M Theory Lesson 204

An octahedron is also known as a rectified tetrahedron, where rectification is the truncation of corners from midpoints along each edge. This construction paints the faces of the octahedron in two colours, depending on whether the face arises from a tetrahedral face or an interior surface. The group of 24 symmetries of this object is isomorphic to the permutation group $S_{4}$. The 48 element quaternionic octahedral group is associated to a double cover of the genus 3 Klein curve. The special quaternion $q = i \textrm{exp}(\frac{\pi j}{4})$ is used to give the relations for this group in terms of the two generators $a$ and $b$. Kneemo pointed out that one can use this representation, along with octonions, to describe the units of the $E8$ lattice.

Now let's have fun rectifying the other polytopes that arise in ternary geometry. A rectified cube has four square and four triangular faces. The dual to a cube, an octahedron, is a birectified cube. A rectified dodecahedron is a icosidodecahedron. An example in the plane turns a heptagon tiling into a tiling with heptagons and triangles.