In the MonCat column, is the categorified version of the tensor product of monoids.

For any set of jobs that must be done, there exists a fun choice function defined on

This axiom asserts that one can always find the fun in any job that must be done; a theorem of Poppins deduces from this that all such jobs are games.

Better than Feynman’s graffiti,
Better than Calabi-Yau,
Better than Pauli’s neutrino,
Better than mu and than tau,

Better than Curie, Poincaré,
And Niels and Albert at Solvay
Better than anything except being in love.

Better than blade runners dreaming,
Better than spice that must flow,
Better than River and reavers,
Better than lightsaber glow,

Better than “Dammit, I’m a doc!”
Or simply “fascinating” Spock,
Better than anything except being in love.

[Edit] See also these better versions.

This image uses a logarithmic spiral to zoom through a factor of 2048 after one turn. There’s a little bit of misalignment between the largest scale and the smallest one, but I think it still turned out pretty well (click for a larger view):

Lovers and thinkers, into the earth with you.
Be one with the dull, the indiscriminate dust.
A fragment of what you felt, of what you knew,
A formula, a phrase remains – but the best is lost.

The answers quick and keen, the honest look, the laughter, the love -
They are gone. They have gone to feed the roses. Elegant and curled
Is the blossom. Fragrant is the blossom. I know. But I do not approve.
More precious was the light in your eyes than all the roses in the world.

Down, down, down into the darkness of the grave
Gently they go, the beautiful, the tender, the kind:
Quietly they go, the intelligent, the witty, the brave.
I know. But I do not approve. And I am not resigned.

- Edna St Vincent Millay

The “cubic” tilings (where eight polyhedra meet around a vertex like cubes do in Euclidean space) are amenable to taking cross-sections; this tiling of hyperbolic space with dodecahedra

has a cross section with a tiling of the hyperbolic plane with pentagons:

Say you’re me and you just watched Vi Hart’s video on infinity elephants and you totally missed the joke about Mr. Tusks even though you read Dinosaur Comics all the time but you liked the bit about Apollonian gaskets, which don’t blow out in Battle Mountain like the one in your car did but rather on the way to the L1 point and then you need Richard Feynman to tell you why. You thought she was going to draw the tiniest camels going through the eye of a needle, but you suppose that would ruin the hyperbole in the parable, so the ellipsis was justified. Anyway, you decide to avoid circular reasoning and doodle rectangles instead, filling them up with squares.

Eventually you wonder which ones you can fill up with finitely many squares and which ones you need infinitely many for, and so you start with squares and build up some rectangles. One square can only make one rectangle, itself. Two squares can only make one rectangle, but it can be lying down or standing up so you decide to say they’re different. Three squares make four rectangles and four squares make eight rectangles, and then you start thinking about Vi Hart’s video on binary trees. So you put the numbers into a tree, but it looks kind of stern, so you add some fareys to cheer it up. Then you see that the height of a rectangle is just the sum of its neighbors’ heights, and similarly for the width. You see lots of nice patterns in the dimensions involving flipping things over or running them backwards, kind of like the Blues Brothers’ police car when it was being chased by the Neo Nazis or that V6 racecar that Johann sent to Frederick.

Now instead of breadth, you decide to go for depth. Making the rectangles very long or very tall is too boring, so you add one square each time, alternately making it longer and taller. 1 1 2 3 5 8 13 21… You get the Fibonacci numbers; the limiting ratio is the golden ratio, [1+sqrt(5)]/2 to 1. This rectangle is the worst at being approximated by repeated squares so it shows up in systems where repetition is bad, like the angle at which plant leaves grow so they overlap the least and gather the most sunlight or how sunflowers pack the most seeds into a flowerhead and Roger Penrose thinks the Fibonacci spirals in the microtubules in the neurons in your brain are doing quantum error correction.

