My 2019 Mathematics A To Z: Platonic


Today’s A To Z term is another from goldenoj. It was just the proposal “Platonic”. Most people, prompted, would follow that adjective with one of three words. There’s relationship, ideal, and solid. Relationship is a little too far off of mathematics for me to go into here. Platonic ideals run very close to mathematics. Probably the default philosophy of western mathematics is Platonic. At least a folk Platonism, where the rest of us follow what the people who’ve taken the study of mathematical philosophy seriously seem to be doing. The idea that mathematical constructs are “real things” and have some “existence” that we can understand even if we will never see a true circle or an unadulterated four. Platonic solids, though, those are nice and familiar things. Many of them we can find around the house. That’s one direction to go.

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.

Platonic.

Before I get to the Platonic Solids, though, I’d like to think a little more about Platonic Ideals. What do they look like? I gather our friends in the philosophy department have debated this question a while. So I won’t pretend to speak as if I had actual knowledge. I just have an impression. That impression is … well, something simple. My reasoning is that the Platonic ideal of, say, a chair has to have all the traits that every chair ever has. And there’s not a lot that every chair has. Whatever’s in the Platonic Ideal chair has to be just the things that every chair has, and to omit things that non-chairs do not.

That’s comfortable to me, thinking like a mathematician, though. I think mathematicians train to look for stuff that’s very generally true. This will tend to be things that have few properties to satisfy. Things that look, in some way, simple.

So what is simple in a shape? There’s no avoiding aesthetic judgement here. We can maybe use two-dimensional shapes as a guide, though. Polygons seem nice. They’re made of line segments which join at vertices. Regular polygons even nicer. Each vertex in a regular polygon connects to two edges. Each edge connects to exactly two vertices. Each edge has the same length. The interior angles are all congruent. And if you get many many sides, the regular polygon looks like a circle.

So there’s some things we might look for in solids. Shapes where every edge is the same length. Shapes where every edge connects exactly two vertices. Shapes where every vertex connects to the same number of edges. Shapes where the interior angles are all constant. Shapes where each face is the same polygon as every other face. Look for that and, in three-dimensional space, we find nine shapes.

Yeah, you want that to be five also. The four extra ones are “star polyhedrons”. They look like spikey versions of normal shapes. What keeps these from being Platonic solids isn’t a lack of imagination on Plato’s part. It’s that they’re not convex shapes. There’s no pair of points in a convex shape for which the line segment connecting them goes outside the shape. For the star polyhedrons, well, look at the ends of any two spikes. If we decide that part of this beautiful simplicity is convexity, then we’re down to five shapes. They’re famous. Tetrahedron, cube, octahedron, icosahedron, and dodecahedron.

I’m not sure why they’re named the Platonic Solids, though. Before you explain to me that they were named by Plato in the dialogue Timaeus, let me say something. They were named by Plato in the dialogue Timaeus. That isn’t the same thing as why they have the name Platonic Solids. I trust Plato didn’t name them “the me solids”, since if I know anything about Plato he would have called them “the Socratic solids”. It’s not that Plato was the first to group them either. At least some of the solids were known long before Plato. I don’t know of anyone who thinks Plato particularly advanced human understanding of the solids.

But he did write about them, and in things that many people remembered. It’s natural for a name to attach to the most famous person writing them. Still, someone had the thought which we follow to group these solids together under Plato’s name. I’m curious who, and when. Naming is often a more arbitrary thing than you’d think. The Fibonacci sequence has been known at latest since Fibonacci wrote about it in 1204. But it could not have that name before 1838, when historian Guillaume Libri gave Leonardo of Pisa the name Fibonacci. I’m not saying that the name “Platonic Solid” was invented in, like, 2002. But traditions that seem age-old can be surprisingly recent.

What is an age-old tradition is looking for physical significance in the solids. Plato himself cleverly matched the solids to the ancient concept of four elements plus a quintessence. Johannes Kepler, whom we thank for noticing the star polyhedrons, tried to match them to the orbits of the planets around the sun. Wikipedia tells me of a 1980s attempt to understand the atomic nucleus using Platonic solids. The attempt even touches me. Along the way to my thesis I looked at uniform charges free to move on the surface of a sphere. It was obvious if there were four charges they’d move to the vertices of a tetrahedron on the sphere. Similarly, eight charges would go to the vertices of the cube. 20 charges to the vertices of the icosahedron. And so on. The Platonic Solids seem not just attractive but also of some deep physical significance.

There are not the four (or five) elements of ancient Greek atomism. Attractive as it is to think that fire is a bunch of four-sided dice. The orbits of the planets have nothing to do with the Platonic solids. I know too little about the physics of the atomic nucleus to say whether that panned out. However, that it doesn’t even get its own Wikipedia entry suggests something to me. And, in fact, eight charges on the sphere will not settle at the vertices of a cube. They’ll settle on a staggered pattern, two squares turned 45 degrees relative to each other. The shape is called a “square antiprism”. I was as surprised as you to learn that. It’s possible that the Platonic Solids are, ultimately, pleasant to us but not a key to the universe.

