Diary — November 2017

November 11, 2017

A big machine to weigh a tiny particle

This is a huge vacuum chamber, bigger than a blue whale, being carried through the streets of a German town. By now it's been buried underground as part of the Karlsruhe Tritium Neutrino Experiment, or KATRIN. It aims to measure the mass of a very light particle!

There are 3 kinds of neutrinos, each with their own antiparticle. Amazingly, they're all so light that we don't know how heavy they are. The KATRIN experiment is trying to measure the mass of the electron antineutrino, which is formed whenever a neutron decays into a proton and electron.

Around 2000, another German experiment showed that the mass of this particle is no more than 0.0000043 times the mass of an electron. That's the mass equivalent of 2.2 electron volts, or eV. The new experiment should be able to measure the electron antineutrino's mass if it's more than 0.2 eV. It works the same basic way: let tritium, with one proton and two neutrons in its nucleus, decay to an element with two protons and one neutron. In this process a neutron turns into a proton, releasing an electron and an electron antineutrino. By very carefully measuring all the stuff you can see, you can estimate the mass of the electron antineutrino... even though that particle is almost impossible to see.

There's a chance the experiment will only put a better upper bound on the mass of the electron antineutrino. To see why, take a look at this article:

Wikipedia says that the difference in the squares of the masses between neutrino mass eigenstates 1 and 2 is about 0.000079 eV2, while the difference in the squares of the masses between eigenstates 2 and 3 is about 0.0027 eV2.

None of this says anything about the actual masses. But then, they say:

In 2009, lensing data of a galaxy cluster were analyzed to predict a neutrino mass of about 1.5 eV. This surprisingly high value requires that the three neutrino masses be nearly equal, with neutrino oscillations on the order of milli electron-Volts. In 2016 this was updated to a mass of 1.85 eV.
But then, they say:
In July 2010, the 3-D MegaZ DR7 galaxy survey reported that they had measured a limit of the combined mass of the three neutrino varieties to be less than 0.28 eV. A tighter upper bound yet for this sum of masses, 0.23 eV, was reported in March 2013 by the Planck collaboration, whereas a February 2014 result estimates the sum as 0.320 B1 0.081 eV based on discrepancies between the cosmological consequences implied by Planck's detailed measurements of the cosmic microwave background and predictions arising from observing other phenomena, combined with the assumption that neutrinos are responsible for the observed weaker gravitational lensing than would be expected from massless neutrinos.

So, while the astronomical estimates are quite different from each other, and some of them must be wrong, they seem to point to neutrino masses that are considerably larger than the neutrino mass differences.

Unfortunately, if the sum of all 3 neutrino masses is about 0.3 eV, and their masses are close, each individual mass is about 0.1 eV, which is below the 0.2 eV that the Karlsruhe Tritium Neutrino Experiment can measure.

However, it's gotta be wrong where it says this:

First efforts, made after the second world war, placed an upper limit on its mass at around 500 electron volts (eV). This figure is about 1/500th of the mass of the electron, itself a relatively tiny particle. (Using a unit of energy to describe the mass of an object may seem strange but all subatomic particles are measured in electron volts, which can also be used as a unit of mass because energy and mass are convertible concepts according to Einstein's $$E = mc^2$$.
The problem is that the electron's mass is about 511,000 eV, so 500 eV would be 1/1000th of that. Math.

For more, see the discussion on my G+ post.

November 12, 2017

How to make a zero-calorie doughnut

Take a doughnut. Remove most of it, leaving a thin tube of dough that wraps twice around the original doughnut hole.

Then remove most of what's left, leaving a thinner tube of dough that wraps 4 times around the original hole.

Then remove most of what's left, leaving an even thinner tube of dough that wraps 8 times around the original hole.

Keep repeating this forever... and then stop. Now you have a zero-calorie doughnut!

They'll never become popular at bakeries, but mathematicians love 'em. You can't really make one out of dough, but it's a perfectly fine mathematical set of points in 3-dimensional space. When I said it has no calories, what I really mean is that it has zero volume . Mathematicians call it a solenoid.

The solenoid was first invented by the topologist Vietoris in 1927. But it also showed up as an attractor in a certain dynamical systems studied by the mathematician Smale.

One cool thing about the solenoid is that it's connected, but not 'path-connected'. In other words, given two different points in the solenoid, you can't connect them with a path that stays in the solenoid. At any finite stage of its construction you could... but as you continue the construction, after a while the path you'd need to take typically keeps roughly doubling in length!

