My All 2020 Mathematics A to Z: Möbius Strip


Jacob Siehler suggested this topic. I had to check several times that I hadn’t written an essay about the Möbius strip already. While I have talked about it some, mostly in comic strip essays, this is a chance to specialize on the shape in a way I haven’t before.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Möbius Strip.

I have ridden at least 252 different roller coasters. These represent nearly every type of roller coaster made today, and most of the types that were ever made. One type, common in the 1920s and again since the 70s, is the racing coaster. This is two roller coasters, dispatched at the same time, following tracks that are as symmetric as the terrain allows. Want to win the race? Be in the train with the heavier passenger load. The difference in the time each train takes amounts to losses from friction, and the lighter train will lose a bit more of its speed.

There are three special wooden racing coasters. These are Racer at Kennywood Amusement Park (Pittsburgh), Grand National at Blackpool Pleasure Beach (Blackpool, England), and Montaña Rusa at La Feria Chapultepec Magico (Mexico City). I’ve been able to ride them all. When you get into the train going up, say, the left lift hill, you return to the station in the train that will go up the right lift hill. These racing roller coasters have only one track. The track twists around itself and becomes a Möbius strip.

Picture of some of the Grand National track, at Blackpool Pleasure Beach, showing off particularly a sign with an illustration of the launch station and 'What The Papers Say', showing the British national papers' 'reviews' of the roller coaster.
I’m not surprised I don’t have good pictures of Grand National. I’ve only been able to visit Blackpool Pleasure Beach the one time. And a few days before my camera had just been thoroughly soaked in heavy rain. I could only take a picture by turning it on; the camera would power up, take a picture as soon as it was ready, and power down again. It’s the other parks I’m surprised I lack better pictures for.

This is a fun use of the Möbius strip. The shape is one of the few bits of advanced mathematics to escape into pop culture. Maybe dominates it, in a way nothing but the blackboard full of calculus equations does. In 1958 the public intellectual and game show host Clifton Fadiman published the anthology Fantasia Mathematica. It’s all essays and stories and poems with some mathematical element. I no longer remember how many of the pieces were about the Möbius strip one way or another. The collection does include A J Deutschs’s classic A Subway Named Möbius. In this story the Boston subway system achieves hyperdimensional complexity. It does not become a Möbius strip, though, in that story. It might be one in reality anyway.

The Möbius strip we name for August Ferdinand Möbius, who in 1858 was the second person known to have noticed the shape’s curious properties. The first — to notice, in 1858, and to publish, in 1862 — was Johann Benedict Listing. Listing seems to have coined the term “topology” for the field that the Möbius strip would be emblem for. He wrote one of the first texts on the field. He also seems to have coined terms like “entrophic phenomena” and “nodal points” and “geoid” and “micron”, for a millionth of a meter. It’s hard to say why we don’t talk about Listing strips instead. Mathematical fame is a strange, unpredictable creature. There is a topological invariant, the Listing Number, named for him. And he’s known to ophthalmologists for Listing’s Law, which describes how human eyes orient themselves.

The Möbius strip is an easy thing to construct. Loop a ribbon back to itself, with an odd number of half-twist before you fasten the ends together. Anyone could do it. So it seems curious that for all recorded history nobody thought to try. Not until 1858 when Lister and then Möbius hit on the same idea.

Kennywood Racer blue roller coaster train, leaving the station on the (camera) left track.
It’s Kennywood’s Racer I’m surprised I don’t have a picture that shows what I really want. We get to Kennywood basically every year, 2020 excepted for obvious reasons. But there also really aren’t good observation tower-type attractions that let you get a photograph of Racer as a whole, I’ll tell myself is the problem.

An irresistible thing, while riding these roller coasters, is to try to find the spot where you “switch”, where you go from being on the left track to the right. You can’t. The track is — well, the track is a series of metal straps bolted to a base of wood. (The base the straps are bolted to is what makes it a wooden roller coaster. The great lattice holding the tracks above ground have nothing to do with it.) But the path of the tracks is a continuous whole. To split it requires the same arbitrariness with which mapmakers pick a prime meridian. It’s obvious that the “longitude” of a cylinder or a rubber ball is arbitrary. It’s not obvious that roller coaster tracks should have the same property. Until you draw the shape in that ∞-loop figure we always see. Then you can get lost imagining a walk along the surface.

And it’s not true that nobody thought to try this shape before 1858. Julyan H E Cartwright and Diego L González wrote a paper searching for pre-Möbius strips. They find some examples. To my eye not enough examples to support their abstract’s claim of “lots of them”, but I trust they did not list every example. One example is a Roman mosaic showing Aion, the God of Time, Eternity, and the Zodiac. He holds a zodiac ring that is either a Möbius strip or cylinder with artistic errors. Cartwright and González are convinced. I’m reminded of a Looks Good On Paper comic strip that forgot to include the needed half-twist.

Mobius Trip. A car, loaded for vacation, with someone in it asking 'Are we there yet? Are we there yet? Are we there yet?' The road is along a Mobius strip, with roadside bits like deer or road signs or opossums crossing the road on the margins.
Dan Collins’s Looks Good on Paper for the 10th of September, 2018. It originally ran the 10th of September, 2016. It falls a half-twist short of being the Möbius strip it intends to be. Essays in which I discuss Looks Good On Paper are at this link.

Islamic science gives us a more compelling example. We have a book by Ismail al-Jazari dated 1206, The Book of Knowledge of Ingenious Mechanical Devices. Some manuscripts of it illustrate a chain pump, with the chain arranged as a Möbius strip. Cartwright and González also note discussions in Scientific American, and other engineering publications in the United States, about drive and conveyor belts with the Möbius strip topology. None of those predate Lister or Möbius, or apparently credit either. And they do come quite soon after. It’s surprising something might leap from abstract mathematics to Yankee ingenuity that fast.

If it did. It’s not hard to explain why mechanical belts didn’t consider Möbius strip shapes before the late 19th century. Their advantage is that the wear of the belt distributes over twice the surface area, the “inside” and “outside”. A leather belt has a smooth and a rough side. Many other things you might make a belt from have a similar asymmetry. By the late 19th century you could make a belt of rubber. Its grip and flexibility and smoothness is uniform on all sides. “Balancing” the use suddenly could have a point.

I still find it curious almost no one drew or speculated about or played with these shapes until, practically, yesterday. The shape doesn’t seem far away from a trefoil knot. The recycling symbol, three folded-over arrows, suggests a Möbius strip. The strip evokes the ∞ symbol, although that symbol was not attached to the concept of “infinity” until John Wallis put it forth in 1655.

Kennywood Racer blue roller coaster train arriving at the station on the (camera) right track; the red train is approaching the (camera) left track.
Also Racer is usually quite packed, and we’d rather put time into Jackrabbit (100 years old this year) and Thunderbolt (originally built in 1924 as Pippin and extensively rebuilt in 1967). Still, people attending as part of special events can get more revealing photographs.

Even with the shape now familiar, and loved, there are curious gaps. Consider game design. If you play on a board that represents space you need to do something with the boundaries. The easiest is to make the boundaries the edges of playable space. The game designer has choices, though. If a piece moves off the board to the right, why not have it reappear on the left? (And, going off to the left, reappear on the right.) This is fine. It gives the game board, a finite rectangle, the topology of a cylinder. If this isn’t enough? Have pieces that go off the top edge reappear at the bottom, and vice-versa. Doing this, along with matching the left to the right boundaries, makes the game board a torus, a doughnut shape.

A Möbius strip is easy enough to code. Make the top and bottom impenetrable borders. And match the left to the right edges this way: a piece going off the board at the upper half of the right edge reappears at the lower half of the left edge. Going off the lower half of the right edge brings the piece to the upper half of the left edge. And so on. It isn’t hard, but I’m not aware of any game — board or computer — that uses this space. Maybe there’s a backgammon variant which does.

Still, the strip defies our intuition. It has one face and one edge. To reflect a shape across the width of the strip is the same as sliding a shape along its length. Cutting the strip down the center unfurls it into a cylinder. Cutting the strip down, one-third of the way from the edge, divides it into two pieces, a skinnier Möbius strip plus a cylinder. If we could extract the edge we could tug and stretch it until it was a circle.

And it primes our intuition. Once we understand there can be shapes lacking sides we can look for more. Anyone likely to read a pop mathematics blog about the Möbius strip has heard of the Klein bottle. This is a three-dimensional surface that folds back on itself in the fourth dimension of space. The shape is a jug with no inside, or with nothing but inside. Three-dimensional renditions of this get suggested as gifts to mathematicians. This for your mathematician friend who’s already got a Möbius scarf.

Log flume car splashing down at the bottom of its hill. In the background is the rust-red track and structure of the Racer roller coaster.
Anyway the roller coaster in background is Racer. This is a view from the Log Jammer, a log flume ride that Kennywood removed in 2018.

Though a Möbius strip looks — at any one spot — like a plane, the four-color map theorem doesn’t hold for it. Even the five-color theorem won’t do. You need six colors to cover maps on such a strip. A checkerboard drawn on a Möbius strip can be completely covered by T-shape pentominoes or Tetris pieces. You can’t do this for a checkerboard on the plane. In the mathematics of music theory the organization of dyads — two-tone “chords” — has the structure of a Möbius strip. I do not know music theory or the history of music theory. I’m curious whether Möbius strips might have been recognized by musicians before the mathematicians caught on.

And they inspire some practical inventions. Mechanical belts are obvious, although I don’t know how often they’re used. More clever are designs for resistors that have no self-inductance. They can resist electric flow without causing magnetic interference. I can look up the patents; I can’t swear to how often these are actually used. There exist — there are made — Möbius aromatic compounds. These are organic compounds with rings of carbon and hydrogen. I do not know a use for these. That they’ve only been synthesized this century, rather than found in nature, suggests they are more neat than practical.

Perhaps this shape is most useful as a path into a particular type of topology, and for its considerable artistry. And, with its “late” discovery, a reminder that we do not yet know all that is obvious. That is enough for anything.

Photograph of a model of the Montana Rusa roller coaster, under glass, and watched over by a Alebrije-styled dragon-ish figure. It sits on the platform for a roller coaster.
Montaña Rusa, sad to say, when we visited La Feria Chapultepec Magico in 2018 was running only one train at a time. It did have this gorgeous model of the roller coaster, although under plexiglass and roped off so that it was difficult to photograph.

There are three steel roller coasters with a Möbius strip track. That is, the metal rail on which the coaster runs is itself braced directly by metal. One of these is in France, one in Italy, and one in Iran. One in Liaoning, China has been under construction for five years. I can’t say when it might open. I have yet to ride any of them.


This and all the other 2020 A-to-Z essays should be at this link. Both the 2020 and all past A-to-Z essays should be at this link. I am hosting the Playful Math Education Blog Carnival at the end of September, so appreciate any educational or recreational or simply fun mathematics material you know about. And, goodness, I’m actually overdue to ask for topics for the latters P through R; I’ll have a post for that tomorrow, I hope. Thank you for your reading and your help.

Meanwhile, in sandwich news


This is a slight thing that crossed my reading yesterday. You might enjoy. The question is a silly one: what’s the “optimal” way to slice banana onto a peanut-butter-and-banana sandwich?

Here’s Ethan Rosenthal’s answer. The specific problem this is put to is silly. The optimal peanut butter and banana sandwich is the one that satisfies your desire for a peanut butter and banana sandwich. However, the approach to the problem demonstrates good mathematics, and numerical mathematics, practices. Particularly it demonstrates defining just what your problem is, and what you mean by “optimal”, and how you can test that. And then developing a numerical model which can optimize it.

And the specific question, how much of the sandwich can you cover with banana slices, one of actual interest. A good number of ideas in analysis involve thinking of cover sets: what is the smallest collection of these things which will completely cover this other thing? Concepts like this give us an idea of how to define area, also, as the smallest number of standard reference shapes which will cover the thing we’re interested in. The basic problem is practical too: if we wish to provide something, and have units like this which can cover some area, how can we arrange them so as to miss as little as possible? Or use as few of the units as possible?

My All 2020 Mathematics A to Z: K-Theory


I should have gone with Vayuputrii’s proposal that I talk about the Kronecker Delta. But both Jacob Siehler and Mr Wu proposed K-Theory as a topic. It’s a big and an important one. That was compelling. It’s also a challenging one. This essay will not teach you K-Theory, or even get you very far in an introduction. It may at least give some idea of what the field is about.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

K-Theory.

This is a difficult topic to discuss. It’s an important theory. It’s an abstract one. The concrete examples are either too common to look interesting or are already deep into things like “tangent bundles of Sn-1”. There are people who find tangent bundles quite familiar concepts. My blog will not be read by a thousand of them this month. Those who are familiar with the legends grown around Alexander Grothendieck will nod on hearing he was a key person in the field. Grothendieck was of great genius, and also spectacular indifference to practical mathematics. Allegedly he once, pressed to apply something to a particular prime number for an example, proposed 57, which is not prime. (One does not need to be a genius to make a mistake like that. If I proposed 447 or 449 as prime numbers, how long would you need to notice I was wrong?)

K-Theory predates Grothendieck. Now that we know it’s a coherent mathematical idea we can find elements leading to it going back to the 19th century. One important theorem has Bernhard Riemann’s name attached. Henri Poincaré contributed early work too. Grothendieck did much to give the field a particular identity. Also a name, the K coming from the German Klasse. Grothendieck pioneered what we now call Algebraic K-Theory, working on the topic as a field of abstract algebra. There is also a Topological K-Theory, early work on which we thank Michael Atiyah and Friedrick Hirzebruch for. Topology is, popularly, thought of as the mathematics of flexible shapes. It is, but we get there from thinking about relationships between sets, and these are the topologies of K-Theory. We understand these now as different ways of understandings structures.

