Though I’m shifting toward applied math, I find myself unable to quit explaining pure math to people—stuff that’s fun purely for its own sake. So, I’ve been posting about symmetry and the fourth dimension over on Google+. Now I want to take those posts, polish them up a bit, and combine them into blog articles.
The idea is to start with something very familiar and then take it a little further than most people have seen… without getting so technical that only people with PhDs understand what’s going on. I’m more interested in communicating with ordinary folks than in wowing the experts.
So, I’ll assume you know and love the five Platonic solids:
• The tetrahedron, with 4 triangular faces, 6 edges and 4 vertices:
• The cube, with 6 square faces, 12 edges and 8 vertices:
• The octahedron, with 8 triangular faces, 12 edges and 6 vertices:
• The dodecahedron, with 12 pentagonal faces, 30 edges and 20 vertices:
• The icosahedron, with 20 triangular faces, 30 edges and 12 vertices:
Starting from these, we’ll build the six Platonic solids that exist in 4 dimensions, and the various fancier shapes we can get from these by cutting off corners, edges and so on.
Luckily, a lot of heroic mathematicians and programmers have made pictures of these shapes freely available online. For example, the rotating Platonic solids above were made by Tom Ruen, who put them on Wiki Commons. It wouldn’t be so bad if all I did is show you lots of these pictures and explain them. But there are some underlying themes that make the story deeper, so I thought I’d reveal those now. As the series marches on, I’ll try to make it easy to ignore these themes or pay attention to them, depending on what you want.
One theme is the quaternions. This is a number system introduced by the famous mathematician William Rowan Hamilton back in 1843. A typical quaternion looks like this:
where are ordinary real numbers and are square roots of -1 that ‘anticommute’:
As their name indicates, the quaternions are a 4-dimensional number system. We can use them to relate rotations in 3 dimensions to rotations in 4 dimensions… and this establishes links between 3d Platonic solids and 4d Platonic solids: special links that just don’t exist in higher dimensions.
For example, the dodecahedron has 60 rotational symmetries, and this fact gives a 4d Platonic solid—or as mathematicians say, a 4d regular polytope—with 120 dodecahedral faces. Getting to understand this in detail will be one of the high points of this series: it’s a really wonderful story!
Another theme is 5-fold symmetry. In 2 dimensions there’s an obvious polygon with 5-fold symmetry: the regular pentagon. In 3 dimensions we have a Platonic solid with pentagonal faces: the regular dodecahedron. In 4 dimensions, as I just mentioned, there’s a regular polytope with regular dodecahedra as faces. But then this pattern ends. There are no higher-dimensional regular polytopes with pentagons in them! Only squares and triangles show up.
But the biggest unifying theme is ‘finite reflection groups’. These show up as symmetry groups of Platonic solids and 4d regular polytopes. Technically, a finite reflection group is a finite group of transformations of n-dimensional Euclidean space that’s generated by reflections. Some examples in 3 dimensions will illustrate the idea: it’s not as scary as it might sound.
Take a regular dodecahedron, for example:
This has lots of rotations and reflections as symmetries—but a finite number of them. Each reflection corresponds to a mirror: a plane through the center of the dodecahedron. The reflection switches points on opposite sides of this mirror. We can get every symmetry by doing a bunch of these reflections, one after another: that’s what we mean by saying a group of symmetries is ‘generated by reflections’. So, the symmetry group of a dodecahedron is a finite reflection group.
But the fun starts when we take a sphere centered at the center of the dodecahedron, and slice it with all these mirrors. We get a picture like this, called a Coxeter complex:
The great circles in this picture are where the mirrors intersect the sphere.
You’ll notice there are 120 triangles in this picture: each of the 12 pentagonal faces of the dodecahedron has been subdivided into 10 right triangles. You should be able to see that if we pick one of these triangles, there’s a symmetry carrying it to any other. So, the symmetry group of the dodecahedron has 120 elements!
By the way: earlier I mentioned the 60 rotational symmetries of the dodecahedron; now I’m talking about 120 symmetries including rotations and reflections. There’s no contradiction. If we start by picking a black triangle, a rotation will take it to another black triangle, while a reflection will take it to be a blue one. There are 60 of each, for a total of 120.
We can play the same game starting with any other Platonic solid. If we start with the icosahedron, nothing really new happens. It has the same symmetry group, so we get the same Coxeter complex. Indeed, if you look carefully here:
you can see a bunch of equilateral triangles, each containing 6 right triangles. There are 20 of these equilateral triangles, and they’re the faces of an icosahedron:
The corners of the icosahedron are located at the centers of the faces of a dodecahedron, and vice versa. So we say these Platonic solids are dual to each other. Dual polyhedra, or more generally dual polytopes, have the same symmetry group and the same Coxeter complex.
