I'm reading about dihedral groups right now, and I'm being asked to find the orders of various groups of rigid motions of 3-dimensional polygons. What I want to know, though, is if there's a nice presentation of the group of symmetries of any 3-dimensional polygons.

I'm curious in particular about each of the platonic solids. I don't really expect there to be a nice rule like there is in $D_{2n}$, but I'm still curious.


a There is indeed a procedure to write down a presentation for the symmetry group of a platonic solid. The key word is ``Coxeter groups''.

Take the cube, for example. The first step is geometric: subdivide the surface of the cube into fundamental domains by subdividing each square face into eight isosceles right triangles. There are $6 \cdot 8 = 48$ triangles in all (and that number equals the order of the symmetry group). Fix one of those triangles $T$. Reflecting the cube across those three sides of $T$ gives three generators $a,b,c$ of the symmetry group. Each of those three reflections has order two, which gives three relators $a^2 = Id$, $b^2 = Id$, $c^2 = Id$. For each of the three vertices $x$ of $T$ there is a relator as well: the product of the two reflections across the sides of $T$ incident to $x$ has finite order. For example, if $x$ is the center of the square face containing $T$, if $a$ is the reflection across the hypotenuse of $T$, and if $b$ is the reflection across the leg of $T$ incident to $x$, then $ab$ is an order 4 rotation around the point $x$, giving a relator $(ab)^4=Id$.

You get a presentation $$\langle a,b,c \, | \, a^2 = b^2 = c^2 = (ab)^4 = (bc)^2 = (ac)^3 = Id\rangle $$ This is one example of a Coxeter presentation. Each Platonic solid gives you a different Coxeter presentation, by following the same procedure.


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