Still, one text I found described (topological) K-Theory as “the first generalized cohomology theory to be studied thoroughly”. I remember how much handwaving I had to do to explain what a cohomology is. The subject looks intimidating because of the depth of technical terms. Every field is deep in technical terms, though. These look more rarefied because we haven’t talked much, or deeply, into the right kinds of algebra and topology.

You find at the center of K-Theory either “coherent sheaves” or “vector bundles”. Which alternative depends on whether you prefer Algebraic or Topological K-Theory. Both alternatives are ways to encode information about the space around a shape. Let me talk about vector bundles because I find that easier to describe. Take a shape, anything you like. A closed ribbon. A torus. A Möbius strip. Draw a curve on it. Every point on that curve has a tangent plane, the plane that just touches your original shape, and that’s guaranteed to touch your curve at one point. What are the directions you can go in that plane? That collection of directions is a fiber bundle — a tangent bundle — at that point. (As ever, do not use this at your thesis defense for algebraic topology.)

Now: what are all the tangent bundles for all the points along that curve? Does their relationship tell you anything about the original curve? The question is leading. If their relationship told us nothing, this would not be a subject anyone studies. If you pick a point on the curve and look at its tangent bundle, and you move that point some, how does the tangent bundle change?

If we start with the right sorts of topological spaces, then we can get some interesting sets of bundles. What makes them interesting is that we can form them into a ring. A ring means that we have a set of things, and an operation like addition, and an operation like multiplication. That is, the collection of things works somewhat like the integers do. This is a comfortable familiar behavior after pondering too much abstraction.

Why create such a thing? The usual reasons. Often it turns out calculating something is easier on the associated ring than it is on the original space. What are we looking to calculate? Typically, we’re looking for invariants. Things that are true about the original shape whatever ways it might be rotated or stretched or twisted around. Invariants can be things as basic as “the number of holes through the solid object”. Or they can be as ethereal as “the total energy in a physics problem”. Unfortunately if we’re looking at invariants that familiar, K-Theory is probably too much overhead for the problem. I confess to feeling overwhelmed by trying to learn enough to say what it is for.

There are some big things which it seems well-suited to do. K-Theory describes, in its way, how the structure of a set of items affects the functions it can have. This links it to modern physics. The great attention-drawing topics of 20th century physics were quantum mechanics and relativity. They still are. The great discovery of 20th century physics has been learning how much of it is geometry. How the shape of space affects what physics can be. (Relativity is the accessible reflection of this.)

And so K-Theory comes to our help in string theory. String theory exists in that grand unification where mathematics and physics and philosophy merge into one. I don’t toss philosophy into this as an insult to philosophers or to string theoreticians. Right now it is very hard to think of ways to test whether a particular string theory model is true. We instead ponder what kinds of string theory could be true, and how we might someday tell whether they are. When we ask what things could possibly be true, and how to tell, we are working for the philosophy department.

My reading tells me that K-Theory has been useful in condensed matter physics. That is, when you have a lot of particles and they interact strongly. When they act like liquids or solids. I can’t speak from experience, either on the mathematics or the physics side.

I can talk about an interesting mathematical application. It’s described in detail in section 2.3 of Allen Hatcher’s text Vector Bundles and K-Theory, here. It comes about from consideration of the Hopf invariant, named for Heinz Hopf for what I trust are good reasons. It also comes from consideration of homomorphisms. A homomorphism is a matching between two sets of things that preserves their structure. This has a precise definition, but I can make it casual. If you have noticed that, every (American, hourlong) late-night chat show is basically the same? The host at his desk, the jovial band leader, the monologue, the show rundown? Two guests and a band? (At least in normal times.) Then you have noticed the homomorphism between these shows. A mathematical homomorphism is more about preserving the products of multiplication. Or it preserves the existence of a thing called the kernel. That is, you can match up elements and how the elements interact.

What’s important is Adams’ Theorem of the Hopf Invariant. I’ll write this out (quoting Hatcher) to give some taste of K-Theory:

The following statements are true only for n = 1, 2, 4, and 8:
a. R^n is a division algebra.
b. S^{n - 1} is parallelizable, ie, there exist n – 1 tangent vector fields to S^{n - 1} which are linearly independent at each point, or in other words, the tangent bundle to S^{n - 1} is trivial.

This is, I promise, low on jargon. “Division algebra” is familiar to anyone who did well in abstract algebra. It means a ring where every element, except for zero, has a multiplicative inverse. That is, division exists. “Linearly independent” is also a familiar term, to the mathematician. Almost every subject in mathematics has a concept of “linearly independent”. The exact definition varies but it amounts to the set of things having neither redundant nor missing elements.

The proof from there sprawls out over a bunch of ideas. Many of them I don’t know. Some of them are simple. The conditions on the Hopf invariant all that S^{n - 1} stuff eventually turns into finding values of n for for which 2^n divides 3^n - 1 . There are only three values of ‘n’ that do that. For example.

What all that tells us is that if you want to do something like division on ordered sets of real numbers you have only a few choices. You can have a single real number, R^1 . Or you can have an ordered pair, R^2 . Or an ordered quadruple, R^4 . Or you can have an ordered octuple, R^8 . And that’s it. Not that other ordered sets can’t be interesting. They will all diverge far enough from the way real numbers work that you can’t do something that looks like division.

And now we come back to the running theme of this year’s A-to-Z. Real numbers are real numbers, fine. Complex numbers? We have some ways to understand them. One of them is to match each complex number with an ordered pair of real numbers. We have to define a more complicated multiplication rule than “first times first, second times second”. This rule is the rule implied if we come to R^2 through this avenue of K-Theory. We get this matching between real numbers and the first great expansion on real numbers.

The next great expansion of complex numbers is the quaternions. We can understand them as ordered quartets of real numbers. That is, as R^4 . We need to make our multiplication rule a bit fussier yet to do this coherently. Guess what fuss we’d expect coming through K-Theory?

R^8 seems the odd one out; who does anything with that? There is a set of numbers that neatly matches this ordered set of octuples. It’s called the octonions, sometimes called the Cayley Numbers. We don’t work with them much. We barely work with quaternions, as they’re a lot of fuss. Multiplication on them doesn’t even commute. (They’re very good for understanding rotations in three-dimensional space. You can also also use them as vectors. You’ll do that if your programming language supports quaternions already.) Octonions are more challenging. Not only does their multiplication not commute, it’s not even associative. That is, if you have three octonions — call them p, q, and r — you can expect that p times the product of q-and-r would be different from the product of p-and-q times r. Real numbers don’t work like that. Complex numbers or quaternions don’t either.

Octonions let us have a meaningful division, so we could write out p \div q and know what it meant. We won’t see that for any bigger ordered set of R^n . And K-Theory is one of the tools which tells us we may stop looking.

This is hardly the last word in the field. It’s barely the first. It is at least an understandable one. The abstractness of the field works against me here. It does offer some compensations. Broad applicability, for example; a theorem tied to few specific properties will work in many places. And pure aesthetics too. Much work, in statements of theorems and their proofs, involve lovely diagrams. You’ll see great lattices of sets relating to one another. They’re linked by chains of homomorphisms. And, in further aesthetics, beautiful words strung into lovely sentences. You may not know what it means to say “Pontryagin classes also detect the nontorsion in \pi_k(SO(n)) outside the stable range”. I know I don’t. I do know when I hear a beautiful string of syllables and that is a joy of mathematics never appreciated enough.


Thank you for reading. The All 2020 A-to-Z essays should be available at this link. The essays from all A-to-Z sequence, 2015 to present, should be at this link. And I am still open for M, N, and O essay topics. Thanks for your attention.

My All 2020 Mathematics A to Z: Michael Atiyah


To start this year’s great glossary project Mr Wu, author of the MathTuition88.com blog, had a great suggestion: The Atiyah-Singer Index Theorem. It’s an important and spectacular piece of work. I’ll explain why I’m not doing that in a few sentences.

Mr Wu pointed out that a biography of Michael Atiyah, one of the authors of this theorem, might be worth doing. GoldenOj endorsed the biography idea, and the more I thought it over the more I liked it. I’m not able to do a true biography, something that goes to primary sources and finds a convincing story of a life. But I can sketch out a bit, exploring his work and why it’s of note.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Michael Atiyah.

Theodore Frankel’s The Geometry of Physics: An Introduction is a wonderful book. It’s 686 pages, including the index. It all explores how our modern understanding of physics is our modern understanding of geometry. On page 465 it offers this:

The Atiyah-Singer index theorem must be considered a high point of geometrical analysis of the twentieth century, but is far too complicated to be considered in this book.

I know when I’m licked. Let me attempt to look at one of the people behind this theorem instead.

The Riemann Hypothesis is about where to find the roots of a particular infinite series. It’s been out there, waiting for a solution, for a century and a half. There are many interesting results which we would know to be true if the Riemann Hypothesis is true. In 2018, Michael Atiyah declared that he had a proof. And, more, an amazing proof, a short proof. Albeit one that depended on a great deal of background work and careful definitions. The mathematical community was skeptical. It still is. But it did not dismiss outright the idea that he had a solution. It was plausible that Atiyah might solve one of the greatest problems of mathematics in something that fits on a few PowerPoint slides.

So think of a person who commands such respect.

His proof of the Riemann Hypothesis, as best I understand, is not generally accepted. For example, it includes the fine structure constant. This comes from physics. It describes how strongly electrons and photons interact. The most compelling (to us) consequence of the Riemann Hypothesis is in how prime numbers are distributed among the integers. It’s hard to think how photons and prime numbers could relate. But, then, if humans had done all of mathematics without noticing geometry, we would know there is something interesting about π. Differential equations, if nothing else, would turn up this number. We happened to discover π in the real world first too. If it were not familiar for so long, would we think there should be any commonality between differential equations and circles?

I do not mean to say Atiyah is right and his critics wrong. I’m no judge of the matter at all. What is interesting is that one could imagine a link between a pure number-theory matter like the Riemann hypothesis and a physical matter like the fine structure constant. It’s not surprising that mathematicians should be interested in physics, or vice-versa. Atiyah’s work was particularly important. Much of his work, from the late 70s through the 80s, was in gauge theory. This subject lies under much of modern quantum mechanics. It’s born of the recognition of symmetries, group operations that you can do on a field, such as the electromagnetic field.

In a sequence of papers Atiyah, with other authors, sorted out particular cases of how magnetic monopoles and instantons behave. Magnetic monopoles may sound familiar, even though no one has ever seen one. These are magnetic points, an isolated north or a south pole without its opposite partner. We can understand well how they would act without worrying about whether they exist. Instantons are more esoteric; I don’t remember encountering the term before starting my reading for this essay. I believe I did, encountering the technique as a way to describe the transitions between one quantum state and another. Perhaps the name failed to stick. I can see where there are few examples you could give an undergraduate physics major. And it turns out that monopoles appear as solutions to some problems involving instantons.

This was, for Atiyah, later work. It arose, in part, from bringing the tools of index theory to nonlinear partial differential equations. This index theory is the thing that got us the Atiyah-Singer Index Theorem too complicated to explain in 686 pages. Index theory, here, studies questions like “what can we know about a differential equation without solving it?” Solving a differential equation would tell us almost everything we’d like to know, yes. But it’s also quite hard. Index theory can tell us useful things like: is there a solution? Is there more than one? How many? And it does this through topological invariants. A topological invariant is a trait like, for example, the number of holes that go through a solid object. These things are indifferent to operations like moving the object, or rotating it, or reflecting it. In the language of group theory, they are invariant under a symmetry.

It’s startling to think a question like “is there a solution to this differential equation” has connections to what we know about shapes. This shows some of the power of recasting problems as geometry questions. From the late 50s through the mid-70s, Atiyah was a key person working in a topic that is about shapes. We know it as K-theory. The “K” from the German Klasse, here. It’s about groups, in the abstract-algebra sense; the things in the groups are themselves classes of isomorphisms. Michael Atiyah and Friedrich Hirzebruch defined this sort of group for a topological space in 1959. And this gave definition to topological K-theory. This is again abstract stuff. Frankel’s book doesn’t even mention it. It explores what we can know about shapes from the tangents to the shapes.

And it leads into cobordism, also called bordism. This is about what you can know about shapes which could be represented as cross-sections of a higher-dimension shape. The iconic, and delightfully named, shape here is the pair of pants. In three dimensions this shape is a simple cartoon of what it’s named. On the one end, it’s a circle. On the other end, it’s two circles. In between, it’s a continuous surface. Imagine the cross-sections, how on separate layers the two circles are closer together. How their shapes distort from a real circle. In one cross-section they come together. They appear as two circles joined at a point. In another, they’re a two-looped figure. In another, a smoother circle. Knowing that Atiyah came from these questions may make his future work seem more motivated.

But how does one come to think of the mathematics of imaginary pants? Many ways. Atiyah’s path came from his first research specialty, which was algebraic geometry. This was his work through much of the 1950s. Algebraic geometry is about the kinds of geometric problems you get from studying algebra problems. Algebra here means the abstract stuff, although it does touch on the algebra from high school. You might, for example, do work on the roots of a polynomial, or a comfortable enough equation like x^2 + y^2 = 1 . Atiyah had started — as an undergraduate — working on projective geometries. This is what one curve looks like projected onto a different surface. This moved into elliptic curves and into particular kinds of transformations on surfaces. And algebraic geometry has proved important in number theory. You might remember that the Wiles-Taylor proof of Fermat’s Last Theorem depended on elliptic curves. Some work on the Riemann hypothesis is built on algebraic topology.