Yet another cool thing about the solenoid is that you can give it the structure of an abelian group. For this it's better to use a more abstract construction — so, get ready for some more serious math. Take the circle, which is an abelian group. Mathematicians call it $$S^1$$. There's a function $$f \colon S^1 \to S^1$$ that doubles angles, and wraps the circle around itself twice. Take the set of infinite sequences of points in the circle where each point is $$f$$ of the next point. You can make this set into a group where $$(x_1, x_2, x_3, \dots ) + (y_1, y_2, y_3, \dots ) = (x_1 + y_1, x_2 + y_2, x_3 + y_3, \dots )$$

This is the solenoid!

To make this more precise we need to get topology into the game. The circle is not only an abelian group, it's also a compact topological space — and the group operations in the circle are continuous, so we call it a 'compact abelian group'. And the solenoid, being built from the circle as above, is also a compact abelian group! Mathematicians call it the 'limit' of the sequence $$S^1 \stackrel{f}{\longleftarrow} S^1 \stackrel{f}{\longleftarrow} S^1 \stackrel{f}{\longleftarrow} S^1 \stackrel{f}{\longleftarrow} \cdots$$

where $$f$$, the map vaguely described above, is a homomorphism of compact abelian groups. The idea is simply that a point in the limit is a sequence of points, one in each copy of $$S^1$$, such that each point is $$f$$ of the next.

In fact, the limit of any sequence of compact abelian groups is again a compact abelian group: this is a spinoff of Tychnonoff's theorem, which says that any product of compact topological spaces is again compact. So, we can build lots of compact abelian groups as limits, for example of $$S^1 \stackrel{f}{\longleftarrow} S^1 \stackrel{g}{\longleftarrow} S^1 \stackrel{f}{\longleftarrow} S^1 \stackrel{h}{\longleftarrow} \cdots$$

where each map $$f, g, h$$ etc. could wrap the circle around itself any number of times. All these are called 'solenoids'.

Puzzle. When are two of these solenoids isomorphic as compact abelian groups?

The answer involves prime numbers! For clues, see this:

• Encyclopedia of Mathematics, Solenoid.
and also the comments on my G+ post.

The above animated gif of the solenoid was created by Jim Belk, and it's part of the Wikipedia article:

November 16, 2017

Wada lived on a white island in a red sea. On the island there was a blue lake and a green lake.

Wada led a peaceful life. Some days he would sit by the red sea. Some days he would sit by the blue lake. And some days he would sit by the green lake.

But eventually he became bored of watching just one color of water at a time. So he decided to dig canals.

On the first day, Wada dug a canal from the red sea so that every piece of land was within 1 mile of some red water.

In the next 1/2 day, Wada dug a canal from the blue lake so that every piece of land was within 1/2 mile of some blue water. The picture here shows what his island looked like then.

In the next 1/4 day, Wada dug a canal from the green lake so that every piece of land was within 1/4 mile of some green water. Can you draw it?

Wada continued this way, digging more and more canals. They got thinner and thinner, so there was always plenty of land left.

By the end of the second day, every piece of land touched the red sea, the blue lake and the green lake! Now he could watch all three bodies of water at once.

He had built the famous Lakes of Wada, which people visit even today.

You can create the Lakes of Wada effect by looking at the reflections in three mirrored spheres that touch each other.

You can also create this effect by applying Newton's method to a cubic polynomial with 3 distinct roots in the complex plane, such as $$z^3 - 1$$:

Newton's method is a simple way of solving equations like $$f(z) = 0$$. You start by making a guess for $$z$$, you figure out $$f(z)$$ and its derivative $$f'(z)$$ at your guess, and you use those to figure out where $$f(z)$$ would equal zero if $$f$$ were a linear function. This is your new guess for $$z$$. Then you repeat this. In good situations your guesses will quickly converge to a value of $$z$$ with $$f(z) = 0$$. In worse situations your guesses will hop around in a complicated way. If you take $$f(z) = z^3 - 1$$, there are 3 solutions. If you start near one of those solutions, your guess will converge to that solution. This defines three basins of attraction. But these basins of attraction are not connected, and each touches the other two at every point of its boundary!

I got the picture in my post from here:

The Lakes of Wada were actually discovered by the Japanese mathematicia Takeo Wada, who lived from 1882 to 1944 and worked on analysis and topology at Kyoto University.

