Wednesday, December 31, 2008

Wednesday Math, Vol. 54: The Truncated Platonic Solids

Last week, we took a look at the Platonic Solids, three dimensional shapes where every face is the same sized regular polygon. About two months ago, we looked at tilings of the plane with regular polygons, but mixing and matching the polygons, like the way square and octagonal linoleum tiles have been used together to make floor or wall patterns for a very long time.

(Boy, I bet you didn't realize you had to take notes to read this blog.)

The shapes we will explore in this post are the truncated Platonic solids. The idea is to take one of the five Platonic solids and shave off a flat surface from each corner, shaving off the same amount from each corner so we get a polygonal cut of the same shape and all the edges are the same length.
The last sentence is easier to type than to understand. Looking at a picture helps. Here in red and yellow with nice shiny silver edges and gold vertices is the truncated tetrahedron. The tetrahedron has four triangular faces, but by shaving off a triangle from each corner, what used to be triangles become the red hexagons, and all the yellow triangles are the faces created by the process of shaving off the original corners.

Slightly harder to visualize is that if we cut deeper into a tetrahedron, the red faces could be the sliced off corner parts and the yellow faces could be what is left of the original tetrahedron's faces.

These next three solids can be thought of as truncated cubes OR truncated octahedra. Recall that the cube and the octahedron are duals of each other, and the act of truncating one can be duplicated by truncating a correct sized version of the other. The first of the shapes is kind of like what most dice look like, where each of the corners is rounded slightly to avoid sharp edges.

Faces on shape #1: Octagons and triangles
Faces on shape #2: Squares and triangles
Faces on shape #3: Hexagons and squares.

We can also truncate dodecahedra (twelve sided polyhedra) and icosahedra (twenty sided polyhedra), and because they too are duals of each other, it depends on your perspective to answer the question which faces are the sliced off parts and which are the original faces.

Faces on shape #1: Decagons and triangles
Faces on shape #2: Pentagons and hexagons

If all the pentagons are given one color and all the hexagons another, the second shape is recognizable as the pattern of leather patches on a soccer ball. If there was a carbon atom at each corner and the edges were the bonds between them, the second shape is also C60, the molecule known either as a fullerene or a buckyball, both names in honor of Buckminister Fuller.

I like these shapes because of their symmetry. This isn't just Matty Boy being weird. Symmetry catches the eye not only of humans but of many species. In math, group theory is the algebraic study of symmetry, and there are many symmetries inside the symmetries. A group theory class in my junior year of college is the reason I changed from an English major to a math major.

Yay, Flags of many Lands™! Yay, Gabon!

Not too go into too great detail, but it appears that in Gabon, you will find at least one of... My People.

Welcome, brother! Or sister, that can happen, too. No judgments.