(I would like to trace things farther back. But the public record of Atiyah’s work doesn’t offer hints. I can find amusing notes like his father asserting he knew he’d be a mathematician. He was quite good at changing local currency into foreign currency, making a profit on the deal.)

It’s possible to imagine this clear line in Atiyah’s career, and why his last works might have been on the Riemann hypothesis. That’s too pat an assertion. The more interesting thing is that Atiyah had several recognizable phases and did iconic work in each of them. There is a cliche that mathematicians do their best work before they are 40 years old. And, it happens, Atiyah did earn a Fields Medal, given to mathematicians for the work done before they are 40 years old. But I believe this cliche represents a misreading of biographies. I suspect that first-rate work is done when a well-prepared mind looks fresh at a new problem. A mathematician is likely to have these traits line up early in the career. Grad school demands the deep focus on a particular problem. Getting out of grad school lets one bring this deep knowledge to fresh questions.

It is easy, in a career, to keep studying problems one has already had great success in, for good reason and with good results. It tends not to keep producing revolutionary results. Atiyah was able — by chance or design I can’t tell — to several times venture into a new field. The new field was one that his earlier work prepared him for, yes. But it posed new questions about novel topics. And this creative, well-trained mind focusing on new questions produced great work. And this is one way to be credible when one announces a proof of the Riemann hypothesis.


Here is something I could not find a clear way to fit into this essay. Atiyah recorded some comments about his life for the Web of Stories site. These are biographical and do not get into his mathematics at all. Much of it is about his life as child of British and Lebanese parents and how that affected his schooling. One that stood out to me was about his peers at Manchester Grammar School, several of whom he rated as better students than he was. Being a good student is not tightly related to being a successful academic. Particularly as so much of a career depends on chance, on opportunities happening to be open when one is ready to take them. It would be remarkable if there wre three people of greater talent than Atiyah who happened to be in the same school at the same time. It’s not unthinkable, though, and we may wonder what we can do to give people the chance to do what they are good in. (I admit this assumes that one finds doing what one is good in particularly satisfying or fulfilling.) In looking at any remarkable talent it’s fair to ask how much of their exceptional nature is that they had a chance to excel.

Reading the Comics, May 29, 2020: Slipping Into Summer More Edition


This is the slightly belated close of last week’s topics suggested by Comic Strip Master Command. For the week we’ve had, I am doing very well.

Werner Wejp-Olsen’s Inspector Danger’s Crime Quiz for the 25th of May sees another mathematician killed, and “identifying” his killer in a dying utterance. Inspector Danger has followed killer mathematicians several times before: the 9th of July, 2012, for instance. Or the 4th of July, 2016, for a case so similar that it’s almost a Slylock Fox six-differences puzzle. Apparently realtors and marine biologists are out for mathematicians’ blood. I’m not surprised by the realtors, but hey, marine biology, what’s the deal? The same gimmick got used the 15th of May, 2017, too. (And in fairness to the late Wejp-Olsen, who could possibly care that similar names are being used in small puzzles used years apart? It only stands out because I’m picking out things that no reasonable person would notice.)

Monty, to his robot pal: 'During a plague, Sir Isaac Newton invented calculus! Shakespeare wrote Lear and Macbeth!' (Two panels of Monty thinking hard and struggling to compose anything.) Monty: 'Maybe I'm more like Charles Darwin. I think he wrote 'On the Origin of Species' when everything was pretty normal.'
Jim Meddick’s Monty for the 25th of May, 2020. Essays with some mention of topics from Monty are at this link.

Jim Meddick’s Monty for the 25th has the title character inspired by the legend of genius work done during plague years. A great disruption in life is a great time to build new habits, and if Covid-19 has given you the excuse to break bad old habits, or develop good new ones, great! Congratulations! If it has not, though? That’s great too. You’re surviving the most stressful months of the 21st century, I hope, not taking a holiday.

Anyway, the legend mentioned here includes Newton inventing Calculus while in hiding from the plague. The actual history is more complicated, and ambiguous. (You will not go wrong supposing that the actual history of a thing is more complicated and ambiguous than you imagine.) The Renaissance Mathematicus describes, with greater authority and specificity than I could, what Newton’s work was more like. And some of how we have this legend. This is not to say that the 1660s were not astounding times for Newton, nor to deny that he worked with a rare genius. It’s more that we are lying to imagine that Newton looked around, saw London was even more a deathtrap than usual, and decided to go off to the country and toss out a new and unique understanding of the infinitesimal and the continuum.

Classroom. The teacher has drawn a geometric ray on the blackboard. Student: 'So that goes on forever? Should we warn people in the hallway?!'
Mark Anderson’s Andertoons rerun for the 27th of May, 2020. It ran at least as recently as the 3rd of August, 2017. and I noticed it then. This, that, and other essays featuring Andertoons can be found at this link.

Mark Anderson’s Andertoons for the 27th is the Mark Anderson’s Andertoons for the week. One of the students — not Wavehead — worries that a geometric ray, going on forever, could endanger people. There’s some neat business going on here. Geometry, like much mathematics, works on abstractions that we take to be universally true. But it also seems to have a great correspondence to ordinary real-world stuff. We wouldn’t study it if it didn’t. So how does that idealization interact with the reality? If the ray represented by those marks on the board goes on to do something, do we have to take care in how it’s used?

Olivia Jaimes’s Nancy for the 29th is set in a (virtual) arithmetic class. It builds on the conflation between “nothing” and “zero”.


And that wraps up my week in comic strips. I keep all my Reading the Comics posts at this link. I am also hoping to start my All 2020 Mathematics A-to-Z shortly, and am open for nominations for topics for the first couple letters. Thank you for reading.

Reading the Comics, May 9, 2020: Knowing the Angles Edition


There were a couple more comic strips in the block of time I want to write about. Only one’s got some deeper content and, I admit, I had to work to find it.

Bob Scott’s Bear With me for the 7th has Bear offering the answer from mathematics class, late.

Jerry Bittle’s Shirley and Sons Classic rerun for the 7th has Louis struggling on an arithmetic test.

Olivia Jaimes’s Nancy for the 8th has Nancy and Sluggo avoiding mathematics homework. Or, “practice”, anyway. There’s more, though; Nancy and Sluggo are doing some analysis of viewing angles. That’s actual mathematics, certainly. Computer-generated imagery depends on it, just like you’d imagine. There are even fun abstract questions that can give surprising insights into numbers. For example: imagine that space were studded, at a regular square grid spacing, with perfectly reflective marbles of uniform size. Is there, then, a line of sight between any two points outside any marbles? Even if it requires tens of millions of reflections; we’re interested in what perfect reflections would give us.

Aunt Fritzi: 'You two were supposed to be doing math practice, not playing cards.' Nancy, holding a fan of cards out and showing a geometric figure with several lines marked off: 'For your information, we were using these to measure angles.' [ Earlier ] Nancy and Sluggo look over the chart; the cards are spread out from a post-it note with a sketch of Aunt Frizi in it. It shows lines of sight. Nancy, in flashback: 'At this angle, she won't be able to see us playing cards.'
Olivia Jaimes’s Nancy for the 8th of May, 2020. When I have reason to discuss Nancy in a Reading the Comics post, I try to tag it so it’ll appear here.

Using playing cards as a makeshift protractor is a creative bit of making do with what you have. The cards spread in a fanfold easily enough and there’s marks on the cards that you can use to keep your measurements reasonably uniform. Creating ad hoc measurement tools like this isn’t mathematics per se. But making a rough tool is a first step to making a precise tool. And you can use reason to improve your estimates.

It’s not on-point, but I did want to share the most wondrous ad hoc tool I know of: You can use an analog clock hand, and the sun, as a compass. You don’t even need a real clock; you can draw the time on a sheet of paper and use that. It’s not a precise measure, of course. But if you need some help, here you go. You’ve got it.

Tony Rubino and Gary Markstein’s Daddy’s Home for the 9th has Elliot avoiding doing his mathematics homework.


And that’s got the last week covered. Some more comic strips should follow at a link here, soon. And I hope to have some other stuff to announce here, soon.

Reading the Comics, May 2, 2020: What Is The Cosine Of Six Edition


The past week was a light one for mathematically-themed comic strips. So let’s see if I can’t review what’s interesting about them before the end of this genially dumb movie (1940’s Hullabaloo, starring Frank Morgan and featuring Billie Burke in a small part). It’ll be tough; they’re reaching a point where the characters start acting like they care about the plot either, which is usually the sign they’re in the last reel.

Patrick Roberts’s Todd the Dinosaur for the 26th of April presents mathematics homework as the most dreadful kind of homework.

Jenny Campbell’s Flo and Friends for the 26th is a joke about fumbling a bit of practical mathematics, in this case, cutting a recipe down. When I look into arguments about the metric system, I will sometimes see the claim that English traditional units are advantageous for cutting down a recipe: it’s quite easy to say that half of “one cup” is a half cup, for example. I doubt that this is much more difficult than working out what half of 500 ml is, and my casual inquiries suggest that nobody has the faintest idea what half of a pint would be. And anyway none of this would help Ruthie’s problem, which is taking two-fifths of a recipe meant for 15 people. … Honestly, I would have just cut it in half and wonder who’s publishing recipes that serve 15.

Bear dressed kind of as Flash Gordon: 'Sorry, Tofu, but there ain't no controlling these muscles!' Cat dressed as a wizard, 'Without a rested mind, you cannot visualize the future.' He sighs, takes out a sheet of paper, and thinks hard; he's surrounded by algebraic equations. Then he flips and folds and bends the paper over and over until it turns into an origami car that looks like the Monopoly game piece. The bear is amazed; the cat says, 'Visualization. Come find me when you've rested your mind.'
Ed Bickford and Aaron Walther’s American Chop Suey for the 28th of April, 2020. I don’t seem to have ever written about this strip before, which does not surprise me. So I have a new tag, then. This and any future essays about American Chop Suey should appear at this link.

Ed Bickford and Aaron Walther’s American Chop Suey for the 28th uses a panel of (gibberish) equations to represent deep thinking. It’s in part of a story about an origami competition. This interests me because there is serious mathematics to be done in origami. Most of these are geometry problems, as you might expect. The kinds of things you can understand about distance and angles from folding a square may surprise. For example, it’s easy to trisect an arbitrary angle using folded squares. The problem is, famously, impossible for compass-and-straightedge geometry.

Origami offers useful mathematical problems too, though. (In practice, if we need to trisect an angle, we use a protractor.) It’s good to know how to take a flat, or nearly flat, thing and unfold it into a more interesting shape. It’s useful whenever you have something that needs to be transported in as few pieces as possible, but that on site needs to not be flat. And this connects to questions with pleasant and ordinary-seeming names like the map-folding problem: can you fold a large sheet into a small package that’s still easy to open? Often you can. So, the mathematics of origami is a growing field, and one that’s about an accessible subject.

Nate Fakes’s Break of Day for the 29th is the anthropomorphic-symbols joke for the week, with an x talking about its day job in equations and its free time in games like tic-tac-toe.

Bill Holbrook’s On The Fastrack for the 2nd of May also talks about the use of x as a symbol. Curt takes eagerly to the notion that a symbol can represent any number, whether we know what it is or not. And, also, that the choice of symbol is arbitrary; we could use whatever symbol communicates. I remember getting problems to work in which, say, 3 plus a box equals 8 and working out what number in the box would make the equation true. This is exactly the same work as solving 3 + x = 8. Using an empty box made the problem less intimidating, somehow.

Students taking a math test. One is demanding of his phone, 'Siri, what is the cosine of 174 degrees?' The teacher looks astonished. In the corner joke a squirrel says, 'It's better than waiting for some kind of cosine from above.'
Dave Whamond’s Reality Check for the 2nd of May, 2020. Essays discussing something mentioned in Reality Check are gathered at this link.

Dave Whamond’s Reality Check for the 2nd is, really, a bit baffling. It has a student asking Siri for the cosine of 174 degrees. But it’s not like anyone knows the cosine of 174 degrees off the top of their heads. If the cosine of 174 degrees wasn’t provided in a table for the students, then they’d have to look it up. Well, more likely they’d be provided the cosine of 6 degrees; the cosine of an angle is equal to minus one times the cosine of 180 degrees minus that same angle. This allows table-makers to reduce how much stuff they have to print. Still, it’s not really a joke that a student would look up something that students would be expected to look up.

… That said …

If you know anything about trigonometry, you know the sine and cosine of a 30-degree angle. If you know a bit about trigonometry, and are willing to put in a bit of work, you can start from a regular pentagon and work out the sine and cosine of a 36-degree angle. And, again if you know anything about trigonometry, you know that there are angle-addition and angle-subtraction formulas. That is, if you know the cosine of two angles, you can work out the cosine of the difference between them.

So, in principle, you could start from scratch and work out the cosine of 6 degrees without using a calculator. And the cosine of 174 degrees is minus one times the cosine of 6 degrees. So it could be a legitimate question to work out the cosine of 174 degrees without using a calculator. I can believe in a mathematics class which has that as a problem. But that requires such an ornate setup that I can’t believe Whamond intended that. Who in the readership would think the cosine of 174 something to work out by hand? If I hadn’t read a book about spherical trigonometry last month I wouldn’t have thought the cosine of 6 a thing someone could reasonably work out by hand.

I didn’t finish writing before the end of the movie, even though it took about eighteen hours to wrap up ten minutes of story. My love came home from a walk and we were talking. Anyway, this is plenty of comic strips for the week. When there are more to write about, I’ll try to have them in an essay at this link. Thanks for reading.