November 19, 2017

This is not an animated gif

This is Akiyoshi Kitaoka messing with your brain. He's a professor of psychology at Ritsumeikan University in Kyoto. He's spent a long time collecting and perfecting illusions. You can see them on his website:
where he writes

You can also see them on Twitter, where there is no such warning. And he has a book, The Oxford Compendium of Illusions.

November 29, 2017

Criminally cute! Ocelots are small wild cats that live in Mexico, Central America and South America. There are even a few in the southwestern United States. This one is just a kitten.

Hilary Swarts has been involved in setting up an ocelot reserve in south Texas, the Laguna Atascosa National Wildlife Refuge. Here's some of a story about her:

Survival can be a real cat fight when you get squeezed out of your rightful home. When your food supply dwindles. When you are small and cute and easy to run down. Even though you are standoffish and try to keep to yourself.

In 22 countries, from Uruguay to south Texas, the ocelot (Leopardus pardalis), one of smallest and most secretive of all wild cat species, is facing this sad plight. Its habitat — thorn scrub, coastal marshes, tropical and pine-oak forests — has shrunk alarmingly, swaths destroyed by building and farming and other human activity. With diminished space in which to establish territories, find secure denning sites and and forage for rodents, birds, snakes, lizards and other prey — plus the increased threat of becoming road kill as highway construction boomed in the 20th century — the ocelot has been in the fight of its life.

Back in the 1960s and early '70s, ocelots were nearly loved to death. Laws then did not prohibit taking them for exotic pets or hunting them for their beautiful, dramatically marked fur. Babou, Salvador Dali's frequent sidekick, may have been the most famous of captive ocelots.

In the U.S., as the wild population of these little cats became depleted under development pressures, the fashion industry turned to import, reaching a peak of 140,000 pelts from Central and South American countries in 1970. Toward the end of the century, all these human endeavors had chipped away at the historic U.S. ocelot range — which once stretched from Louisiana to Arizona — cornering the few known remaining individuals in the Lower Rio Grande Valley, where Texas meets the Mexican border and the Gulf of Mexico. Wildlife biologists, scientists, researchers, conservationists and other experts started running the numbers and saw that time was running out. Now, even after several decades of legal protection and some active conservation projects, only 55 or so known individual ocelots remain in the U.S.

There are few rays of sunshine in this grim picture, but one of the brightest landed at Laguna Atascosa National Wildlife Refuge a little over three years ago in the form of wildlife biologist Hilary Swarts '94.

Swarts is a graduate of Pomona College, whose magazine I got this story from. The picture for the ocelot is not quite as grim as this story makes it sound: there are about 40,000 mature ocelots in the world, their population is considered stable, and they're listed as being of Least Concern by the IUCN Red List of Threatened Species. In the US, ocelots are indeed endangered, especially since with the new border walls they're getting cut off from the larger population in Mexico. But maybe we don't need ocelots. Maybe we don't deserve ocelots.

Swarts has a different opinion:

Entering the ecology program at the University of California, Davis, she earned a Ph.D. in ecology with an emphasis on conservation. Then, shrugging off that "never working for the government" notion, she took a job with the U.S. Fish and Wildlife Service, working on regulatory projects involving endangered species. "Regulatory work is so important," she emphasizes. But after a while, the day-to-day responsibilities of what she terms "desk biology" began to wear. "It's soul-crushing work," she explains. "You know exactly what each day, a month ahead, will be."

So, when a job opening in the wilds of south Texas popped up in her email for a wildlife biologist charged with leading the hands-on effort to save the ocelot in the U.S., she leapt at the challenge.

The Laguna Atacosa National Wildlife Refuge is a flat, sunbaked remnant of coastal prairie mixed with thorn bush, bordering on a vast hypersaline lagoon across from South Padre Island. Its dense thicket of low scrub is home to — at last count — 15 of the remaining ocelots still living in the U.S., and for Swarts, it's where the fight to save them from extinction is being waged.

Meeting with her here can feel like a bracing seminar in All Things Ocelot. For starters, she'll whip her refuge pickup into her driveway (on Ocelot Road, of course) and say, pointing at the license plate on her 2000 Buick LeSabre, "Look!" The plate says "OCELOT" (of course), and the vanity fee collected by the State of Texas goes to Friends of Laguna Atascosa for outreach programs.

More important, it quickly becomes clear that she's a walking compendium of information about the species she's working to rescue. "We think that these Texas ocelots may have developed great fidelity to thick underbrush because of pursuit by hunters back in the 1960s," she explains. More facts come tumbling out: Two-thirds of births are single, after a gestation of 79 to 82 days. Kittens stay with their mothers, to learn survival and hunting skills, for up to two years. "Although," she adds, "I'm beginning to think it may be closer to a year and a half, if the teaching goes well and there is a reliable prey base. And the past two winters have been super wet, so there's been prey out the wazoo."