But we get something different if we start with the cube:
This gives a Coxeter complex with 48 triangles, formed by subidividing each of the 6 square faces of the cube into 8 right triangles:
There’s a symmetry of the cube carrying any of these right triangles to any other, so its symmetry group has 48 elements.
The octahedron has the same symmetry group as the cube, because they’re duals. But we get something different if we start with the regular tetrahedron:
This gives a Coxeter complex with 24 triangles, formed by subidividing each of the 4 triangular faces of the tetrahedron into 6 right triangles:
There’s a symmetry of the tetrahedron carrying any of these right triangles to any other, so its symmetry group has 24 elements.
The tetrahedron is its own dual. So the Platonic solids only give us three finite reflection groups in 3 dimensions. There are also some some infinite sequences of less interesting ones, like the symmetry groups of the hosohedra, which are funny degenerate Platonic solids whose faces have just 2 sides… these faces need to be curved:
But in general, the possibilities are quite restricted. So, finite reflection groups are not only beautiful: they’re a bit rare. This makes them doubly precious. People have written books about them:
• James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge U. Press, 1992.
And as we’re beginning to see, the Coxeter complex is a vivid picture of the finite reflection group it comes from. We can already see that in 3 dimensions, it has one black triangle for each reflection in the group, and one blue triangle for each rotation. But it contains much more information than this, in neatly visible form… and it works well not just in 3 dimensions, but any dimension.
That’s enough for now: I want to keep these blog articles bite-sized, rather than letting them grow jaw-breakingly big. But if you’re hungry for more right now, try this:
• John Baez, Platonic solids in all dimensions.
I’ll also leave you with this:
Puzzle: How many great circles are there in these Coxeter complexes?
• The Coxeter complex of the tetrahedral finite reflection group, also known to mathematicians as A3:
• The Coxeter complex of the octahedral finite reflection group, also known as B3:
• The Coxeter complex of the icosahedral finite reflection group, also known as H3:
The answers will say how many reflections there are in the finite reflection groups we’ve met today.
Salut! V interesting post. Will plod through, reread, and steep in these.
Salut! Glad to see you here.
Thank you John. If there is a space for wishes in your planned posts, I’d wish a glimpse of how the matters could be presented to junior E.T. who would be familiar with ℂ but would not have heard of ℝ. Assuming of course such beings aren’t excluded by natural law – I’d be fascinated by theorems to the effect they can’t exist :)
I need to acknowledge that on second look, my wish is too late on the beat so to say, since I guess the right step for attempting such a variation on your narrative is at the definition of the quaternions themselves. Already past in your first post above !
I wonder what could be the right domain or branch to submit a research or creative paper built around the corresponding synthetical mathematical narrative ? – Which would be the exposition to fictional students initially familiar only with complex numbers (perhaps under a more appropriate name) first of all of quaternions and then only of real numbers. The latter naturally presented as _”numbers that are most to complex numbers, like complex numbers are to quaternions”_ :) A fun part would be inventing elegant theorems to derive “unfamiliar” properties of real numbers from “familiar” properties of complex numbers.
My best response to your challenge is this:
• John Baez, Division algebras and quantum theory.
People often ask why quantum theory chose to use complex Hilbert spaces instead of real or quaternionic ones. I argue that it uses all three…
… and moreover, I explain how we can think of a real Hilbert space as a complex Hilbert space with extra structure, or a quaternionic Hilbert space as a real one with extra structure, or a quaternionic one as a complex one with extra structure, etcetera… going through all 6 cases. There are good reasons why we’ve chosen complex Hilbert spaces as the ‘default’ case, but there really is complete flexibility if we want it.
I think you meant to show an octahedral complex for the cube, not a tetrahedral one. (Also, the cube itself doesn’t have any triangular faces to subdivide – probably you meant square?)
Yes, I switched the tetrahedron and the cube at the last minute and forgot to make these changes. They should be fixed now!
I think the picture for the Coxeter complex of the cube is wrong — it looks identical to the picture below the tetrahedron.
If you will talk about 4d solids and Quaternions — why aren’t there more 8d solids, given the Octonions? The octonions are enough, for example, to make the seven-dimensional sphere parallelizable (trivial tangent bundle).
It’s often tricky, in mathematics, to say ‘why’ things don’t exist. You have to imagine an impossible situation in as much detail as you can and then explain why it’s impossible. But I can explain one thing that works in 4 dimensions, which doesn’t in 8.
The key fact linking 3d and 4d Platonic solids is this: the quaternions of length 1 form a group. This group is the double cover of the group of rotations in 3d space.
So, there are two points in the sphere in 4d Euclidean space for each rotation of a sphere in 3d Euclidean space!
Thus, if you take a dodecahedron, for example, and look at its rotational symmetries (of which there are 60), there will be a very symmetrical arrangement of twice as many points in the sphere in 4d space. These are the centers of the faces of the 120-cell… and also the corners of another Platonic solid in 4 dimensions, which we’ll discuss later.