Reading the Comics, April 6, 2020: My Perennials Edition


As much as everything is still happening, and so much, there’s still comic strips. I’m fortunately able here to focus just on the comics that discuss some mathematical theme, so let’s get started in exploring last week’s reading. Worth deeper discussion are the comics that turn up here all the time.

Lincoln Peirce’s Big Nate for the 5th is a casual mention. Nate wants to get out of having to do his mathematics homework. This really could be any subject as long as it fit the word balloon.

John Hambrock’s The Brilliant Mind of Edison Lee for the 6th is a funny-answers-to-story-problems joke. Edison Lee’s answer disregards the actual wording of the question, which supposes the group is travelling at an average 70 miles per hour. The number of stops doesn’t matter in this case.

Mark Anderson’s Andertoons for the 6th is the Mark Anderson’s Andertoons for the week. In it Wavehead gives the “just use a calculator” answer for geometry problems.

On the blackboard: Perimeter, with a quadrilateral drawn, the sides labelled A, B, C, and D, and the formula A + B + C + D on the board. Wavehead asks the teacher, 'Or you could just walk around thet edge and let your fitness tracker tell you the distance.'
Mark Anderson’s Andertoons for the 6th of April, 2020. I haven’t mentioned this strip in two days. Essays featuring Andertoons are at this link, though.

Not much to talk about there. But there is a fascinating thing about perimeters that you learn if you go far enough in Calculus. You have to get into multivariable calculus, something where you integrate a function that has at least two independent variables. When you do this, you can find the integral evaluated over a curve. If it’s a closed curve, something that loops around back to itself, then you can do something magic. Integrating the correct function on the curve around a shape will tell you the enclosed area.

And this is an example of one of the amazing things in multivariable calculus. It tells us that integrals over a boundary can tell us something about the integral within a volume, and vice-versa. It can be worth figuring out whether your integral is better solved by looking at the boundaries or at the interiors.

Heron’s Formula, for the area of a triangle based on the lengths of its sides, is an expression of this calculation. I don’t know of a formula exactly like that for the perimeter of a quadrilateral, but there are similar formulas if you know the lengths of the sides and of the diagonals.

Richard Thompson’s Cul de Sac rerun for the 6th sees Petey working on his mathematics homework. As with the Big Nate strip, it could be any subject.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th depicts, fairly, the sorts of things that excite mathematicians. The number discussed here is about algorithmic complexity. This is the study of how long it takes to do an algorithm. How long always depends on how big a problem you are working on; to sort four items takes less time than sorting four million items. Of interest here is how much the time to do work grows with the size of whatever you’re working on.

Caption: 'Mathematicians are weird.' Mathematician: 'You know that thing that was 2.3728642?' Group of mathematicians: 'Yes?' Mathematician; 'I got it down to 2.3728639.' The mathematicians burst out into thunderous applause.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th of April, 2020. I haven’t mentioned this strip in two days. Essays featuring Saturday Morning Breakfast Cereal are at this link, though.

The mathematician’s particular example, and I thank dtpimentel in the comments for finding this, is about the Coppersmith–Winograd algorithm. This is a scheme for doing matrix multiplication, a particular kind of multiplication and addition of squares of numbers. The squares have some number N rows and N columns. It’s thought that there exists some way to do matrix multiplication in the order of N2 time, that is, if it takes 10 time units to multiply matrices of three rows and three columns together, we should expect it takes 40 time units to multiply matrices of six rows and six columns together. The matrix multiplication you learn in linear algebra takes on the order of N3 time, so, it would take like 80 time units.

We don’t know the way to do that. The Coppersmith–Winograd algorithm was thought, after Virginia Vassilevska Williams’s work in 2011, to take something like N2.3728642 steps. So that six-rows-six-columns multiplication would take slightly over 51.796 844 time units. In 2014, François le Gall found it was no worse than N2.3728639 steps, so this would take slightly over 51.796 833 time units. The improvement doesn’t seem like much, but on tiny problems it never does. On big problems, the improvement’s worth it. And, sometimes, you make a good chunk of progress at once.


I’ll have some more comic strips to discuss in an essay at this link, sometime later this week. Thanks for reading.

Reading the Comics, April 4, 2020: Ruling Things Out Edition


This little essay should let me wrap up the rest of the comic strips from the past week. Most of them were casual mentions. At least I thought they were when I gathered them. But let’s see what happens when I actually write my paragraphs about them.

Darrin Bell and Theron Heir’s Rudy Park rerun for the 1st of April uses arithmetic as emblematic of things which we know with certainty to be true.

Thaves’s Frank and Ernest for the 2nd is a bit of wordplay, having Euclid and Galileo talking about parallel universes. I’m not sure that Galileo is the best fit for this, but I’m also not sure there’s another person connected who could be named. It’d have to be a name familiar to an average reader as having something to do with geometry. Pythagoras would seem obvious, but the joke is stronger if it’s two people who definitely did not live at the same time. Did Euclid and Pythagoras live at the same time? I am a mathematics Ph.D. and have been doing pop mathematics blogging for nearly a decade now, and I have not once considered the question until right now. Let me look it up.

It doesn’t make any difference. The comic strip has to read quickly. It might be better grounded to post Euclid meeting Gauss or Lobachevsky or Euler (although the similarity in names would be confusing) but being understood is better than being precise.

Stephan Pastis’s Pearls Before Swine for the 2nd is a strip about the foolhardiness of playing the lottery. And it is foolish to think that even a $100 purchase of lottery tickets will get one a win. But it is possible to buy enough lottery tickets as to assure a win, even if it is maybe shared with someone else. It’s neat that an action can be foolish if done in a small quantity, but sensible if done in enough bulk.

Chalkboard problem 10 - 7, with answers given and crossed out of 0, 5, 7, 4, 17, 9, 1, 2, and 70. Wavehead, to teacher: 'OK, the good news is we've ruled these out.'
Mark Anderson’s Andertoons for the 3rd of April, 2020. This is actually the first time I’ve mentioned this strip in two months. But any time I discuss a topic raised by Andertoons should appear at this link.

Mark Anderson’s Andertoons for the 3rd is the Mark Anderson’s Andertoons for the week. Wavehead has made a bunch of failed attempts at subtracting seven from ten, but claims it’s at least progress that some thing have been ruled out. I’ll go along with him that there is some good in ruling out wrong answers. The tricky part is in how you rule them out. For example, obvious to my eye is that the correct answer can’t be more than ten; the problem is 10 minus a positive number. And it can’t be less than zero; it’s ten minus a number less than ten. It’s got to be a whole number. If I’m feeling confident about five and five making ten, then I’d rule out any answer that isn’t between 1 and 4 right away. I’ve got the answer down to four guesses and all I’ve really needed to know is that 7 is greater than five but less than ten. That it’s an even number minus an odd means the result has to be odd; so, it’s either one or three. Knowing that the next whole number higher than 7 is an 8 says that we can rule out 1 as the answer. So there’s the answer, done wholly by thinking of what we can rule out. Of course, knowing what to rule out takes some experience.

Mark Parisi’s Off The Mark for the 4th is roughly the anthropomorphic numerals joke for the week. It’s a dumb one, but, that’s what sketchbooks are for.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th is the Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th for the week. It shows in joking but not wrong fashion a mathematical physicist’s encounters with orbital mechanics. Orbital mechanics are a great first physics problem. It’s obvious what they’re about, and why they might be interesting. And the mathematics of it is challenging in ways that masses on springs or balls shot from cannons aren’t.

How To Learn Orbital Mechanics. Step 1: Gauge Difficulty. Person reading a text: 'It's Newtonian! Piece of cake. Just a bunch of circles and dots.' Step 2: Correction. 'OK, *ellipses* and dots.' Step 3: Concern. 'Oh, Christ, sometimes there are more than two dots.' Step 4: Pick an easier subject. 'I'm gonna go study quantum computing.' The textbook is in the trash.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th of April, 2020. This is actually the first time I’ve mentioned this strip ina week. But any time I discuss a topic raised in Saturday Morning Breakfast Cereal should appear at this link.

A few problems are very easy, like, one thing in circular orbit of another. A few problems are not bad, like, one thing in an elliptical or hyperbolic orbit of another. All our good luck runs out once we suppose the universe has three things in it. You’re left with problems that are doable if you suppose that one of the things moving is so tiny that it barely exists. This is near enough true for, for example, a satellite orbiting a planet. Or by supposing that we have a series of two-thing problems. Which is again near enough true for, for example, a satellite travelling from one planet to another. But these is all work that finds approximate solutions, often after considerable hard work. It feels like much more labor to smaller reward than we get for masses on springs or balls shot from cannons. Walking off to a presumably easier field is understandable. Unfortunately, none of the other fields is actually easier.

Pythagoras died somewhere around 495 BC. Euclid was born sometime around 325 BC. That’s 170 years apart. So Pythagoras was as far in Euclid’s past as, oh, Maria Gaetana Agnesi is to mine.

I did a little series looking into orbital mechanics, not necessarily ones that look like planetary orbits, a couple years ago. You might enjoy that. And I figure to have more mathematically-themed comic strips in the near future. Thanks for reading.

The Playful Math Education Blog Carnival #136


Greetings, friends, and thank you for visiting the 136th installment of Denise Gaskins’s Playful Math Education Blog Carnival. I apologize ahead of time that this will not be the merriest of carnivals. It has not been the merriest of months, even with it hosting Pi Day at the center.

Playful Math Education Blog Carnival banner, showing a coati dressed in bright maroon ringmaster's jacket and top hat, with multiplication and division signs sitting behind atop animal-training podiums; a greyscale photograph audience is in the far background.
Banner art again by Thomas K Dye, creator of Newshounds, Infinity Refugees, Something Happens, and his current comic strip, Projection Edge. You can follow him on Patreon and read his comic strip nine months ahead of its worldwide publication. The banner art was commissioned several weeks ago when I expected I would be in a more playful mood this week.

In consideration of that, let me lead with Art in the Time of Transformation by Paula Beardell Krieg. This is from the blog Playful Bookbinding and Paper Works. The post particularly reflects on the importance of creating a thing in a time of trouble. There is great beauty to find, and make, in symmetries, and rotations, and translations. Simple polygons patterned by simple rules can be accessible to anyone. Studying just how these symmetries and other traits work leads to important mathematics. Thus how Kreig’s page has recent posts with names like “Frieze Symmetry Group F7” but also to how symmetry is for five-year-olds. I am grateful to Goldenoj for the reference.

Kreig’s writing drew the attention of another kind contributor to my harvesting. Symmetry and Multiplying Negative Numbers explores one of those confusing things about negative numbers: how can a negative number times a negative number be positive? One way to understand this is to represent arithmetic operations as geometric operations. Particularly, we can see negation as a reflection.

That link was brought to my attention by Iva Sallay, another longtime friend of my little writings here. She writes fun pieces about every counting number, along with recreational puzzles. And asked to share 1458 Tangrams Can Be A Pot of Gold, as an example of what fascinating things can be found in any number. This includes a tangram. Tangrams we see in recreational-mathematics puzzles based on ways that you can recombine shapes. It’s always exciting to be able to shift between arithmetic and shapes. And that leads to a video and related thread again pointed to me by goldenoj …

This video, by Mathologer on YouTube, explains a bit of number theory. Number theory is the field of asking easy questions about whole numbers, and then learning that the answers are almost impossible to find. I exaggerate, but it does often involve questions that just suppose you understand what a prime number should be. And then, as the title asks, take centuries to prove.

Fermat’s Two-Squares Theorem, discussed here, is not the famous one about a^n + b^2 = c^n . Pierre de Fermat had a lot of theorems, some of which he proved. This one is about prime numbers, though, and particularly prime numbers that are one more than a multiple of four. This means it’s sometimes called Fermat’s 4k+1 Theorem, which is the name I remember learning it under. (k is so often a shorthand for “some counting number” that people don’t bother specifying it, the way we don’t bother to say “x is an unknown number”.) The normal proofs of this we do in the courses that convince people they’re actually not mathematics majors.

What the video offers is a wonderful alternate approach. It turns key parts of the proof into geometry, into visual statements. Into sliding tiles around and noticing patterns. It’s also a great demonstration of one standard problem-solving tool. This is to look at a related, different problem that’s easier to say things about. This leads to what seems like a long path from the original question. But it’s worth it because the path involves thinking out things like “is the count of this thing odd or even”? And that’s mathematics that you can do as soon as you can understand the question.

Iva Sallay also brought up Jenna Laib’s Making Meaning with Arrays: More Preschooler Division which similarly sees numerical truths revealed through geometric reasoning. Here, particularly, by the problem of baking muffins and thinking through how to divide them up. A key piece here, for a particular child’s learning, was being able to pick up and move things around. Often in shifting between arithmetic and geometry we suppose that we can rearrange things without effort. As adults it’s easy to forget that this is an abstraction that we need to learn.

Sharing of food, in this case cookies, appears in Helena Osana’s Mathematical thinking begins in the early years with dialogue and real-world exploration. Mathematic, Osana notes, is primarily about thinking. An important part in mathematics education is working out how the thinking children most like to do can also find mathematics.

I again thank Iva Sallay for that link, as well as this essay. Dan Meyer’s But Artichokes Aren’t Pinecones: What Do You Do With Wrong Answers? looks at the problem of students giving wrong answers. There is no avoiding giving wrong answers. A parent’s or teacher’s response to wrong answers will vary, though, and Meyer asks why that is. Meyer has some hypotheses. His example notes that he doesn’t mind a child misidentifying an artichoke as a pinecone. Not in the same way identifying the sum of 1 and 9 as 30 would. What is different about those mistakes?

