Dork Scratchings

A while ago I was pondering the symmetries of Platonic solids and a strange thing occurred to me.  The number of rotational symmetries for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron are 12, 24, 24, 60, and 60 respectively $^\dagger$.  These happen to match the number of hours in an afternoon (I'm told mornings have the same number), the number of hours in a day, the number of minutes in an hour,  and the number of seconds in a minute. This is probably a coincidence due to the fact that the Babylonians (who created our time system) used base 60, and 60 is divisible by a lot of small numbers.  But it's nice to imagine, and just possible, that they were thinking about regular solids when then came up with the system for measuring time.  It's just turned midnight When this occurred to me a picture jumped into my head, which I've tried to recreate above.  It's a clock built from platonic solids.  Every second one or more of them rotate...

Generator symmetries c, b, and a, applied sequentially to the octahedron, resulting in a composite operation of order 2 I've become a bit obsessed with Cayley graphs recently. I figured out a way to construct a graph for the rotational symmetries of the 5 Platonic solids, and the result was quite elegant, I thought.  And it seemed to me that extending this to the complete set of  orthogonal symmetries (which includes reflections) should be quite simple - it is just twice the number of nodes after all.  However,  it took a surprising amount of staring into the middle distance and mumbling to myself to come up with the answer. And I'm going to describe its construction in this post.  As an example I'm using the octahedron here,  but the construction works in exactly the same way whichever solid you choose.  My generators I'm calling $a$, $b$, and $c$, where $a$ is a clockwise rotation of a face,  $b$ is a clockwise rotation around an adjoining verte...

In the previous post I used a short python program to draw a Cayley graph for the rotational symmetries of the cube.  The result was a rhombicuboctahedron, a kind of cross between a cube and an octahedron. Now,  these two platonic solids are "duals" of each other, which is to say that if you start with one and draw a node for each face,  and an edge for each pair of faces that meet, you end up with the other! Is there a pattern here? Can we choose generators for the rotational symmetry groups of the other platonic solids so that the resulting Cayley graphs look like crosses between the original solids and their duals? First let's see if we can find a generic representation for the rotational symmetry group.  Let's assume our solid has $n$ sided faces and the vertices are all of degree $m$. If $a$ is a clockwise rotation about one face and $b$ is a clockwise rotation about an adjoining vertex then $ab$ flips an edge about its midpoint, which implies $(ab)^2=1$. This s...

Definitions: A Cayley graph is a directed graph showing how to get from any element of a group to any other using only a finite number of "generator" elements.  For example, in the group $\mathbb{Z}_2\times\mathbb{Z}_2$ we could use the generators $a = (1,0)$ and $b = (0,1)$ and we'd end up with a graph that looks like a square with arrows going round the edges. A Group presentation defines a group by specifying generators and relations between them.  For example the presentation $\langle a,b \vert a^2=1,b^2=1,aba^{-1}b^{-1}=1\rangle$ specifies a group.  In fact the group it specifies is isomorphic to  $\mathbb{Z}_2\times\mathbb{Z}_2$ and we can prove this using the facts that $a$ and $b$ commute and both have order 2.  In this case it's easy to see what group we have but in general it's a bit more difficult.  A useful first step would be to be able to draw the Cayley graph automatically. Question: Is it possible to automatically generate a Cayley graph f...