You decide to look at other irrational numbers to see if they have any nice patterns. e does. pi doesn’t. square roots of positive integers give repeating palindromes! You wonder whether all palindromes occur and if not which of the lyrics to Weird Al’s song Bob are special that way. And maybe then you make up a palindrome with vi hart’s name in it and turn it into a square root.

sqrt(1770203383334463140868642687939525148769043583402360581400094929361780283347187467842099172837131164923233584044530)

Maybe you decide that you want to doodle some circles after all, so you start with this gasket and figure out where the circles touch the line. The numbers look very familiar. You wonder what the areas of the circles are and how the gasket relates to the modular group and Poincare’s half-plane model of the hyperbolic plane and wish you had time to just sit in math class and doodle…

There are those who name the stars, who watch the sky by night,
Seeking out the darkest place, to better see the light.
Long ago, when torture broke the remnant of his will,
Galileo recanted, but the Earth is moving still.
High above the mountaintops, where only distance bars,
The truth has left its footprints in the dust between the stars.
We may watch and study or may shudder and deny,
Humans wrote the Bible; God wrote the sky.

By stem and root and branch we trace, by feather, fang and fur,
How the living things that are descend from things that were.
The moss, the kelp, the zebrafish, the very mice and flies,
These tiny, humble, wordless things–how shall they tell us lies?
We are kin to beasts; no other answer can we bring.
The truth has left its fingerprints on every living thing.
Remember, should you have to choose between them in the strife,
Humans wrote the Bible; God wrote life.

And we who listen to the stars, or walk the dusty grade,
Or break the very atoms down to see how they are made,
Or study cells, or living things, seek truth with open hand.
The profoundest act of worship is to try to understand.
Deep in flower and in flesh, in star and soil and seed,
The truth has left its living word for anyone to read.
So turn and look where best you think the story is unfurled.
Humans wrote the Bible; God wrote the world.

-Catherine Faber, The Word of God

Once, when the clock read noon, it traveled without hesitation in a straight path to my retina. Once it took another course, only to bend around a molecule of nitrogen and reach the same destination.

Once it traced the signature of a man no one remembers.

Once, at half past three, it decayed into a tiny spark and its abominable opposite image; their mutual horrified fascination drew them together, each a moth in the other’s flame. The last visible flash of light from their fiery consummation was indistinguishable from the one that spawned them.

Once, when the clock ticked in the silence just before dawn, the light decayed into two mirrored worlds, somewhat better than ours due to the fact that I was never born there. Both worlds were consumed by mirrored dragons before collapsing back into the chaos from which they arose; all that remained was an orange flash of light.

Once it traveled to a far distant galaxy, reflected off the waters of a billion worlds and witnessed the death of a thousand stars before returning to the small room in which we sat.

Once it transcribed a short story of Borges in his own cramped scrawl, the six versions he discarded, the corrupt Russian translation by Nabokov, and a version in which the second-to-last word was illegible.

Once it traveled every path, each in its time; once it became and destroyed every possible world. All these summed to form what was: I saw an orange flash, and in that moment, I was enlightened.

Kublai Khan died in 1294 and his successor Temur Khan was convinced to reinstate the astrologers. Despite this, the young mathematician Zhu Shijie took up the old Khan’s challenge in 1305. Zhu had already completed two enormously influential mathematical texts: Introduction to Mathematical Studies, published in 1299, and True reflections of the four unknowns, published in 1303. This latter work included a table of “the ancient method of powers”, now known as Pascal’s triangle, and Zhu used it extensively in his analysis of polynomials in up to four unknowns.

In turning to the analysis of divination, Zhu naturally focused his attention on the I Ching. The first step in performing an I Ching divination is casting coins or yarrow stalks to construct a series of hexagrams. In 1308, Zhu published his treatise on probability theory, Path of the falling stone. It included an analysis of the probability for generating each hexagram as well as betting strategies for several popular games of chance. Using his techniques, Zhu became quite wealthy and began to travel; it was during this period that he was exposed to the work of the mathematicians in northern China. In the preface to True reflections, Mo Ruo writes that “Zhu Shijie of Yan-shan became famous as a mathematician. He travelled widely for more than twenty years and the number of those who came to be taught by him increased each day.”