Jessannwa’s Soft Start In The Intermediate Classroom looks to the teaching of older students. No muffins and cookies here. That the students might be more advanced doesn’t change the need to think of what they have energy for, and interest in. She discusses a class setup that’s meant to provide structure in ways that don’t feel so authority-driven. And ways to turn practicing mathematics problems into optimizing game play. I will admit this is a translation of the problem which would have worked well for me. But I also know that not everybody sees a game as, in part, something to play at maximum efficiency. It depends on the game, though. They’re on Twitter as @jesannwa.

Speaking of the game, David Coffey’s Creating Positive Change in Math Class was written in anticipation of the standardized tests meant to prove out mathematics education. Coffey gets to thinking about how to frame teaching to more focus on why students should have a skill, and how they can develop it. How to get students to feel involved in their work. Even how to get students to do homework more reliably. Coffey’s scheduled to present at the Michigan Council of Teachers of Mathematics conference in Grand Rapids this July. This if all starts going well. And this is another post I know of thanks to Goldenoj.

These are thoughts about how anyone can start learning mathematics. What does it look like to have learned a great deal, though, to the point of becoming renowned for it? Life Through A Mathematician’s Eyes posted Australian Mathematicians in late January. It’s a dozen biographical sketches of Australian mathematicians. It also matches each to charities or other public-works organizations. They were trying to help the continent through the troubles it had even before the pandemic struck. They’re in no less need for all that we’re exhausted. The page’s author is on Twitter as @lthmath.

Mathematical study starts small, though. Often it starts with games. There are many good ones, not least Iva Sallay’s Find the Factors puzzles.

Besides that, Dads Worksheets has provided a set of Math Word Search Puzzles. It’s a new series from people who create worksheets for many grade levels and many aspects of mathematics. They’re on Twitter as @dadsworksheets.

Mr Wu, of the Singapore Math Tuition blog, has also begun a new series of recreational mathematics puzzles. He lays out the plans for this, puzzles aimed at children around eight to ten years old. One of the early ones is the Stickers Math Question. A more recent one is The Secret of the Sweets (Sweet Distribution Problem). Mr Wu can be found on Twitter as @mathtuition88.

Denise Gaskins, on Twitter as @letsplaymath, and indefatigable coordinator for this carnival, offers the chance to Play Math with Your Kids for Free. This is an e-book sampler of mathematics gameplay.

I have since the start of this post avoided mentioning the big mathematical holiday of March. Pi Day had the bad luck to fall on a weekend this year, and then was further hit by the Covid-19 pandemic forcing the shutdown of many schools. Iva Sallay again helped me by noting YummyMath’s activities page It’s Time To Gear Up For Pi Day. This hosts several worksheets, about the history of π and ways to calculate it, and several formulas for π. This even gets into interesting techniques like how to use continued fractions in finding a numerical value.

The Guys and Good Health blog presented Happy Pi Day on the 14th, with — in a move meant to endear the blog to me — several comic strips. This includes one from Grant Snider, who draws lovely strips. I’m sad that his Incidental Comics has left GoComics.com, so I can’t feature it often during my Reading the Comics roundups anymore.

Virtual Brush Box, meanwhile, offers To Celebrate Pi Day, 10 Examples of Numbers and 10 Examples of Math Involved with Horses which delights me by looking at π, and mathematics, as they’re useful in horse-related activities. This may be the only blog post written specifically for me and my sister, and I am so happy that there is the one.

There’s a bit more, a bit of delight. It was my greatest surprise in looking for posts for this month. That is poetry. I mean this literally.

Whimsy-Mimsy wrote on Pi Day a haiku.

D Avery, on Shift N Shake, wrote the longer Another Slice of Pi Day, the third year of their composing poems observing the day.

Rolands Rag Bag shared A Pi-Ku for Pi-Day featuring a poem written in a form I wasn’t aware anyone did. The “Pi-Ku” as named here has 3 syllables for the first time, 1 syllable in the second line, 4 syllables in the third line, 1 syllable the next line, 5 syllables after that … you see the pattern. (One of Avery’s older poems also keeps this form.) The form could, I suppose, go on to as many lines as one likes. Or at least to the 40th line, when we would need a line of zero syllables. Probably one would make up a rule to cover that.

Blind On The Light Side similarly wrote Pi poems, including a Pi-Ku, for March 12, 2020. These poems don’t reach long enough to deal with the zero-syllable line, but we can forgive someone not wanting to go on that long.

As a last note, I have joined Mathstodon, the Mastodon instance with a mathematics theme. You can follow my shy writings there as @nebusj@mathstodon.xyz, or follow a modest number of people talking, largely, about mathematics. Mathstodon is a mathematically-themed microblogging site. On WordPress, I do figure to keep reading the comics for their mathematics topics. And sometime this year, when I feel I have the energy, I hope to do another A to Z, my little glossary project.

And this is what I have to offer. I hope the carnival has brought you some things of interest, and some things of delight. And, if I may, please consider this Grant Snider cartoon, Hope.

Life Through A Mathematician’s Eyes is scheduled to host the 137th installment of the Playful Math Education Blog Carnival, at the end of April. I look forward to seeing it. Good luck to us all.

Reading the Comics, February 1, 2020: I Never Talk About Marvin Edition


There’s some comic strips that get mentioned here all the time. Then there’s comic strips that I have been reading basically my whole life, and that never give me a thread to talk about. Although I’ve been reading comic strips for their mathematics content for a long while now, somehow, I am still surprised when these kinds of comic strip are not the same thing. So here’s the end of last week’s comics, almost in time for next week to start:

Kevin Fagan’s Drabble for the 28th has Penny doing “math” on colors. Traditionally I use an opening like this to mention group theory. In that we study things that can be added together, in ways like addition works on the integers. Colors won’t quite work like this, unfortunately. A group needs an element that’s an additive identity. This works like zero: it can be added to anything without changing its value. There isn’t a color that you can mix with other colors that leaves the other color unchanged, though. Even white or clear will dilute the original color.

Mom: 'How was school today, Penny?' Penny: 'Great, Mommy! I learned how to do math! Want me to show you? Blue plus red equals purple!'
Kevin Fagan’s Drabble for the 28th of January, 2020. It doesn’t come up often, but when it does, Drabble appears in essays at this link.

If you’ve thought of the clever workaround, that each color can be the additive identity to itself, you get credit for ingenuity. Unfortunately, to be a group there has to be a lone additive identity. Having more than one makes a structure that’s so unlike the integers that mathematicians won’t stand for it. I also don’t know of any interesting structures that have more than one additive identity. This suggests that nobody has found a problem that they represent well. But the strip suggests maybe it could tell us something useful for colors. I don’t know.

Marvin: 'After all the talk about 'fake news' I'm starting to question EVERYTHING big people tell me.' He's looking at a teacher holding up the flashcard 1 + 1 = 2.
Tom Armstrong’s Marvin for the 28th of January, 2020. I don’t think it has ever come up before, but what the heck. Any essays which mention Marvin should be at this link.

Tom Armstrong’s Marvin for the 28th is a strip which follows from the discovery that “fake news” is a thing that people say. Here the strip uses a bit of arithmetic as the sort of incontrovertibly true thing that Marvin is dumb to question. Well, that 1 + 1 equals 2 is uncontrovertibly true, unless we are looking at some funny definitions of ‘1’ or ‘plus’ or something. I remember, as a kid, being quite angry with a book that mentioned “one cup of popcorn plus one cup of water does not give us two cups of soggy popcorn”, although I didn’t know how to argue the point.

Title: 'The Math Homework.' Dad, in the kitchen, to kid: 'What's surface area? Ask your mother.' The mother is in the kitchen, working, and has every bit of surface area that isn't being used for homework with cooking tools. Footer joke: Mom asks, 'Can you please move? I need this space.'
Hilary Price and Rina Piccolo’s Rhymes with Orange for the 30th of January, 2020. Essays with some mention of Rhymes With Orange should be at this link.

Hilary Price and Rina Piccolo’s Rhymes with Orange for the 30th is … well, I’m in this picture and I don’t like it. I come from a long line of people who cover every surface with stuff. But as for what surface area is? … Well, there’s a couple of possible definitions. One that I feel is compelling is to think of covering sets. Take a shape that’s set, by definition, to have an area of 1 unit of area. What is the smallest number of those unit shapes which will cover the original shape? Cover is a technical term here. But also, here, the ordinary English word describes what we need it for. How many copies of the unit shape do you need to exactly cover up the whole original shape? That’s your area. And this fits to the mother’s use of surfaces in the comic strip neatly enough.

Mutt: 'What's the matter, you stuck?' Jeff, looking at his car: 'Yes and no! I tried the cary products they advertise on TV. They claimed this car would use 50% less gas. Then I bought a carburettor which saves 30%, special spark plugs which save 20% and a new brand of gas which saved 10%! Now when I drive the gas tank overflows!' Jeff shows gas pouring out of the tank.
Bud Fisher’s Mutt and Jeff for the 31st of January, 2020. And the essays which have mentioned Mutt and Jeff comics appear at this link.

Bud Fisher’s Mutt and Jeff for the 31st is a rerun of vintage unknown to me. I’m not sure whether it’s among the digitally relettered strips. The lettering’s suspiciously neat, but, for example, there’s at least three different G’s in there. Anyway, it’s an old joke about adding together enough gas-saving contraptions that it uses less than zero gas. So far as it’s tenable at all, it comes from treating percentage savings from different schemes as additive, instead of multiplying together. Also, I suppose, that the savings are independent, that (in this case) Jeff’s new gas saving ten percent still applies even with the special spark plugs or the new carburettor [sic]. The premise is also probably good for a word problem, testing out understanding of percentages and multiplication, which is just a side observation here.


This wraps up last week’s mathematically-themed comic strips. This week I can tell you already was a bonanza week. When I start getting to its comics I should have an essay at this link. Thanks for reading.

Reading the Comics, January 4, 2020: The Little Things Edition


Today’s essay is just to mention the comic strips which, last week, said mathematics but in some incidental way. Or some way that I can’t write a reasonable blog entry for.

Gary Larson’s The Far Side reruns for the 30th of December, 2019, included this classic about curiosity killing cats. This 1985 strip rates a mention because a blackboard of mathematical symbols gets used to represent their intellectual inquiries.

Bill Amend’s FoxTrot for the 29th, a Sunday and thus new strip, is some wordplay based on the Disney+ line of entertainment product.

Jim Meddick’s Monty for the 29th has the time-travelling Professor Xemit (get it?) show a Times Square Ball Drop of the future. The ball gets replaced with a “demihypercube”, the idea being that the future will have some more complicated geometry than a mere “ball”. There is no such thing as “a” demihypercube, in the same way there is not “a” pentagon. There is a family of shapes, all called demihypercubes. There’s a variety of ways to represent them. A reasonable one, though, is a roughly spherical shape made of pointy triangles all over. It wouldn’t look absurd. There are probably time ball drops that use something like a demihypercube already.

Ruben Bolling’s Super-Fun-Pak Comix rerun for the 1st of January, 2020 features a Comics For The Elderly speaking of the advantages an abacus has over a spreadsheet.

Neal Rubin and Rod Whigham’s Gil Thorp for the 2nd has one of the student athletes working on calculus. And coach Mimi Thorp is doing the mathematics of studying athlete performance. If this strip makes you curious, too, my other blog should this Sunday recap what’s going on in Gil Thorp.

Also this coming Sunday I should look at more mathematically-themed comic strips. That should appear at this link, unless something urgent commands my attention first. Thank you.

Reading the Comics, December 25, 2019: Running Out The Year Edition


The last full week of the year had, again, comic strips that mostly mention mathematics without getting into detail. That’s all right. I have a bit of a cold so I’m happy not to have to compose thoughts about too many of them.

John Zakour and Scott Roberts’s Maria’s Day for the 22nd has Maria finishing, and losing, her mathematics homework. I suppose the implication’s that she couldn’t hope to reconstruct it before class. It’s not like she could re-write a short essay for history, though.

Percy Crosby’s Skippy for the 23rd has Skippy and Sookie doing the sort of story problem arithmetic of working out a total bill. The strip originally ran the 11th of August, 1932.

Cy Olson’s Office Hours for the 24th, which originally ran the 14th of October, 1971, comes the nearest to having enough to talk about here. The secretary describes having found five different answers in calculating the profits and so used the highest one. The joke is on incompetent secretaries, yes. But it is respectable, if trying to understand something very complicated, to use several different models for what one wants to know. These will likely have different values, although how different they are, and how changes in one model tracks changes in another, can be valuable. We’re accustomed to this, at least in the United States, by weather forecasts: any local weather report will describe expected storms by different models. These use different ideas about how much moisture moves into the air, how fast raindrops will form (a very difficult problem), how winds will shift, that sort of thing. It’s defensible to make similar different models for reporting the health of a business, particularly if company owns things with a price that can’t be precisely stated.

Marguerite Dabaie and Tom Hart’s Ali’s House for the 24th continues a story from the week before in which a character imagines something tossing us out of three-dimensional space. A seven-dimensional space is interesting mathematically. We can define a cross product between vectors in three-dimensional space and in seven-dimensional space. Most other spaces don’t allow something like a cross product to be coherently defined. Seven-dimensional space also allows for something called the “exotic sphere”, which I hadn’t heard of before either. It’s a structure that’s topologically a sphere, but that has a different kind of structure. This isn’t unique to seven-dimensional space. It’s not known whether four-dimensional space has exotic spheres, although many spaces higher than seven dimensions have them.

Gordon Bess’s Redeye for the 25th of December has Pokey asking his horse Loco to do arithmetic. There’s a long history of animals doing, or seeming to do, arithmetic. The strip originally ran the 23rd of August, 1973.

