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Polytope LibraryThe polytope library is a collection of polyhedra and polychora that are nice and symmetrical in various ways. The whole thing is here thanks to H. S. Teoh, who knew which shapes to include and gave me the raw data for all of them. He has also written about all the shapes at length on his wonderful web site (here). The pictures and animations there are not only gorgeous but also very useful for learning how to think in terms of solid blocks of color.
I have more to say about the polytopes, but if you'd rather just jump in and start looking at them, here's some quick advice.
Next, here's what the subdirectory names mean.
Finally, I'd like to tell you about truncation and all the other crazy operations that you'll see mentioned in the "uni4" directory. Let's start with an easy example, the truncation of a cube in 3D.
Imagine a cube, and imagine eight diagonal planes just touching the corners of the cube. Now we're going to bring those planes gradually inward. First they just cut off the corners of the cube, leaving little triangles. As we move them further inward, the original square faces of the cube become more and more octagonal, until at a certain point they become regular octagons. That shape is called a truncated cube.
As we go further, the octagons change back into squares (in a different orientation), and the triangular faces touch at the corners. That shape is called a rectified cube, or cuboctahedron.
Although the triangles are touching, we can continue to bring the planes inward, and that turns the triangles into hexagons, and eventually into regular hexagons. That shape is called a truncated octahedron.
Then, finally, the planes reach the centers of the original square faces, and we're left with an octahedron, the dual of the cube. There's no point in moving the planes any further, as it would just make the octahedron smaller.
So, that's the process of truncation, which produces not only truncated shapes but also rectified and dual ones. The word "truncate" is thus slightly ambiguous, but once you understand the process, you should be able to make sense of everything.
Now let's go back and look at the same thing in a different way. Imagine a cube again, and imagine a triangle that connects a corner, an edge midpoint, and a face center. Let's number the vertices of that triangle: 0 for the corner because it's 0-dimensional, 1 for the edge midpoint because edges are 1-dimensional, and 2 for the face center because faces are 2-dimensional. Now I want to make a diagram out of those vertices. The idea is that it's 0-1-2, but because the numbers will always be in the same order, I'll just leave them out and use dots instead: o-o-o.
So far so good? Now imagine all the other similar triangles on the cube. There are eight per face, and six faces per cube, for a total of 48 triangles. If we put big black dots on all the 0-vertices, we get a cube shape. What diagram should we use to represent that shape? Well, we could put a big black dot on the 0-vertex: @-o-o. In the same way, we can draw diagrams for other shapes.
o-o-@ : If we put dots on all the 2-vertices (face centers) and connect them up, we get an octahedron.
So, every shape we could produce by truncation, we can produce by drawing a diagram. The reverse isn't true, though! There are a couple of shapes that we can only produce by drawing a diagram.
@-o-@ : If we put dots the correct distance along the lines between the 0-vertices and the 2-vertices, we get a shape made of triangles and squares. It's called a rhombicuboctahedron.
There are even a few shapes of interest that can't be produced by drawing a diagram: in 3D, the snub cube and snub dodecahedron, and in 4D, the snub 24-cell and grand antiprism. I'll let you read about those on Teoh's site.
Now let's stop and make a table of what we've learned about diagrams and operations in 3D. The first three can be produced by truncation.
@-o-o : the original shape
We don't need to include o-o-@ because that's the dual shape, and we don't need to include o-@-@ because that's the truncated form of the dual shape. Does that mean that taking the dual is another operation, one that flips diagrams from left to right? Not exactly. Taking the dual is an operation that you can perform on any shape, but it doesn't just flip diagrams. For example, the dual of the cuboctahedron (o-@-o) is the rhombic dodecahedron, a nice shape but not one we're talking about here. The operation that flips diagrams is taking the dual of the original shape.
What about 4D? Well, you can do exactly the same thing there. Instead of a triangle with three vertices, you have to imagine a kind of right-angled tetrahedron with four vertices, and so on, and here's the result. The first four can be produced by truncation.
@-o-o-o : the original shape
The words "rectified" and "cantellated" are troublesome in the same way that "face" is. When we talk about faces in 3D, we mean two-dimensional bounding elements, but in 4D the bounding elements and the two-dimensional elements are different things, so part of the meaning gets lost. In the same way, when we move from 3D to 4D, part of the meaning of "cantellated" goes to @-o-@-o and part goes to @-o-o-@.
By the way, my current favorite uniform polychoron is the bitruncated tesseract. The truncation process has reduced the eight cubic faces of the tesseract to truncated octahedra, but they're all still in solid contact with each other via their square subfaces. At the same time, the sixteen new faces produced by truncation, originally tetrahedra, have come into solid contact with each other and are now truncated tetrahedra! It's very 4D.
That's about all I wanted to say about truncation and other operations. It's a reasonable self-contained explanation, but in case you want to do further reading elsewhere, here's some information that should help get you going.
As one idea for further reading, the Wikipedia page about Schläfli symbols (here) has some nice tables of all these operations (about halfway down).