Truncating Fruit Polyhedra

If, by chance, you have a pile of malic Platonic solids, you may be wondering what to do with them next. Luckily I have your answer: you can truncate them to get a whole new set of polyhedra! Truncating a polyhedron means cutting off its corners, and while usually this is a descriptive term used to help name a shape, in this case we get to literally cut the corners off of the polyhedra.

The tetrahedron only has four corners, so it's the simplest to start with. The key to a good truncation is to cut the corners evenly, so that the leftover pieces are all little pyramids of the same size. In this case, the bits you cut off will be triangular pyramids, also known as tetrahedra!

After cutting off the corners, there will be four new triangular faces, and the old faces of the tetrahedron will be hexagons. Depending on how much you cut, the hexagons will be more or less regular. The polyhedron that people usually think of when you say "truncated tetrahedron" is the state where the original faces are regular. If you keep cutting slices off the new triangles, you'll get to a point where they meet. The hexagon faces become degenerate—pairs of vertices coincide so that you get triangles. This shape might look familiar to you; four old triangular faces plus four new ones make an octahedron!

Starting from an octahedron, cutting off corners leaves little squares, because four faces meet at each vertex. Like the tetrahedron, the original faces are also triangles, and so they also become hexagons—only this time there's eight of them.

Like the tetrahedron, if you keep paring down the new cuts, eventually they meet. This shape, with eight triangles (like an octahedron) and six squares (like a cube) is aptly called the cuboctahedron.

The cube has three faces meeting at each vertex, so cutting the corners off makes new triangular faces—eight to be exact. Like the tetrahedron, the bits chopped off the corners are little tetrahedra, but they are not regular tetrahedra because three of the four triangles are right rather than equilateral. The old square faces become octagons, but if you keep truncating they degenerate back into squares. This shape should look familiar: it's the cuboctahedron again! This happens because the cube and octahedron have a special relationship: they're duals.

The truncated icosahedron is another familiar polyhedron: it's the shape of a soccerball (or football). It may be difficult to see here because my apple icosahedron was not all to perfect to start with, but it has twelve pentagonal faces surrounded by twenty hexagonal ones.

Overly-truncating will lead to the icosadodecahedron, with twelve pentagons and twenty triangles.

The dodecahedron overtruncates (I don't know if there's a technical term out there for this) into the icosadodecahedron also, for the same reason that the cube and octahedron both overtruncate to the same thing.

Extra cuboctahedra? No problem! We can truncate those too!

Polyhedral Fun Fact: the shape commonly known as the truncated cuboctahedron is not actually a truncated cuboctahedron like we see above. My apples may not be perfect, but the new faces really are supposed to be rectangles, when truncating a cuboctahedron. The solid known as the truncated cuboctahedron is an Archimedean solid which has squares rather than rectangles, and is in my opinion better called the great rhombicuboctahedron. Below, we see what happens if we overtruncate, which is a polyhedron similar to the rhombicuboctahedron, but once again with rectangles instead of being regularized with squares. (Does anyone know a better name for this shape than "the union of the cuboctahedron and its dual the rhombic dodecahedron"?)

I've still got another extra cuboctahedron, so I decided to indulge in my habit of cutting things in half to find hexagons:

Polyhedral Fun Fact: the two halves of the cuboctahedron, if put back together with a 60-degree twist, make the Johnson solid called the triangular orthobicupola!

Finally, I cut the icosadodecahedron in half across the core of the apple not to show any property of the symmetry of the icosadodecahedron, but to show off the lovely five-fold symmetry inherent in the apple:

Most apples have five-fold symmetry, but I've also seen examples of 7-fold symmetry. I have yet to know whether this has to do with random variation or whether some varieties of apple tend to have different symmetry. I also wonder: is the symmetry always an odd number? A prime number?