Zhu worked for nearly a decade on the subsequent problem, that of interpreting a series of hexagrams. Hexagrams themselves are generated one bit at a time by looking at remainders modulo four of random handfuls of yarrow stalks; the four outcomes either specify the next bit directly or in terms of the previous bit. These latter rules give I Ching its subtitle, The Book of Changes. For mystical reasons, Zhu asserted that the proper interpretation of a series of hexagrams should also be given by a set of changes, but for years he could find no reason to prefer one set of changes to any other. However, in 1316, Zhu wrote to Yang Hui:

“I dreamed that I was summoned to the royal palace. As I stepped upon the threshold, the sun burst forth over the gilded tile; I was blinded and, overcome, I fell to my knees. I lifted my hand to shield my eyes from its brilliance, and the Emperor himself took it and raised me up. To my surprise, he changed his form as I watched; he became so much like me that I thought I was looking in a mirror.

“‘How can this be?’ I cried. He laughed and took the form of a phoenix; I fell back from the flames as he ascended to heaven, then sorrowed as he dove toward the Golden Water River, for the water would surely quench the bird. Yet before reaching the water, he took the form of an eel, dove into the river and swam to the bank; he wriggled ashore, then took the form of a seed, which sank into the earth and grew into a mighty tree. Finally he took his own form again and spoke to me: ‘I rule all things; things above the earth and in the earth and under the earth, land and sea and sky. I can rule all these because I rule myself.’

“I woke and wondered at the singularity of the vision; when my mind reeled in amazement and could stand no more, it retreated to the familiar problem of the tables of changes. It suddenly occurred to me that as the Emperor could take any form, there could be a table of changes that could take the form of any other. Once I had conceived the idea, the implementation was straightforward.”

The rest of the letter has been lost, but Yang Hui described the broad form of the changes in a letter to a friend; the Imperial Changes were a set of changes that we now recognize as a Turing-complete programming language, nearly seven hundred years before Turing. It was a type of register machine similar to Melzak’s model, where seeds were ‘planted’ in pits; the lists of hexagrams generated by the yarrow straws were the programs, and the result of the computation was taken as the interpretation of the casting. Zhu recognized that some programs never stopped–some went into infinite loops, some grew without bound, and some behaved so erratically he couldn’t decide whether they would ever give an interpretation.

Given his fascination with probabilities, it was natural that Zhu would consider the probability that a string of hexagrams had an interpretation. We do not have Zhu’s reasoning, only an excerpt from his conclusion: “The probability that a list of hexagrams has an interpretation is a secret beyond the power of fate to reveal.” It may be that Zhu anticipated Chaitin’s proof of the algorithmic randomness of this probability as well.

All of Zhu’s works were lost soon after they were published; True reflections survived in a corrupted form through Korean (1433 AD) and Japanese (1658 AD) translations and was reintroduced to China only in the nineteenth century. One wonders what the world might have been like had the Imperial Changes been understood and exploited. We suppose it is a secret beyond the power of fate to reveal.

Thermometer (unitless temperature):
	[1]	inverse temperature (unitless)
	[m]	y coordinate
	[kg/s^2 K]	spring constant * temp unit conversion
	[m]	how position changes with (inverse) temperature
	[kg m/s^2 K]	force per Kelvin
	[kg m^2/s^2 K]	stretching energy per Kelvin
	[kg m^2/s^2 K]	potential energy per Kelvin
	[kg m^2/s^2 K]	entropy
Thermometer:
	[1/K]	inverse temperature
	[m]	y coordinate
	[kg/s^2 K^2 = bits/m^2 K]	how information density changes with temp
	[m K]	how position changes with (inverse) temperature
	[kg m/s^2 K = bits/m]	force per Kelvin
	[kg m^2/s^2 = bits K]	stretching energy = change in stretching information with invtemp
	[kg m^2/s^2 = bits K]	potential energy = change in potential information with invtemp
	[bits]	entropy

I assume that the dynamics of such a system would follow a path where is that a minimum-entropy path or a maximum?

