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.

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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.

Slowly Rotating Hyperdodecahedron


Here’s an engaging moving picture from RobertLovesPi. The Platonic solids — cubes, pyramids, octahedrons, icosahedrons, and dodecahedrons — are five solid shapes each with the same regular convex polygon as their face. This is a nice two-dimensional rendering of a three-dimensional projection of a “hyperdodecahedron”. It’s made of 120 dodecahedrons, in a four-dimensional space. And it’s got the same kind of structure that Platonic solids have, being made of the same regular convex polyhedron for each face.

Remarkably, I learn from Mathworld, the shape is three-colorable. That is, suppose you wanted to assign colors to each of the corners in this four-dimensional shape. They’re all green circles here, but they don’t have to be. There are a lot of these corners, and they’re connected in complicated ways to one another. But you could color in every one of them, so that none if them is connected directly to another of the same color, using only three different colors.

RobertLovesPi.net

This is the hyperdodecahedron, or 120-cell, one of the six four-dimensional analogs of the Platonic solids. It’s been shown on this blog before, but this image has one major change: a much slower rotational speed. It is my hope that this will help people, including myself, with the difficult task of understanding four-dimensional objects.

5-Hi, 120-cell, Hecatonicosachoron

This image was created using Stella 4d, a program you can try, as a free trial download, at this website.)

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