As you hint, the octonions of length 1 are closed under multiplication, so the sphere in 8-dimensional space also has special properties. But it’s not a group—multiplication is not associative—so we can’t play a trick like I just described.
However, there are certain ‘integer octonions’ called Cayley integers, also closed under multiplication. If we take the Cayley integers of length 1 we get a very special arrangement of 240 points in the sphere in 8d space.
These are not associated to a regular polytope, but they’re the corners of a ‘uniform’ polytope, which is sort of the next best thing. It’s called the E8 Gosset polytope. It has important connections to topology, string theory and so on, and it looks like this:
A polytope is uniform if it has enough symmetries to map any vertex to any other. A polytope is regular if it has enough symmetries to map any flag to any other. When I talk about ‘Platonic solids in higher dimensions’, I’m using poetic license to refer to regular polytopes in higher dimensions.
To answer the puzzle, I think there are 6 planes of symmetry for a tetrahedron (one passing through each edge), 9 for a cube (3 that swap opposite pairs of faces plus 6 that swap opposite pairs of edges), and 15 for an icosahedron (one for each pair of edges).
Here’s another way to derive/check those answers.
Generically, planes through the origin will intersect along lines, which in turn will intersect a sphere centred on the origin at points. Now, obviously in these cases the intersections actually overlap a lot, but if we imagine the results of perturbing all the planes slightly we can equate the total counts that result to .
For the tetrahedron, three planes intersect at the centre of every face, and if we perturbed them slightly they’d intersect at 3(3-1)/2=3 different points. So that’s 3 times 4 = 12 from faces. Two planes intersect at each edge, so that’s 6 from edges. And three planes intersect at every vertex, so that’s 3 times 4 = 12 from vertices. So we have
For the cube, four planes intersect at the centre of every face, and if we perturbed them slightly they’d intersect at 4(4-1)/2=6 different points. So that’s 6 times 6 = 36 from faces. We get 12 from edges. And three planes intersect at every vertex, so that’s 3 times 8 = 24 from vertices. From
For the icosahedron, we have 3 times 20 = 60 from faces and 30 from edges. Five planes intersect at every vertex, and if they were perturbed they’d intersect at 5(5-1)/2=10 separate points, giving us 120 for the 12 vertices. So we have
That’s nice! And when you’re counting mirror planes, you’re also counting elements in the Coxeter group whose square is 1.
It’s interesting how the three cases are so different. For example, for the tetrahedron and icosahedron all these elements are conjugate to each other, since all reflection symmetries ‘look alike’. But for the the cube they come in two conjugacy classes: reflections that switch 2 opposite faces, and reflections that switch two opposite edges. That’s a purely group-theoretic difference. For the icosahedron each reflection switches two opposite edges, while for the tetrahedron there aren’t reflections that switch opposite edges. That’s not a purely group-theoretic difference, at least not in the way I’ve described it.
Not all elements whose square is 1, but only the elements that act as reflections. (In B_3, for example, there are some rotations whose square is 1.)
Keep up the explaining pure math bit! It’s great.
Oh, whoops! And in A3, too!
(By the way, for other people listening in, here I don’t mean the alternating group on 3 letters: I mean the full symmetry group of a regular simplex in 3 dimensions. There’s an annoying conflict of notation here.)
This gives a Coxeter complex with 48 triangles, formed by subidividing each of the 6 square faces of the cube into *8* right triangles
Whoops! I’ll fix that. Yes, this stuff is so fun that if I weren’t careful I could easily spend all my time on it. I might, in fact, write a book on this stuff someday. I’ve been wanting to for many years. The problem is figuring out how much it should cover. Coxeter diagrams link discrete groups with Lie groups in a wonderful way, so they let us tell a tale that starts with discrete symmetries (and crystallography) and then goes on to continuous symmetries (and particle physics). The road leads ever on…
And I I’ve been waiting for many years that you actually start writing this book! I have a John Baez folder where I have copied all the scatttered pieces where you talk about the octonions, Dynkin diagrams, Platonic solids, Klein quartic, E8, Mathieu groups, Leech lattice, icosians, etc. But it would be of course so much nicer to have that in one coherent book. I am sure it will be an international bestseller that will make you rich and famous. Write it!
Rich and famous? Okay, then I’ll write it!
Hi John, are you familiar with the spinning dancer illusion? It’s an animation of a spinning dancer that looks at first to be spinning the one way, but, with some practice, can be gotten to spin the other way instead – although most people find this impossible to believe at first. It’s an animated version of an ambiguous figure, in essence:
Interestingly, your rotating solids are subject to this illusion as well. The cube, in particular, looks like a good old cube if seen spinning the “right” way, but if you can get it to spin the “other” way, it becomes grossly deformed, having the shape of an Ames room, more or less. When I first saw this post, the cube was spinning the “wrong” way, and I thought, “How on earth is that a cube?” The spinning dancer illusion is well-worth one’s acquaintance if one hasn’t seen it before. This is the first time I’m seeing it in a completely unintended context! I guess this comment has nothing to do with the math but I know you’d be interested anyway!