Say we have two boxes, one inside the other:

+---------------+
|               |
| +----------+  |
| |          |  |
| |          |  |
| |          |  |
| +----------+  |
+---------------+

Say the inner box has room for ten bits on its surface and the outer one room for twenty. Each box can use as many “1″s as there are particles inside it:

+---------------+
|      X        |
| +----------+  |
| |          |  |
| |  X       |  |
| |          |  |
| +----------+  |
+---------------+

In this case, the inner box has only one particle inside, so there are 10 choose 1 = 10 ways to choose a labeling of the inner box; the outer box has two particles inside, so there are 20 choose 2 = 190 ways. Thus there are 1900 ways to label the system in all.

If both particles are in the inner box, though, the number of ways increases:

+---------------+
|               |
| +----------+  |
| |          |  |
| |  X  X    |  |
| |          |  |
| +----------+  |
+---------------+

The inner box now has 10 choose 2 ways = 45, while the outer box still has 190. So using the standard assumption that all labelings are equally likely, it’s 4.5 times as likely to find both particles in the inner box, and we get an entropic force drawing them together.

The best explanation of Verlinde’s paper I’ve seen is Sabine Hossenfelder’s Comments on and Comments on Comments on Verlinde’s paper “On the Origin of Gravity and the Laws of Newton”.

And he dreamed, and behold a ladder set up on the earth, and the top of it reached to heaven: and behold the angels of God ascending and descending on it.

Edit (May 20):

Even though it’s not a conformal transformation, this version looks better in a lot of ways.

Rather than cramming the whole picture into a single window frame, it presumes there’s a concentric set of these castles, each half as small as the previous, and built within its open internal patio. Doing it really well would involve extending the walls out to the edge of the outer wall that obscures them.

1. Count the number of dice held by the students and write it on the board.
2. Have everyone roll each die once.
3. Collect all the dice that show a ‘one’, count them, write the sum on the board, then set them aside.
4. Go back to step 1.

A run with 30 dice will look something like this:

Point out how the number of dice rolling a one on each turn is about one sixth of the dice that hadn’t yet rolled a one on the previous turn. Also, that you lose about half of the remaining dice after about four turns.

Send someone out of the room; do either four or eight turns, then bring them back and ask them to guess how many turns the group took. The student should be able to see that if half the dice are left, there were only four turns, but if a quarter of the dice are left, there were eight turns.

If the students are advanced enough to use logarithms, try the above with some number other than four or eight and have the student use logarithms to calculate the number of turns:

turns = log(number remaining/total) / log(5/6),

or, equivalently, in terms of the half-life (which is really closer to 3.8 than 4):

turns = 3.8 * log(number remaining/total) / log(1/2).

When Zircon crystals form, they strongly reject lead atoms: new zircon crystals have no lead in them. They easily accept uranium atoms. Each die represents a uranium atom, and rolling a one represents decaying into a lead atom: because uranium atoms are radioactive, they can lose bits of their nucleus and turn into lead–but only randomly, like rolling a die. Instead of four turns, the half-life of U238 is 4.5 billion years.

Zircon forms in almost all rocks and is hard to break down. So to judge the age of a rock, you get the zircon out, throw it in a mass spectrometer, look at the proportion of uranium to (lead plus uranium) and calculate

years = 4.5 billion * log(mass of uranium/mass of (lead+uranium)) / log(1/2).

Problem: given a zircon crystal where there’s one lead atom for every ninety-nine uranium atoms, how long ago was it formed?

4.5 billion * log(99/100) / log(1/2) = 65 million years ago.

In reality, it’s slightly more complicated: there are two isotopes of uranium and several of lead. But this is a good thing, since we know the half-lives of both isotopes and can use them to cross-check each other; it’s as though each student had both six- and twenty-sided dice, and the student guessing the number of turns could use information from both groups to refine her guess.

z ↦ z² + c

does pretty much the same thing, but adds wiggles to the curve. Since the iterations stop when |z| > 2, the boundary at the zeroth iteration is a circle; after the first it’s a pear shape, and so on. We can map any point in the region between bands to a point in a rectangular tile that’s periodic once along the outside edge and twice along the inside edge. Here’s a site with a few different examples.