Working with ocelots, because they stay so well hidden, is different from her previous fieldwork, when she could watch the animals she was studying in their own environment (such as following gorillas around as they nosed about on their daily routines, which she describes as "total soap opera"). In fact, the only time Swarts and her small staff of interns actually see ocelots in the flesh is during trapping season, from October to May, when the little cats are lured by caged pigeons posing as an easy meal, then sedated long enough for blood and genetic samples to be taken. After a quick exam and insertion of a microchip, they are photographed, fitted with a GPS collar, given reversal drugs and released.

"With the ocelots, I'm essentially doing detective work," she explains. Across the refuge, there are more than 50 cameras tucked into the thorn scrub, monitoring animal activity night and day. Using cameras and GPS collars may not be as immediately satisfying as shadowing gorillas, but it's the only way she can keep tabs on the elusive little creatures she's trying to save.

For instance, last year, on March 25, 2016, a heavily pregnant female was captured for routine data collection and then released. On the following two days, GPS signals from her collar indicated that she was staying put, likely in a den. After a few weeks, GPS showed more activity — she was almost certainly leaving the den for water, repeat behavior that is usual for a lactating female. "On April 15, when we knew she was away and couldn't detect us, we found the little kitten, tucked under some Spartina. A male, healthy, weighing less than a pound, with his eyes just opened." Swarts, who took hair samples, DNA swabs and his baby picture, was ecstatic to document and report this first confirmed ocelot den at the refuge in 20 years.

"From my perspective they are doing their job — reproducing," she says. "And ecologically we are in great shape." However, she has grave concerns that the confirmed refuge population of 15, including kittens, may be approaching capacity. Home range for a female varies from one to nine square miles, depending on the availability of water and prey. For a male, figure four to 25 square miles.

That brings us to exhibit one for the three top threats to survival of the species — habitat loss. Hemmed in by agriculture, highways and industry, the refuge itself can't be greatly expanded. The other Texas ocelots, about 40 individuals, live on limited private lands in neighboring Willacy County, with no safe passage connecting the populations.

And that leads directly to the second threat — vehicular mortality, which stands at an astounding 40 percent. Swarts cites the ugly statistics that piled up between June 2015 and April 2016, when seven ocelots, including six males, were killed by vehicles on roads adjacent to fragile ocelot territory.

Which brings us to the third item on Swarts' list of top threats to the ocelot's long-term survival: in-breeding, which occurs when populations are so isolated that no new genes can get into the mix. Even before her arrival in Texas, efforts to freshen the gene pool by bringing in a female ocelot from Tamaulipas, Mexico, had started and stopped several times, partly due to cartel violence. Still, she remains optimistic that, with research and negotiation, a female from Mexico will eventually be allowed to cross the border.

Progress is agonizingly slow — as Swarts stoically puts it, "Conservation is often two steps forward and one step back." However, she has begun to see encouraging signs. The refuge has cranked up an aggressive habitat restoration project — planting ocelot corridors, extensions of the habitat that ocelots are known to use, with the low-growing, bushy native species they prefer. As a precaution against vehicular mortality, the refuge has closed some of its roads and plans to relocate its entrance. Most heartening, the Texas Department of Transportation is installing 12 new underpasses specifically designed for ocelots at known hot spots on two highways where there have been multiple incidents of road kill. "And now it seems likely they will put wildlife crossings into new road design from the start," she adds. "This is a sea change — and for this state agency to come around bodes so well for the state and its environmental future."

One can wonder if it's really worth such a fuss for a few dozen cats, especially when there are many more outside the US. I like to think of it this way: the ocelot is the charismatic representative of a certain ecosystem. The ecosystem, bristling with complex information evolved over millions of years, is valuable in ways we're just beginning to understand. Saving the ocelot is an easily understood stand-in for saving the ecosystem.

We're like kids in a grand library, kids who can barely read. We notice that some of the books have pretty pictures. The ocelot is one of those pretty pictures.

The article I quoted was written by Shakespeare. Margaret Shakespeare, that is. Check out the rest, and the pictures, here:

• Margaret Shakespeare, Ocelot country, Pomona College Magazine, April 10, 2017.
Watch someone play with a young semi-domesticated ocelot in the jungle of Costa Rica: