Symmetry and the Fourth Dimension (Part 7)

Lately I’ve been showing you what happens if you start with a Platonic solid and start chopping off its corners, more and more, until you get the dual Platonic solid. There’s just one I haven’t done yet: the tetrahedron.

This is the simplest Platonic solid, so why did I wait and do this it last?

Because the tetrahedron is self-dual. Remember, the Coxeter diagram of this shape looks like this:

V—3—E—3—F

And remember what this diagram means. It means the tetrahedron has

• 3 vertices and 3 edges touching each face,
• 3 edges and 3 faces touching each vertex.

When we take the dual of a solid, we

• replace vertices by faces;
• replace edges by edges;
• replace faces by vertices.

So when we do this to the tetrahedron, we get back a tetrahedron! This ‘self-duality’ is reflected in the symmetry of its Coxeter diagram. If we switch the letters V and F, we get the same thing back—drawn backwards, but that doesn’t matter.

This self-duality also means that when we take a tetrahedron and keep cutting off the corners more and more deeply, we wind up where we started. And in fact, when we reach the halfway point of this process, we start retracing our steps… going backwards!

Let’s see how it goes. Remember, we have a system of diagrams for drawing the most important solids we meet along the way.

Tetrahedron: •—3—o—3—o

We start with the tetrahedron:

Truncated tetrahedron: •—3—•—3—o

Then we get the truncated tetrahedron:

Octahedron: o—3—•—3—o

Then, halfway through, we get a shape we’ve seen before! It’s our friend the octahedron:

Like the other ‘halfway through’ shapes we’ve seen—the cuboctahedron and icosidodecahedron—every edge of the octahedron lies on a great circle’s worth of edges:

        

Puzzle 1. Why does it always work this way?

I don’t actually know!

Truncated tetrahedron: o—3—•—3—•

Then we get back to the truncated tetrahedron:

Tetrahedron: o—3—o—3—•

At the end, we get back where we started… the tetrahedron:

Where are we?

We’ve begun to explore the three great families of semiregular polyhedra:

• the tetrahedron family shown here,

• the cube/octahedron family shown in Part 5,

• and the dodecahedron/icosahedron family shown in Part 6.

We’ve seen that for each family, we have a Coxeter complex, which is summarized by a Coxeter diagram. By coloring the dots in this diagram either white or black, we get different polyhedra in our family.

Our goal is to explore how this works in 4 dimensions. It’s very similar, but much more rich! We can still use Coxeter diagrams, but they’ll have four dots, so there will be more ways to label them, and we’ll get more shapes. And, of course, we’ll have the fun of learning to visualize 4-dimensional shapes!

But before we can explore the 4d story, there’s a hole in the story so far, that I need to fill.

Puzzle 2. Can you guess what it is?

Maybe you’ll see it if you look over our results so far.

The tetrahedron family

Here are the shapes related to the tetrahedron. It has some repeats, because the tetrahedron is its own dual! It also repeats some shapes we’ll see in other families.

tetrahedron		•—3—o—3—o
truncated tetrahedron		•—3—•—3—o
octahedron		o—3—•—3—o
truncated tetrahedron		o—3—•—3—•
tetrahedron		o—3—o—3—•

And here’s the Coxeter complex that runs the show:

This has one right triangle for each element in the group that acts as symmetries of all these shapes. This group has 24 elements, and it’s called the tetrahedral finite reflection group, or A₃. So, we can also call this collection of polyhedra the A₃ family.

The cube/octahedron family

Here are the shapes related to the cube and the octahedron:

cube		•—4—o—3—o
truncated cube		•—4—•—3—o
cuboctahedron		o—4—•—3—o
truncated octahedron		o—4—•—3—•
octahedron		o—4—o—3—•

And here’s the Coxeter complex:

Again, this has one right triangle for each element in the group that acts as symmetries of all these shapes. This group has 48 elements, and it’s called the octahedral finite reflection group, or B₃. So, we can call this collection of polyhedra the B₃ family.

The dodecahedron/icosahedron family

And here are the shapes related to dodecahedron and icosahedron:

dodecahedron		•—5—o—3—o
truncated dodecahedron		•—5—•—3—o
icosidodecahedron		o—5—•—3—o
truncated icosahedron		o—5—•—3—•
icosahedron		o—5—o—3—•

And here’s the Coxeter complex:

Yet again, this has one right triangle for each element in the group that acts as symmetries of all these shapes. This group has 120 elements, and it’s called the icosahedral finite reflection group, or H₃. So, we can call this collection of polyhedra the H₃ family.