The transformation Escher used in “Print Gallery” takes concentric circles at r=1/rⁿ to a logarithmic spiral. The concentric boundaries between iterations for the Julia set at c = 0 are circles. There ought to be a transformation for Mandelbrot / Julia sets similar to the Droste effect but spiraling inward so that the frequency is smoothly increasing just as the distance can be made to smoothly increase.

Their name derives from the stone lapis lazuli and means, roughly, “little blue stone”. Lazulinos are so named because even though the crystals in which they arise absorb visible light, and would otherwise be jet black, they lose energy through surface plasmons in the form of near-ultraviolet photons, with visible peaks at 380, 402, and 417nm. Optical interference imparts a “laser speckle” quality to the emitted light; Craigie described the effect in a famously poetic way: “Their colour is the blue that we are permitted to see only in our dreams”. What makes lazulinos particularly interesting is that they are massive and macroscopic. Since the number operator does not commute with the Hamiltonian, lazulinos themselves do not have a well-defined mass; if the population is N, then the mass of any particular lazulino is m/N, where m is the total mass of the condensate.

In a recent follow-up to the “quantum mirage” experiment [2], Don Eigler’s group at IBM used a scanning tunneling microscope to implement “quantum mancala”—picking up the lazulino ‘stones’ in a particular location usually changes the number of stones, so the strategy for winning becomes much more complicated. In order to pick up a fixed number of stones, you must choose a superposition of locations [1].

1. C.P. Lutz and D.M. Eigler, “Quantum Mancala: Manipulating Lazulino Condensates,” Nature 465, 132 (2010).
2. H.C. Manoharan, C.P. Lutz and D.M. Eigler, “Quantum Mirages: The Coherent Projection of Electronic Structure,” Nature 403, 512 (2000). Images available at http://www.almaden.ibm.com/almaden/media/image_mirage.html
3. A. Craigie, “Surface plasmons in cobalt-doped Y₃Al₅O₁₂,” Phys. Rev. D 15 (1977). Also available at http://tinyurl.com/35oyrnd.

Let be a small category. The functor assigns

• to each pair of objects the set of morphisms between them, and
• to each pair of morphisms a function that takes a morphism and returns the composite morphism , where

It turns out that given any functor we can make a new category where and are subcategories and is actually the hom functor; some keywords for more information on this are “collages” and “Artin glueing”. So we can also think of as assigning

• to each pair of objects a set of morphisms between them, and
• to each pair of morphisms a function that takes a morphism and returns the composite morphism , where and

We can think of these functors as adjacency matrices, where the two parameters are the row and column, except that instead of counting the number of paths, we’re taking the set of paths. So is kind of like a matrix whose elements are sets, and we want to do something like sum the diagonals.

The coend of is the coequalizer of the diagram

The top set consists of all the pairs where

• the first element is a morphism and
• the second element is a morphism

The bottom set is the set of all the endomorphisms in

The coequalizer of the diagram, the coend of is the bottom set modulo a relation. Starting at the top with a pair the two arrows give the relation

where I’m using the lollipop to mean a morphism from

So this says take all the endomorphisms that can be chopped up into a morphism from going one way and a from going the other, and then set For this to make any sense, it has to identify any two objects related by such a pair. So it’s summing over all the endomorphisms of these equivalence classes.

To get the trace of the hom functor, use in the analysis above and replace the lollipop with a real arrow. If that category is just a group, this is the set of conjugacy classes. If that category is a preorder, then we’re computing the set of isomorphism classes.

The coend is also used when “multiplying matrices”. Let Then the top set consists of triples the bottom set of pairs and the coend is the bottom set modulo

That is, it doesn’t matter if you think of as connected to or to ; the connection is associative, so you can go all the way from to

Notice here how a morphism can turn “inside out”: when and the identities surround a morphism in , it’s the same as being surrounded by morphisms in and ; this is the difference between a trace, where we’re repeating indices on the same matrix, and matrix multiplication, where we’re repeating the column of the first matrix in the row of the second matrix.