I’ll have some more comic strips to close out the year, I expect, which should appear at this link, most like on Tuesday. Thanks for reading.

Why does the Quantum Mechanics Momentum Operator look like that?


I don’t know. I say this for anyone this has unintentionally clickbaited, or who’s looking at a search engine’s preview of the page.

I come to this question from a friend, though, and it’s got me wondering. I don’t have a good answer, either. But I’m putting the question out there in case someone reading this, sometime, does know. Even if it’s in the remote future, it’d be nice to know.

And before getting to the question I should admit that “why” questions are, to some extent, a mug’s game. Especially in mathematics. I can ask why the sum of two consecutive triangular numbers a square number. But the answer is … well, that’s what we chose to mean by ‘triangular number’, ‘square number’, ‘sum’, and ‘consecutive’. We can show why the arithmetic of the combination makes sense. But that doesn’t seem to answer “why” the way, like, why Neil Armstrong was the first person to walk on the moon. It’s more a “why” like, “why are there Seven Sisters [ in the Pleiades ]?” [*]

But looking for “why” can, at least, give us hints to why a surprising result is reasonable. Draw dots representing a square number, slice it along the space right below a diagonal. You see dots representing two successive triangular numbers. That’s the sort of question I’m asking here.

From here, we get to some technical stuff and I apologize to readers who don’t know or care much about this kind of mathematics. It’s about the wave-mechanics formulation of quantum mechanics. In this, everything that’s observable about a system is contained within a function named \Psi . You find \Psi by solving a differential equation. The differential equation represents problems. Like, a particle experiencing some force that depends on position. This is written as a potential energy, because that’s easier to work with. But it’s the kind of problem done.

Grant that you’ve solved \Psi , since that’s hard and I don’t want to deal with it. You still don’t know, like, where the particle is. You never know that, in quantum mechanics. What you do know is its distribution: where the particle is more likely to be, where it’s less likely to be. You get from \Psi to this distribution for, like, particles by applying an operator to \Psi . An operator is a function with a domain and a range that are spaces. Almost always these are spaces of functions.

Each thing that you can possibly observe, in a quantum-mechanics context, matches an operator. For example, there’s the x-coordinate operator, which tells you where along the x-axis your particle’s likely to be found. This operator is, conveniently, just x. So evaluate x\Psi and that’s your x-coordinate distribution. (This is assuming that we know \Psi in Cartesian coordinates, ones with an x-axis. Please let me do that.) This looks just like multiplying your old function by x, which is nice and easy.

Or you might want to know momentum. The momentum in the x-direction has an operator, \hat{p_x} , which equals -\imath \hbar \frac{\partial}{\partial x} . The \partial is partial derivatives. The \hbar is Planck’s constant, a number which in normal systems of measurement is amazingly tiny. And you know how \imath^2 = -1 . That – symbol is just the minus or the subtraction symbol. So to find the momentum distribution, evaluate -\imath \hbar \frac{\partial}{\partial x}\Psi . This means taking a derivative of the \Psi you already had. And multiplying it by some numbers.

I don’t mind this multiplication by \hbar . That’s just a number and it’s a quirk of our coordinate system that it isn’t 1. If we wanted, we could set up our measurements of length and duration and stuff so that it was 1 instead.

But. Why is there a -\imath in the momentum operator rather than the position operator? Why isn’t one \sqrt{-\imath} x and the other \sqrt{-\imath} \frac{\partial}{\partial x} ? From a mathematical physics perspective, position and momentum are equally good variables. We tend to think of position as fundamental, but that’s surely a result of our happening to be very good at seeing where things are. If we were primarily good at spotting the momentum of things around us, we’d surely see that as the more important variable. When we get into Hamiltonian mechanics we start treating position and momentum as equally fundamental. Even the notation emphasizes how equal they are in importance, and treatment. We stop using ‘x’ or ‘r’ as the variable representing position. We use ‘q’ instead, a mirror to the ‘p’ that’s the standard for momentum. (‘p’ we’ve always used for momentum because … … … uhm. I guess ‘m’ was already committed, for ‘mass’. What I have seen is that it was taken as the first letter in ‘impetus’ with no other work to do. I don’t know that this is true. I’m passing on what I was told explains what looks like an arbitrary choice.)

So I’m supposing that this reflects how we normally set up \Psi as a function of position. That this is maybe why the position operator is so simple and bare. And then why the momentum operator has a minus, an imaginary number, and this partial derivative stuff. That if we started out with the wave function as a function of momentum, the momentum operator would be just the momentum variable. The position operator might be some mess with \imath and derivatives or worse.

I don’t have a clear guess why one and not the other operator gets full possession of the \imath though. I suppose that has to reflect convenience. If position and momentum are dual quantities then I’d expect we could put a mere constant like -\imath wherever we want. But this is, mostly, me writing out notes and scattered thoughts. I could be trying to explain something that might be as explainable as why the four interior angles of a rectangle are all right angles.

So I would appreciate someone pointing out the obvious reason these operators look like that. I may grumble privately at not having seen the obvious myself. But I’d like to know it anyway.


[*] Because there are not eight.

My 2019 Mathematics A To Z: The Game of ‘Y’


Today’s A To Z term is … well, my second choice. Goldenoj suggested Yang-Mills and I was so interested. Yang-Mills describes a class of mathematical structures. They particularly offer insight into how to do quantum mechanics. Especially particle physics. It’s of great importance. But, on thinking out what I would have to explain I realized I couldn’t write a coherent essay about it. Getting to what the theory is made of would take explaining a bunch of complicated mathematical structures. If I’d scheduled the A-to-Z differently, setting up matters like Lie algebras, maybe I could do it, but this time around? No such help. And I don’t feel comfortable enough in my knowledge of Yang-Mills to describe it without describing its technical points.

That said I hope that Jacob Siehler, who suggested the Game of ‘Y’, does not feel slighted. I hadn’t known anything of the game going in to the essay-writing. When I started research I was delighted. I have yet to actually play a for-real game of this. But I like what I see, and what I can think I can write about it.

Cartoony banner illustration of a coati, a raccoon-like animal, flying a kite in the clear autumn sky. A skywriting plane has written 'MATHEMATIC A TO Z'; the kite, with the letter 'S' on it to make the word 'MATHEMATICS'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Game of ‘Y’.

This is, as the name implies, a game. It has two players. They have the same objective: to create a ‘y’. Here, they do it by laying down tokens representing their side. They take turns, each laying down one token in a turn. They do this on a shape with three edges. The ‘y’ is created when there’s a continuous path of their tokens that reaches all three edges. Yes, it counts to have just a single line running along one edge of the board. This makes a pretty sorry ‘y’ but it suggests your opponent isn’t trying.

There are details of implementation. The board is a mesh of, mostly, hexagons. I take this to be for the same reason that so many conquest-type strategy games use hexagons. They tile space well, they give every space a good number of neighbors, and the distance from the centers of one neighbor to another is constant. In a square grid, the centers of diagonal neighbors are farther than the centers of left-right or up-down neighbors. Hexagons do well for this kind of game, where the goal is to fill space, or at least fill paths in space. There’s even a game named Hex, slightly older than Y, with similar rules. In that the goal is to draw a continuous path from one end of the rectangular grid to another. The grid of commercial boards, that I see, are around nine hexagons on a side. This probably reflects a desire to have a big enough board that games go on a while, but not so big that they go on forever

Mathematicians have things to say about this game. It fits nicely in game theory. It’s well-designed to show some things about game theory. It’s the kind of game which has perfect information game, for example. Each player knows, at all times, the moves all the players have made. Just look at the board and see where they’ve placed their tokens. A player might have forgotten the order the tokens were placed in, but that’s the player’s problem, not the game’s. Anyway in Y, the order of token-placing doesn’t much matter.

It’s also a game of complete information. Every player knows, at every step, what the other player could do. And what objective they’re working towards. One party, thinking enough, could forecast the other’s entire game. This comes close to the joke about the prisoners telling each other jokes by shouting numbers out to one another.

It is also a game in which a draw is impossible. Play long enough and someone must win. This even if both parties are for some reason trying to lose. There are ingenious proofs of this, but we can show it by considering a really simple game. Imagine playing Y on a tiny board, one that’s just one hex on each side. Definitely want to be the first player there.

So now imagine playing a slightly bigger board. Augment this one-by-one-by-one board by one row. That is, here, add two hexes along one of the sides of the original board. So there’s two pieces here; one is the original territory, and one is this one-row augmented territory. Look first at the original territory. Suppose that one of the players has gotten a ‘Y’ for the original territory. Will that player win the full-size board? … Well, sure. The other player can put a token down on either hex in the augmented territory. But there’s two hexes, either of which would make a path that connects the three edges of the board. The first player can put a token down on the other hex in the augmented territory, and now connects all three of the new sides again. First player wins.

All right, but how about a slightly bigger board? So take that two-by-two-by-two board and augment it, adding three hexes along one of the sides. Imagine a player’s won the original territory board. Do they have to win the full-size board? … Sure. The second player can put something in the augmented territory. But there’s again two hexes that would make the path connecting all three sides of the full board. The second player can put a token in one of those hexes. But the first player can put a token in the other of those. First player wins again.

How about a slightly bigger board yet? … Same logic holds. Really the only reason that the first player doesn’t always win is that, at some point, the first player screws up. And this is an existence proof, showing that the first player can always win. It doesn’t give any guidance into how to play, though. If the first player plays perfectly, she’s compelled to win. This is something which happens in many two-player, symmetric games. A symmetric game is one where either player has the same set of available moves, and can make the same moves with the same results. This proof needs to be tightened up to really hold. But it should convince you, at least, that the first player has an advantage.

So given that, the question becomes why play this game after you’ve decided who’ll go first? The reason you might if you were playing a game is, what, you have something else to do? And maybe you think you’ll make fewer mistakes than your opponent. One approach often used in symmetric games like this is the “pie rule”. The name comes from the story about how to slice a pie so you and your sibling don’t fight over the results. One cuts the pie, the other gets first pick of the slice, and then you fight anyway. In this game, though, one player makes a tentative first move. The other decides whether they will be Player One with that first move made or whether they’ll be Player Two, responding.

There are some neat quirks in the commercial Y games. One is that they don’t actually show hexes, and you don’t put tokens in the middle of hexes. Instead you put tokens on the spots that would be the center of the hex. On the board are lines pointing to the neighbors. This makes the board actually a string of triangles. This is the dual to the hex grid. It shows a set of vertices, and their connections, instead of hexes and their neighbors. Whether you think the hex grid or this dual makes it easier to tell when you’ve connected all three edges is a matter of taste. It does make the edges less jagged all around.

Another is that there will be three vertices that don’t connect to six others. They connect to five others, instead. Their spaces would be pentagons. As I understand the literature on this, this is a concession to game balance. It makes it easier for one side to fend off a path coming from the center.

It has geometric significance, though. A pure hexagonal grid is a structure that tiles the plane. A mostly hexagonal grid, with a couple of pentagons, though? That can tile the sphere. To cover the whole sphere you need something like at least twelve irregular spots. But this? With the three pentagons? That gives you a space that’s topographically equivalent to a hemisphere, or at least a slice of the sphere. If we do imagine the board to be a hemisphere covered, then the result of the handful of pentagon spaces is to make the “pole” closer to the equator.

So as I say the game seems fun enough to play. And it shows off some of the ways that game theorists classify games. And the questions they ask about games. Is the game always won by someone? Does one party have an advantage? Can one party always force a win? It also shows the kinds of approach game theorists can use to answer these questions. This before they consider whether they’d enjoy playing it.


I am excited to say that there’s just the one more time this year that I will realize: it’s Wednesday evening and I’m 1200 words short. Please stop in Thursday when I hope to have the letter Z represented. That, and all of this year’s A-to-Z essays, should appear at this link. And if that isn’t enough, I’ll feature some past essays on Friday and Saturday, and have most of my past A-to-Z essays at this link. Thank you.

A Geometry Thing That’s Left Me Unsettled


I came across a little geometry thing that left me unsettled, even as I have to admit it’s correct. Start with a two-dimensional space, or as the hew-mons call it, a plane. Draw a square with sides of length two and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1), (+1, -1), (-1, +1), and (-1, -1). Around each of these corners draw a circle of radius 1.

There is some largest circle that you can draw, centered on the origin, the dead center of the square, with Cartesian coordinates (0, 0), and that just touches all of the corners’ circles. It has a radius of a little under 0.414.

Now think of the three-dimensional analog. Three-dimensional space. Draw a box with sides all of length two and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1, +1), (+1, +1, -1), (+1, -1, +1), (+1, -1, -1), (-1, +1, +1), (-1, +1, -1), (-1, -1, +1), and (-1, -1, -1). Around each of these eight corners draw a circle of radius 1.

There is some largest sphere that you can draw, centered on the origin, the point with Cartesian coordinates (0, 0, 0), that just touches all of the corners’ circles. It has a radius of a little under 0.732.

Think of the four-dimensional analog. This is harder to sketch. But a four-dimensional hypercube, with each side of length 2 and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1, +1, +1), (+1, +1, +1, -1), (+1, +1, -1, +1), (+1, +1, -1, -1), and you know what? Will you let me pretend we listed all sixteen corners? Thanks. Around each of these corners draw a circle of radius 1.

There is some largest hypersphere you can draw, centered on the origin, the point with Cartesian coordinates (0, 0, 0, 0), and that just touches all of these corners’ circles. It has a radius of 1.

Keep going. Five-dimensional space, with corners like (+1, +1, +1, +1, +1). Six-dimensional space, with corners like (+1, +1, +1, +1, +1, +1). Seven-dimensional space. And so on.

