cold keeps people away: there are just 2,000 people per hundred square kilometers in Sweden.

If you're having trouble seeing the British isles in these pictures, that's because they weren't islands back then! They were connected to continental Europe.

Of course these simulations are insanely hard to do, so I wouldn't trust them too much. But it's still cool to think about.

The paper is not free, but the "supporting information" is, and that has a lot of good stuff:

Abstract: The severe cooling and the expansion of the ice sheets during the Last Glacial Maximum (LGM), 27,000-19,000 y ago (27-19 ky ago) had a major impact on plant and animal populations, including humans. Changes in human population size and range have affected our genetic evolution, and recent modeling efforts have reaffirmed the importance of population dynamics in cultural and linguistic evolution, as well. However, in the absence of historical records, estimating past population levels has remained difficult. Here we show that it is possible to model spatially explicit human population dynamics from the pre-LGM at 30 ky ago through the LGM to the Late Glacial in Europe by using climate envelope modeling tools and modern ethnographic datasets to construct a population calibration model. The simulated range and size of the human population correspond significantly with spatiotemporal patterns in the archaeological data, suggesting that climate was a major driver of population dynamics 30-13 ky ago. The simulated population size declined from about 330,000 people at 30 ky ago to a minimum of 130,000 people at 23 ky ago. The Late Glacial population growth was fastest during Greenland interstadial 1, and by 13 ky ago, there were almost 410,000 people in Europe. Even during the coldest part of the LGM, the climatically suitable area for human habitation remained unfragmented and covered 36% of Europe.

June 25, 2015

This ivory portrait head is at least 25,000 years old! It was found in Dolnm Vestonice in the Czech Republic, and it's a product of the Gravettian culture.

The Gravettian is a phase of European culture that lasted from 30,000 to 22,000 years ago. Since this was a very cold phase of the last glacial period, and game was plentiful, the Gravettians ate a lot of meat. They were better at hunting than previous cultures. They learned to take advantage of animal migration patterns, and they used small pointed blades to hunt bison, horse, reindeer and mammoth. They also used nets to hunt small game. Stone arrowheads were only developed by the later Solutrean culture, which lasted from 22,000 to 17,000 years ago.

Gravettian art includes a lot of round female 'Venus' figures, but this seems to be a more realistic portrait of a woman.

This piece is usually kept at the Brno Museum, but it was part of the show 'Ice Age Art: arrival of the modern mind' at the British Museum in London in 2013.

For more on the Gravettian and other periods of European prehistory, read my August 30, 2009 diary entry, which starts out being about the domestication of wolves.

June 26, 2015


How can you change an electrical circuit made out of resistors without changing what it does? 5 ways are shown here:

  1. You can remove a loop of wire with a resistor on it. It doesn't do anything.
  2. You can remove a wire with a resistor on it if one end is unattached. Again, it doesn't do anything.
  3. You can take two resistors in series — one after the other — and replace them with a single resistor. But this new resistor must have a resistance that's the sum of the old two.
  4. You can take two resistors in parallel and replace them with a single resistor. But this resistor must have a conductivity that's the sum of the old two. Conductivity is the reciprocal of resistance.
  5. Finally, the really cool part: the Y-Δ transform. You can replace a Y made of 3 resistors by a triangle of resistors. But their resistances must be related by the equations shown here.

For circuits drawn on the plane, these are all the rules you need! There's a nice paper on this by three French dudes: Yves Colin de Verdihre, Isidoro Gitler and Dirk Vertigan.

Today I'm going to Warsaw to a workshop on Higher-Dimensional Rewriting. Electrical circuits give a nice example, so I'll talk about them. I'm also giving a talk on control theory — a related branch of engineering.

You can see my talk slides, and much more, here:

I'll be staying in downtown Warsaw in the Polonia Palace Hotel.

For my July 2015 diary, go here.

© 2015 John Baez