The example of the Platonic Solids does give us the cue to look for other families of solids. There are many such. The Archimedean Solids, for example, are again convex polyhedrons. They have faces of two or more regular polygons, rather than the lone one of Platonic Solids. There are 13 of these, with names of great beauty like the snub cube or the small rhombicuboctahedron. The Archimedean Solids have duals. The dual of a polyhedron represents a face of the original shape with a vertex. Faces that meet in the original polyhedron have an edge between their dual’s vertices. The duals to the Archimedean Solids get the name Catalan Solids. This for the Belgian mathematician Eugène Catalan, who described them in 1865. These attract names like “deltoidal icositetrahedron”. (The Platonic Solids have duals too, but those are all Platonic solids too. The tetrahedron is even its own dual.) The star polyhedrons hint us to look at stellations. These are shapes we get by stretching out the edges or faces of a polyhedron until we get a new polyhedron. It becomes a dizzying taxonomy of shapes, many of them with pointed edges.

There are things that look like Platonic Solids in more than three dimensions of space. In four dimensions of space there are six of these, five of which look like versions of the Platonic Solids we all know. The sixth is this novel shape called the 24-cell, or hyperdiamond, or icositetrachoron, or some other wild names. In five dimensions of space? … it turns out there are only three things that look like Platonic Solids. There’s versions of the tetrahedron, the cube, and the octahedron. In six dimensions? … Three shapes, again versions of the tetrahedron, cube, and octahedron. And it carries on like this for seven, eight, nine, any number of dimensions of space. Which is an interesting development. If I hadn’t looked up the answer I’d have expected more dimensions of space to allow for more Platonic Solid-like shapes. Well, our experience with two and three dimensions guides us to thinking about more dimensions of space. It doesn’t mean that they’re just regular space with a note in the corner that “N = 8”. Shapes hold surprises.


The essays for the Fall 2019 A To Z should be gathered here. And, in time, every past A to Z essay should be at this link. For now, it’s at least several years’ worth there. Thank you.

How May 2015 Treated My Mathematics Blog


For May 2015 I tried a new WordPress theme — P2 Classic — and I find I rather like it. Unfortunately it seems to be rubbish on mobile devices and I’m not WordPress Theme-equipped-enough to figure out how to fix that. I’m sorry, mobile readers. I’m honestly curious whether the theme change affected my readership, which was down appreciably over May.

According to WordPress, the number of pages viewed here dropped to 936 in May, down just over ten percent from April’s 1047 and also below March’s 1022. Perhaps the less-mobile-friendly theme was shooing people away. Maybe not, though: in March and April I’d posted 14 articles each, while in May there were a mere twelve. The number of views per post increased steadily, from 73 in March to just under 75 in April to 78 in May. I’m curious if this signifies anything. I may get some better idea next month. June should have at least 13 posts from the Mathematics A To Z gimmick, plus this statistics post, and there’ll surely be at least two Reading The Comics posts, or at least sixteen posts. And who knows what else I’ll feel like throwing in? It’ll be an interesting experiment at least.

Anyway, the number of unique visitors rose to 415 in May, up from April’s 389 but still below March’s 468. The number of views per visitor dropped to 2.26, far below April’s 2.68, but closer in line with March’s 2.18. And 2.26 is close to the normal count for this sort of thing.

The number of likes on posts dropped to 259. In April it was 296 likes and in March 265. That may just reflect the lower number of posts, though. Divide the number of likes by the number of posts and March saw an average of 18.9, April 21.14, and May 21.58. That’s all at least consistent, although there’s not much reason to suppose that only things from the current month were liked.

The number of comments recovered also. May saw 83 comments, up from April’s 64, but not quite back to March’s 93. That comes to, for May, 6.9 comments for each post, but that’s got to be counting links to other posts, including pingbacks and maybe the occasional reblogging. I’ve been getting chattier with folks around here, but not seven comments per post chatty.

June starts at 24,820 views, and 485 people following specifically through WordPress.

I’ve got a healthy number of popular posts the past month; all of these got at least 37 page views each. I cut off at 37 because that’s where the Trapezoids one came in and we already know that’s popular. More popular than that were:

I have the suspicion that comics fans are catching on, quietly, to all this stuff.

Now the countries report. The nations sending me at least twenty page views were the United States (476), the United Kingdom (85), Canada (65), Italy (53), and Austria (20).

Sending just a single reader were Belgium, Bulgaria, Colombia, Nigeria, Norway, Pakistan, Romania, and Vietnam. Romania is on a three-month single-reader streak; Vietnam, two. India sent me a mere two readers, down from six last month. The European Union sent me three.

And among the interesting search terms this past month were:

  • origin is the gateway to your entire gaming universe.
  • how to do a cube box (the cube is easy enough, it’s getting the boxing gloves on that’s hard)
  • popeye “computer king” (Remember that comic?)
  • google can you show me in 1 trapezoid how many cat how many can you make of 2 (?, although I like the way Google is named at the start of the query, like someone on Next Generation summoning the computer)
  • plato “divided line” “arthur cayley” (I believe that mathematics comes in on the lower side of the upper half of Plato’s divided line)
  • where did negative numbers originate from

Someday I must work out that “origin is the gateway” thing.

Split Lines


My spouse, the professional philosopher, was sharing some of the engagingly wrong student responses. I hope it hasn’t shocked you to learn your instructors do this, but, if you got something wrong in an amusing way, and it was easy to find someone to commiserate with, yes, they said something.

The particular point this time was about Plato’s Analogy of the Divided Line, part of a Socratic dialogue that tries to classify the different kinds of knowledge. I’m not informed enough to describe fairly the point Plato was getting at, but the mathematics is plain enough. It starts with a line segment that gets divided into two unequal parts; each of the two parts is then divided into parts of the same proportion. Why this has to be I’m not sure (my understanding is it’s not clear exactly why Plato thought it important they be unequal parts), although it has got the interesting side effect of making exactly two of the four line segments of equal length.

Continue reading “Split Lines”