Eventually, the space is vast enough that the radius of this largest-touching hypersphere is bigger than 2. That is, reaching out more than twice as far as the original box goes, this even though the corner hyperspheres line the edges of the box, and touch one another along those edges.

Non-Euclidean geometry has the reputation of holding deep, inscrutable mysteries. To say something is a non-Euclidean space, outside of a mathematical context, is to designate it as a place immune to reason and beyond human comprehension. This is not such a case. This is a completely Euclidean space; it’s just got a lot of dimensions to it. Strange things will happen.

Another weird, but to me not so unsettling matter, concerns the surface (or hypersurface) area and the volume of these spheres. The circumference of a unit circle is, famously, 2π. The area of a unit sphere is 4π. For a four-dimensional hypersphere the surface area is a bit bigger yet. And bigger again for five and six and seven dimensions. But at eight dimensions the surface area starts shrinking again, and it never grows again. Have a great enough number of dimensions and the unit hypersphere has almost zero surface area. The volume of a unit circle is π. Of a unit sphere, \frac43 \pi . For a four-dimensional hypersphere, \frac12 \pi^2 . For a five-dimensional hypersphere, \frac{8}{15}\pi^2 . It is never so large again; for six or more dimensions the volume starts to shrink again. As the number of dimensions of space grows, the volume of the unit hypersphere dwindles to zero.

You know, that’s unsettling me more now that I’m paying attention to it.

Reading the Comics, November 13, 2019: I Could Have Posted This Wednesday Edition


Now let me discuss the comic strips from last week with some real meat to their subject matter. There weren’t many: after Wednesday of last week there were only casual mentions of any mathematics topic. But one of the strips got me quite excited. You’ll know which soon enough.

Mac King and Bill King’s Magic in a Minute for the 10th uses everyone’s favorite topological construct to do a magic trick. This one uses a neat quirk of the Möbius strip: that if sliced along the center of its continuous loop you get not two separate shapes but one Möbius strip of greater length. There are more astounding feats possible. If the strip were cut one-third of the way from an edge it would slice the strip into two shapes, one another Möbius strip and one a simple loop.

Or consider not starting with a Möbius strip. Make the strip of paper by taking one end and twisting it twice around, for a full loop, before taping it to the other end. Slice this down the center and what results are two interlinked rings. Or place three twists in the original strip of paper before taping the ends together. Then, the shape, cut down the center, unfolds into a trefoil knot. But this would take some expert hand work to conceal the loops from the audience while cutting. It’d be a neat stunt if you could stage it, though.

Presenting as a magic trick: cutting a loop of paper in half, along the loop's center, with the result being a single yet larger loop. The trick is to make the paper into a Moebius strip, and conceal the 'twist' so that your audience does not know what they see.
Mac King and Bill King’s Magic in a Minute for the 10th of November, 2019. This and other mathematics-based tricks featured in Magic In A Minute are at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th uses mathematics as obfuscation. We value mathematics for being able to make precise and definitely true statements. And for being able to describe the world with precision and clarity. But this has got the danger that people hear mathematical terms and tune out, trusting that the point will be along soon after some complicated talk.

Confession Tip: Use Statistics. Kid: 'Mom! Dad! Did you know that, in your immediate area, teen pregnancy may be as high as 100 percent?!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th of November, 2019. This and many other essays discussing Saturday Morning Breakfast Cereal are at this link.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 11th would be a Pi Day joke if it hadn’t run in November. But when this strip first ran, in 2010, Pi Day was not such a big event in the STEM/Internet community. The Boychuks couldn’t have known.

The formulas on the blackboard are nearly all legitimate, and correct, formulas for the value of π. The upper-left and the lower-right formulas are integrals, and ones that correspond to particular trigonometric formulas. The The middle-left and the upper-right formulas are series, the sums of infinitely many terms. The one in the upper right, \sum \frac{1}{n^2} = \frac{\pi^2}{6} , was roughly proven by Leonhard Euler. Euler developed a proof that’s convincing, but that assumed that infinitely-long polynomials behave just like finitely-long polynomials. In this context, he was correct, but this can’t be generally trusted to happen. We’ve got proofs that, to our eyes, seem rigorous enough now.

On the blackboard several calculus formulas for the value of pi. At the table a scientist type says to another, while eating a slice of pie, 'I don't know why, Haskins, but I've had a craving for this all day.'
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers rerun for the 11th of November, 2019. It originally ran the 29th of November, 2010. This and other essays mentioning The Chuckle Brothers, which has gone into perpetual reruns, are at this link.

The center-left formula doesn’t look correct to me. To my eye, this looks like a mistaken representation of the formula

\pi = 2 \sum_{k = 0}^{\infty} \frac{2^k \cdot k!^2}{\left(2k + 1\right)!}

But it’s obscured by Haskins’s head. It may be that this formula’s written in a format that, in full, would be correct. There are many, many formulas for π (here’s Mathworld’s page of them and here’s Wikipedia’s page of π formulas); it’s impossible to list them all.

The center-right formula is interesting because, in part, it looks weird. It’s written out as

\pi = \frac{4}{6+}\frac{1^2}{6+}\frac{3^2}{6+}\frac{5^2}{6+}\frac{7^2}{6+} \cdots

That looks at first glance like something’s gone wrong with one of those infinite-product series for π. Not so; this is a notation used for continued fractions. A continued fraction has a string of denominators that are typically some whole number plus another fraction. Often the denominator of that fraction will itself be a whole number plus another fraction. This gets to be typographically challenging. So we have this notation instead. Its syntax is that

a + \frac{b}{c + \frac{d}{e + \frac{f}{g}}} = a + \frac{b}{c+} \frac{d}{e+} \frac{f}{g}

There are many attractive formulas for π. It’s temping to say this is because π is such a lovely number it naturally has beautiful formulas. But more likely humans are so interested in π we go looking for formulas with some appealing sequence to them. There are some awful-looking formulas out there too. I don’t know your tastes, but for me I feel my heart cool when I see that π is equal to four divided by this number:

\sum_{n = 0}^{\infty} \frac{(-1)^n (4n)! (21460n + 1123)}{(n!)^4 441^{2n + 1} 2^{10n + 1}}

however much I might admire the ingenuity which found that relationship, and however efficiently it may calculate digits of π.

Eno, on a coffee date: 'So you claim you're a teacher, huh?' Teacher: 'What do you mean, 'claim'?' Eno; 'What's 30 divided by 5?' Teacher: 'Six!' Eno: 'OK, you check out.'
Glenn McCoy and Gary McCoy’s The Duplex for the 13th of November, 2019. This surprises me by not being a new comic tag. Essays mentioning The Duplex are at this link.

Glenn McCoy and Gary McCoy’s The Duplex for the 13th uses skill at arithmetic as shorthand for proving someone’s a teacher. There’s clearly some implicit idea that this is a school teacher, probably for elementary schools, and doesn’t have a particular specialty. But it is only three panels; they have to get the joke done, after all.


And that’s all for the comic strips this week. Come Sunday I should have another Reading the Comics post. And the Fall 2019 A-to-Z draws closer to its conclusion with two more essays, trusting that I can indeed write them, for Tuesday and Thursday. I also have something disturbing to write about for Wednesday. Can’t wait.

Reading the Comics, November 2, 2019: Eugene the Jeep Edition


I knew by Thursday this would be a brief week. The number of mathematically-themed comic strips has been tiny. I’m not upset, as the days turned surprisingly full on me once again. At some point I would have to stop being surprised that every week is busier than I expect, right?

Anyway, the week gives me plenty of chances to look back to 1936, which is great fun for people who didn’t have to live through 1936.

Elzie Segar’s Thimble Theatre rerun for the 28th of October is part of the story introducing Eugene the Jeep. The Jeep has astounding powers which, here, are finally explained as being due to it being a fourth-dimensional creature. Or at least able to move into the fourth dimension. This is amazing for how it shows off the fourth dimension being something you could hang a comic strip plot on, back in the day. (Also back in the day, humor strips with ongoing plots that might run for months were very common. The only syndicated strips like it today are Gasoline Alley, Alley Oop, the current storyline in Safe Havens where they’ve just gone and terraformed Mars, and Popeye, rerunning old daily stories.) The Jeep has many astounding powers, including that he can’t be kept inside — or outside — anywhere against his will, and he’s able to forecast the future.

Prof Gipf: 'Lady and Gentleman: my test proves my teory was right.' Popeye: ''Splain yerself!' Gipf, holding up Eugene: 'This animal executes his escaping and disappearing stunts in the FOURTH DIMENSION! Proving that all mysteries are simple when solved. Good day, folks.' (He leaves.) Popeye: 'Well, that's that!' Olive and Popeye, trading words: 'But what the heck is the fourt' dimension??' Author's Narration Box: 'Ahoy, children - we haven't room here to explain, and we think you should know --- so as your dad or mother to explain in detail to you all about the Fourth Dimension'
Elzie Segar’s Thimble Theatre rerun for the 28th of October, 2019. It originally ran the 28th of May, 1936. Essays that have some inspiration in things that turn up in Popeye (current syndication strips) or Thimble Theatre (the 1930s reprints) or Popeye’s Cartoon Club (a special Sunday event for this year) should be at this link. Also I know that these 1930s strips are great massive heaps of words, but they are worth looking at. There’s a bunch of funny stuff going on in here, including Professor Gipf addressing Olive Oyl and Popeye as “Lady and Gentleman”, or his declaration that of course all mysteries are simple when solved. And there is a great puckish glee in Elzie Segar’s final panel, urging kids everywhere to ask their parents to explain “in detail” about the fourth dimension.

Could there be a fourth-dimensional animal? I dunno, I’m not a dimensional biologist. It seems like we need a rich chemistry for life to exist. Lots of compounds, many of them long and complicated ones. Can those exist in four dimensions? I don’t know the quantum mechanics of chemical formation well enough to say. I think there’s obvious problems. Electrical attraction and repulsion would fall off much more rapidly with distance than they do in three-dimensional space. This seems like it argues chemical bonds would be weaker things, which generically makes for weaker chemical compounds. So probably a simpler chemistry. On the other hand, what’s interesting in organic chemistry is shapes of molecules, and four dimensions of space offer plenty of room for neat shapes to form. So maybe that compensates for the chemical bonds. I don’t know.

But if we take the premise as given, that there is a four-dimensional animal? With some minor extra assumptions then yeah, the Jeep’s powers fit well enough. Not being able to be enclosed follows almost naturally. You, a three-dimensional being, can’t be held against your will by someone tracing a line on the floor around you. The Jeep — if the fourth dimension is as easy to move through as the third — has the same ability.

Forecasting the future, though? We have a long history of treating time as “the” fourth dimension. There’s ways that this makes good organizational sense. But we do have to treat time as somehow different from space, even to make, for example, general relativity work out. If the Jeep can see and move through time? Well, yeah, then if he wants he can check on something for you, at least if it’s something whose outcome he can witness. If it’s not, though? Well, maybe the flow of events from the fourth dimension is more obvious than it is from a mere three, in the way that maybe you can spot something coming down the creek easily, from above, in a way that people on the water can’t tell.

Olive Oyl and Popeye use the Jeep to tease one another, asking for definite answers about whether the other is cute or not. This seems outside the realm of things that the fourth dimension could explain. In the 1960s cartoons he even picks up the power to electrically shock offenders; I don’t remember if this was in the comic strips at all.

Wimpy: 'My friends, I can't explain the fourth dimension to you in detail, but I believe I can give you an idea as to how the Jeep performs magical tricks. We live in a three-dimensional world. Our minds know but three dimensions: length, breath, and thickness. Our minds cannot even imagine a fourth dimension - just try to do it. Our eyes can see no more than three dimensions. The Jeep has the power to turn into a fourth dimensional animal. And when he does this he is invisible because to us he does not exist. A three-dimensional thing can't hold a fourth-dimensional thing because the third and fourth do not exist to each other. And so it is easy for a fourth-dimensional animal to walk through a thing that does not exist ... isn't it?' Popeye: 'Wimpy, I wants to congratuake ya on account of yer great brain. Ya've explained the whole works an' now ever'thing's as clear as --- MUD!'
Elzie Segar’s Thimble Theatre rerun for the 29th of October, 2019. It originally ran the 29th of May, 1936. Also wait, where did Wimpy pick up all this talk about the fourth dimension? I guess if you’re going to let a line of smooth patter take the place of working you have to be on top of anything that might come up, but it still seems like a lot of work he’s gone to here to use the Jeep to win horse races.

Elzie Segar’s Thimble Theatre rerun for the 29th of October has Wimpy doing his best to explain the fourth dimension. I think there’s a warning here for mathematician popularizers here. He gets off to a fair start and then it all turns into a muddle. Explaining the fourth dimension in terms of the three dimensions we’re familiar with seems like a good start. Appealing to our intuition to understand something we have to reason about has a long and usually successful history. But then Wimpy goes into a lot of talk about the mystery of things, and it feels like it’s all an appeal to the strangeness of the fourth dimension. I don’t blame Popeye for not feeling it’s cleared anything up. Segar would come back, in this storyline, to several other attempted explanations of the Jeep’s powers, although they do come back around to, y’know, it’s a magical animal. They’re all over the place in the Popeye comic universe.