Yes, I’m familiar with that illusion but never had it with these rotating solids. I guess there’s a possible interesting combination of the spinning dancer illusion and the Necker cube lurking around here. The Necker cube is an isometric drawing of the cube without any depth cues, so one can see either of two sides as the ‘front’:
Of course, depth cues resolve this ambiguity:
It’s frustrating at first, but eventually fun, to practice deliberately getting the spinning dancer to switch directions. I’d gotten pretty good at it, but now I’ve completely lost the knack, so it seems completely impossible again. There’s some trick to it…
There’s probably a vaguely similar trick for switching from optimism to pessimism or vice versa—seeing a glass as either ‘half empty’ or ‘half full’.
It does look like a combination of spinning dancer and Necker cube. Here’s a still of the animation. Necker-cube it to see the Ames-room shape (some trapezoidal prism) that appears if one gets the cube to spin the “wrong” way.
The animated version is quite startling. The other solids are not affected much however, just the cube.
Unfortunately, pictures posted in comments get deleted by the blog… before they even reach me! The last time you did this I was able to find the gif and insert by hand. This time that would be more work. If you still have the gif, please email it to me. My email address is at the bottom of my homepage, though using it involves a simple intelligence test.
I had the same experience: it was something I used to be able to do, but now couldn’t switch the spinning dancer’s direction. I can now do it again by a kind of cheat: cover her up with my hand, and then show just the top of her head. It seems easier to get that to switch, and once it does, I can instantly uncover her and she is spinning in the new way.
I guess that doesn’t work for the cube though. The cube seems inherently harder, because there are some kind of depth cues in it, at least that is my impression.
Extending the skill to optimism/pessimism sounds like a more practical talent …
This resurrects an idea for 4d shape perception that came up during the G+ “early release” of the series. The proposal was to decouple rotation-induced depth perception from stereoscopic depth perception, so as to bridge 4D through to the brain with an encoding of 4D it can (perhaps) train itself to decode, as discrepancies between the pair of 3D signals.
[…] Last time I showed you that any Platonic solid has a bunch of symmetries where we reflect it across planes. These planes, called mirrors, all intersect at the center of that Platonic solid. If we take a sphere and slice it with these mirrors, it gets chopped up into triangles, and we get a pattern called a Coxeter complex. […]
In Part 1 we started by looking at the five Platonic solids […]
These images of the Coxeter complexes are very nice. It is sure easier to get the idea from your description of these examples than from the Wikipedia page. I went back to this post looking for the source of these images of the Coxeter complex on a sphere, but I didn’t find it. I know it was mentioned somewhere …
Sorry, I should have provided a clickable link on the images themselves, but this time I forgot. Almost always that means I got them from Wikipedia. Let’s see… you can find images of all 3 Coxeter complexes here, drawn by Tom Ruen: he draws lots of these great geometrical images. But these are marked up in various ways…
Hmm, I can’t find the original source of these images anymore! But I’m sure they were made by Tom Ruen and they reside somewhere on Wikicommons. He’s released them all into the public domain, like a true hero—so feel free to grab them off this blog post and use them as you like!
Thanks John. I did look at Tom Ruen’s page, but couldn’t find them. Even a Google image search for Coxeter complexes only turns up these images on your pages. I started to feel like something has changed, but the history of the Coxeter complex Wikipedia page is short (it was only created this year); I don’t think these images were ever there. I guess they appear somewhere that doesn’t mention Coxeter complexes. I feel like you mentioned their source somewhere, but I only dug up one G+ post with such an image https://plus.google.com/u/0/117663015413546257905/posts/EiWnRcQFtY2
Oh, okay I found them:
and in the links mentioned at the bottom of their wikimedia pages:
I think there is a slight typo here: “Each reflection corresponds to a mirror: a plane through the center of the tetrahedron.”
That part talks about dodecahedra, so I think the reference to a tetrahedron is wrong?
Keep up the good work: fascinating but definitely not facile maths, made accessible :-)
Thanks! I fixed that typo.
[…] Baez’s blog series on platonic solids and Coxeter stuff […]
This is just one of the passages from Schild’s Ladder by Greg Egan, an author of Hard Sci Fi. I’d come across the author while reading a series of posts by John Baez on his blog Azimuth on Platonic Solids and the Fourth dimension (it was part 8 that mentioned Egan). The book opens wonderfully […]
Maybe this deserves to be called a quantum Coxeter complex, since the Coxeter complex for the group of rotations and reflections of the dodecahedron looks like this: