Last time we saw that Platonic solids come in dual pairs, with the tetrahedron being dual to itself:
When you have a dual pair, you can start chopping off the corners of one, more and more, and keep going until you reach the other. Along the way you get some interesting shapes:
At certain points along the way, we get semiregular polyhedra, meaning that:
• all the faces are regular polygons, and
• there’s a symmetry carrying any corner to any other.
Let’s see how it goes with the cube/octahedron pair. And on the way, I’ll show you some diagrams that summarize what’s going on. I’ll explain them later, but you can try to guess the pattern. As a clue, I’ll say they’re based on the Coxeter diagram for the cube, which I explained last time:
Here we go!
First we have the cube, with all square faces:
Truncated cube: •—4—•—3—o
Then we get the truncated cube, with octagons and triangles as faces:
Halfway through we get the aptly named cuboctahedron, with squares and triangles as faces:
Truncated octahedron: o—4—•—3—•
Then we get the truncated octahedron, with squares and hexagons as faces:
Then finally we get the octahedron, with triangles as faces:
Can you see what’s going on with the diagrams here?
Clearly the black dots tend to move from left to right as we move down the chart, but there’s something much cooler and more precise going on. The black dots secretly say where the corners of the shapes are!
Let’s see how quickly I can explain this, and how quickly you can get what I’m talking about. Remember how I defined a ‘flag’ in Part 3?
No? Good, because today I’m going to call that a ‘complete flag’. So, given a Platonic solid, we’ll say a complete flag is a vertex, edge and face where the vertex lies on the edge and the edge lies on the face.
For example, here is a complete flag for a cube:
It’s the black vertex lying on the blue edge lying on the yellow face.
But the term ‘complete flag’ hints that there are also ‘partial flags’. And there are! A vertex-edge flag is a vertex and edge where the vertex lies on the edge. Here’s a vertex-edge flag for the cube:
Similarly, an edge-face flag is an edge and a face where the edge lies on the face. Here’s an edge-face flag for the cube:
You can see why they’re called partial flags: they’re different parts of a complete flag.
Now, fix the Coxeter diagram for the cube firmly in mind:
V for vertex, E for edge and F for face.
• a cube obviously has one corner for each vertex of the cube, so we draw it like this:
• a truncated cube has one corner for each vertex-edge flag of the cube, so we draw it like this:
• a cuboctahedron has one corner for each edge of the cube, so we draw it like this:
• a truncated octahedron has one corner for each edge-face flag of the cube, so we draw it like this:
• an octahedron has one corner for each face of the cube, so we draw it like this:
Alas, I don’t have the patience to draw all the pictures needed to explain this clearly; I’ll just grab the pictures I can get for free on Wikicommons. Here’s how an octahedron has a corner for each face of the cube:
And here’s how the cuboctahedron has a corner for each edge of the cube:
But these are the least interesting cases! It’s more interesting to see how the truncated cube has one corner for each vertex-edge flag of the cube. Do you see how it works? You have to imagine this truncated cube sitting inside a cube:
Then, notice that the truncated cube has 2 corners on each edge of the cube, one near each end. So, it has one corner for each vertex-edge flag of the cube!
Similarly, the truncated octahedron has one corner for each edge-face flag of the cube. But since I don’t have a great picture to help you see that, lets use duality to change our point of view. A face of the cube corresponds to a vertex of the octahedron. So, think of this truncated octahedron as sitting inside an octahedron:
The truncated octahedron has 4 corners near each vertex of the octahedron, one on each edge touching that vertex. In short, it has one corner for each vertex-edge flag of the octahedron. So it’s got one corner for each edge-face flag of the cube!
This change of viewpoint can be justified more thoroughly:
Puzzle. The diagrams we’ve been using were based on the Coxeter diagram for the cube. What would they look like if we based them on the Coxeter diagram for the octahedron instead?
By the way, the pretty pictures of solids with brass balls at the vertices were made by Tom Ruen using Robert Webb’s Stella software. They’re available on Wikicommons, and you can find most of them by clicking on the images here and looking around on the Wikipedia articles you’ll reach that way. On this blog, I try hard to make most images take you to more information when you click on them.