[NORMAL SCIENTIST] Person: 'No mathematics, no science can ever predict the human soul!' Normal Scientist: 'That's not even a specific claim!? What does it even mean?!' [COMPUTER SCIENtiST] Person: 'No mathematics, no science can ever predict the human soul!' Computer Scientist: 'Ooh! We can use it for cryptography!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th of October, 2019. Those occasional times I do think to discuss Saturday Morning Breakfast Cereal are gathered at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th of October is a riff on predictability and encryption. Good encryption schemes rely on randomness. Concealing the content of a message means matching it to an alternate message. Each of the alternate messages should be equally likely to be transmitted. This way, someone who hasn’t got the key would not be able to tell what’s being sent. The catch is that computers do not truly do randomness. They mostly rely on quasirandom schemes that could, in principle, be detected and spoiled. There are ways to get randomness, mostly involving putting in something from the real world. Sensors that detect tiny fluctuations in temperature, for example, or radio detectors. I recall one company going for style and using a wall of lava lamps, so that the rise and fall of lumps were in some way encoded into unpredictable numbers.

Dana: 'It's so freaky you and I have the same birthday!' Marcie: 'It happens, Dana! We employ 615 people at this health pavilion. I'll bet thirty share our birthday.' Marcie's husband, on the phone, later: 'You were gambling at work and lost HOW MUCH?'
Robb Armstrong’s JumpStart for the 2nd of November, 2019. When I have an essay inspired by something in JumpStart it appears at this link.

Robb Armstrong’s JumpStart for the 2nd of November is a riff on the Birthday “Paradox”, the thing where you’re surprised to find someone shares a birthday with you. (I have one small circle of friends featuring two people who share my birthday, neatly enough.) Paradox is in quotes because it defies only intuition, not logic. The logic is clear that you need only a couple dozen people before some pair will probably share a birthday. Marcie goes overboard in trying to guess how many people at her workplace would share their birthday on top of that. Birthdays are nearly uniformly spread across all days of the year. There are slight variations; September birthdays are a little more likely than, say, April ones; the 13th of any month is a less likely birthday than the 12th or the 24th are. But this is a minor correction, aptly ignored when you’re doing a rough calculation. With 615 birthdays spread out over the year you’d expect the average day to be the birthday of about 1.7 people. (To be not silly about this, a ten-day span should see about 17 birthdays.) However, there are going to be “clumps”, days where three or even four people have birthdays. There will be gaps, days nobody has a birthday, or even streaks of days where nobody has a birthday. If there weren’t a fair number of days with a lot of birthdays, and days with none, we’d have to suspect birthdays weren’t random here.


There were also a handful of comic strips just mentioning mathematics, that I can’t make anything in depth about. Here’s two.

T Shepherd’s Snow Sez for the 1st of November nominally talks about how counting can be a good way to meditate. It can also become a compulsion, with hazards, though.

Terri Libenson’s The Pajama Diaries for the 2nd of November uses mathematics as the sort of indisputably safe topic that someone can discuss in place of something awkward.


And that is all I have to say for last week’s comics. Tuesday I should publish the next Fall 2019 A to Z essay. I also figure to open the end of the alphabet up to nominations this week. My next planned Reading the Comic post should be Sunday. Thanks for reading.

By the way, Thimble Theatre is trying to explain the fourth dimension


I hope to have proper comment about it in the usual Sunday Reading the Comics post. But the “current” storyline in Elzie Segar’s Thimble Theatre comic strip — Popeye to normal people — is the 1936 introduction of Eugene the Jeep. If you’ve looked at my user icon here you know I like Eugene.

Anyway, Eugene the Jeep has wondrous powers. These include the power of prophecy and the power to disappear from even enclosed spaces. Segar’s explanation for this was that the Jeep can turn into the fourth dimension and so do things we can’t hope to do. Which is a fun premise, yes. More, though, it’s got to be a pretty early use of the fourth or other high dimensions in pop culture. Yes, there were some things normal people might know that talk about higher dimensions. H G Wells’s The Time Machine starts with talk about time as a dimension like space. Edwin Abbott’s Flatland is explicitly about two- and three-dimensions, although Square thinks of whether there could be four- or more-dimensional spaces.

Professor: 'I am Professor Gipf, solver of scientific mysteries. Have you a mystery to be solved?' Olive Oyl: 'I have! We locked this Jeep in a room and he escaped. He is also able to disappear right before my eyes.' Professor: 'Is Zasso? I suspect he has fourth-dimensional qualities. It's lucky I have my dimension detector.' (He pulls out many bundles of wires, and more and more, making a huger tangle.) Professor: 'Darn it! Oh, darn the luck!' Olive Oyl: 'Is something missing?' Professor: 'Yes. Have you got a piece of wire?'
Elzie Segar’s Thimble Theatre (Popeye) for the 25th of October, 2019. It originally ran the 26th of May, 1936. Fun fact: that last panel shows the extension strip in the living room where we have the TV, DVR, Blu-Ray, Switch, and record player plugged in.

Wikipedia helps me find a few pieces of literature mentioning the fourth dimension before Eugene the Jeep. And a few pieces of visual art as well. No mention of earlier comic strips, although there’s no mention of Eugene the Jeep in either. So, all I can say is this is an early pop cultural appearance of the fourth dimension. I can’t say it’s the first, even among major comic strips.

Do not try to use this to pass your geometry quals.

My 2019 Mathematics A To Z: Quadrature


I got a good nomination for a Q topic, thanks again to goldenoj. It was for Qualitative/Quantitative. Either would be a good topic, but they make a natural pairing. They describe the things mathematicians look for when modeling things. But ultimately I couldn’t find an angle that I liked. So rather than carry on with an essay that wasn’t working I went for a topic of my own. Might come back around to it, though, especially if nothing good presents itself for the letter X, which will probably need to be a wild card topic anyway.

Cartoony banner illustration of a coati, a raccoon-like animal, flying a kite in the clear autumn sky. A skywriting plane has written 'MATHEMATIC A TO Z'; the kite, with the letter 'S' on it to make the word 'MATHEMATICS'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Quadrature.

We like comparing sizes. I talked about that some with norms. We do the same with shapes, though. We’d like to know which one is bigger than another, and by how much. We rely on squares to do this for us. It could be any shape, but we in the western tradition chose squares. I don’t know why.

My guess, unburdened by knowledge, is the ancient Greek tradition of looking at the shapes one can make with straightedge and compass. The easiest shape these tools make is, of course, circles. But it’s hard to find a circle with the same area as, say, any old triangle. Squares are probably a next-best thing. I don’t know why not equilateral triangles or hexagons. Again I would guess that the ancient Greeks had more rectangular or square rooms than the did triangles or hexagons, and went with what they knew.

So that’s what lurks behind that word “quadrature”. It may be hard for us to judge whether this pentagon is bigger than that octagon. But if we find squares that are the same size as the pentagon and the octagon, great. We can spot which of the squares is bigger, and by how much.

Straightedge-and-compass lets you find the quadrature for many shapes. Like, take a rectangle. Let me call that ABCD. Let’s say that AB is one of the long sides and BC one of the short sides. OK. Extend AB, outwards, to another point that I’ll call E. Pick E so that the length of BE is the same as the length of BC.

Next, bisect the line segment AE. Call that point F. F is going to be the center of a new semicircle, one with radius FE. Draw that in, on the side of AE that’s opposite the point C. Because we are almost there.

Extend the line segment CB upwards, until it touches this semicircle. Call the point where it touches G. The line segment BG is the side of a square with the same area as the original rectangle ABCD. If you know enough straightedge-and-compass geometry to do that bisection, you know enough to turn BG into a square. If you’re not sure why that’s the correct length, you can get there quickly. Use a little algebra and the Pythagorean theorem.

Neat, yeah, I agree. Also neat is that you can use the same trick to find the area of a parallelogram. A parallelogram has the same area as a square with the same bases and height between them, you remember. So take your parallelogram, draw in some perpendiculars to share that off into a rectangle, and find the quadrature of that rectangle. you’ve got the quadrature of your parallelogram.

Having the quadrature of a parallelogram lets you find the quadrature of any triangle. Pick one of the sides of the triangle as the base. You have a third point not on that base. Draw in the parallel to that base that goes through that third point. Then choose one of the other two sides. Draw the parallel to that side which goes through the other point. Look at that: you’ve got a parallelogram with twice the area of your original triangle. Bisect either the base or the height of this parallelogram, as you like. Then follow the rules for the quadrature of a parallelogram, and you have the quadrature of your triangle. Yes, you’re doing a lot of steps in-between the triangle you started with and the square you ended with. Those steps don’t count, not by this measure. Getting the results right matters.

And here’s some more beauty. You can find the quadrature for any polygon. Remember how you can divide any polygon into triangles? Go ahead and do that. Find the quadrature for every one of those triangles then. And you can create a square that has an area as large as all those squares put together. I’ll refrain from saying quite how, because realizing how is such a delight, one of those moments that at least made me laugh at how of course that’s how. It’s through one of those things that even people who don’t know mathematics know about.

With that background you understand why people thought the quadrature of the circle ought to be possible. Moreso when you know that the lune, a particular crescent-moon-like shape, can be squared. It looks so close to a half-circle that it’s obvious the rest should be possible. It’s not, and it took two thousand years and a completely different idea of geometry to prove it. But it sure looks like it should be possible.

Along the way to modernity quadrature picked up a new role. This is as part of calculus. One of the legs of calculus is integration. There is an interpretation of what the (definite) integral of a function means so common that we sometimes forget it doesn’t have to be that. This is to say that the integral of a function is the area “underneath” the curve. That is, it’s the area bounded by the limits of integration, by the horizontal axis, and by the curve represented by the function. If the function is sometimes less than zero, within the limits of integration, we’ll say that the integral represents the “net area”. Then we allow that the net area might be less than zero. Then we ignore the scolding looks of the ancient Greek mathematicians.

No matter. We love being able to find “the” integral of a function. This is a new function, and evaluating it tells us what this net area bounded by the limits of integration is. Finding this is “integration by quadrature”. At least in books published back when they wrote words like “to-day” or “coördinate”. My experience is that the term’s passed out of the vernacular, at least in North American Mathematician’s English.

Anyway the real flaw is that there are, like, six functions we can find the integral for. For the rest, we have to make do with approximations. This gives us “numerical quadrature”, a phrase which still has some currency.

And with my prologue about compass-and-straightedge quadrature you can see why it’s called that. Numerical integration schemes often rely on finding a polynomial with a part that looks like a graph of the function you’re interested in. The other edges look like the limits of the integration. Then the area of that polygon should be close to the area “underneath” this function. So it should be close to the integral of the function you want. And we’re old hands at how the quadrature of polygons, since we talked that out like five hundred words ago.

Now, no person ever has or ever will do numerical quadrature by compass-and-straightedge on some function. So why call it “numerical quadrature” instead of just “numerical integration”? Style, for one. “Quadrature” as a word has a nice tone, clearly jargon but not threateningly alien. Also “numerical integration” often connotes the solving differential equations numerically. So it can clarify whether you’re evaluating integrals or solving differential equations. If you think that’s a distinction worth making. Evaluating integrals and solving differential equations are similar together anyway.

And there is another adjective that often attaches to quadrature. This is Gaussian Quadrature. Gaussian Quadrature is, in principle, a fantastic way to do numerical integration perfectly. For some problems. For some cases. The insight which justifies it to me is one of those boring little theorems you run across in the chapter introducing How To Integrate. It runs something like this. Suppose ‘f’ is a continuous function, with domain the real numbers and range the real numbers. Suppose a and b are the limits of integration. Then there’s at least one point c, between a and b, for which:

\int_a^b f(x) dx = f(c) \cdot (b - a)

So if you could pick the right c, any integration would be so easy. Evaluate the function for one point and multiply it by whatever b minus a is. The catch is, you don’t know what c is.

Except there’s some cases where you kinda do. Like, if f is a line, rising or falling with a constant slope from a to b? Then have c be the midpoint of a and b.

That won’t always work. Like, if f is a parabola on the region from a to b, then c is not going to be the midpoint. If f is a cubic, then the midpoint is probably not c. And so on. And if you don’t know what kind of function f is? There’s no guessing where c will be.

But. If you decide you’re only trying to certain kinds of functions? Then you can do all right. If you decide you only want to integrate polynomials, for example, then … well, you’re not going to find a single point c for this. But what you can find is a set of points between a and b. Evaluate the function for those points. And then find a weighted average by rules I’m not getting into here. And that weighted average will be exactly that integral.

Of course there’s limits. The Gaussian Quadrature of a function is only possible if you can evaluate the function at arbitrary points. If you’re trying to integrate, like, a set of sample data it’s inapplicable. The points you pick, and the weighting to use, depend on what kind of function you want to integrate. The results will be worse the less your function is like what you supposed. It’s tedious to find what these points are for a particular assumption of function. But you only have to do that once, or look it up, if you know (say) you’re going to use polynomials of degree up to six or something like that.

And there are variations on this. They have names like the Chevyshev-Gauss Quadrature, or the Hermite-Gauss Quadrature, or the Jacobi-Gauss Quadrature. There are even some that don’t have Gauss’s name in them at all.

Despite that, you can get through a lot of mathematics not talking about quadrature. The idea implicit in the name, that we’re looking to compare areas of different things by looking at squares, is obsolete. It made sense when we worked with numbers that depended on units. One would write about a shape’s area being four times another shape’s, or the length of its side some multiple of a reference length.

We’ve grown comfortable thinking of raw numbers. It makes implicit the step where we divide the polygon’s area by the area of some standard reference unit square. This has advantages. We don’t need different vocabulary to think about integrating functions of one or two or ten independent variables. We don’t need wordy descriptions like “the area of this square is to the area of that as the second power of this square’s side is to the second power of that square’s side”. But it does mean we don’t see squares as intermediaries to understanding different shapes anymore.


Thank you again for reading. This essay and all the others written for the Fall 2019 A to Z should be at this link. This should include, later this week, something for the letter R. And all of the A to Z essays ought to be at this link.