### 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.

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.

## 3 Comments:

Kea, about "... rectification ... the truncation of corners from midpoints along each edge ..." of a polytope,

you can also "Golden Rectify" by using Golden Ratio points (instead of midpoints) along each edge.

For examples:

consistent Golden Ratio points on each of the 12 edges of the Octahedron give the 12 vertices of an icosahedron;

consistent Golden Ratio points on each of the 96 edges of the 24-cell give 96 of the 120 vertices of the 600-cell (the other 120-96 = 24 being the vertices of the 24-cell itself).

Since the 120 vertices of the 4-dim 600-cell contain some coordinates with sqrt(5) (due to the Golden Ratio),

you can use a sqrt(5) extension to expand the 4-dim space to an 8-dim space for the 120 vertices,

and

then you see that the 120 vertices are half of the 240 vertices of an E8 root vector polytope.

My guess is that the different ways to choose consistent Golden Ratio points on the 96 edges of the 24-cell

look like the 7 ways you can construct the 240-vertex E8 polytope

which in turn look like the 7 imaginary octonions and the 7 E8 lattices.

Tony Smith

Great drawing, reminds me of constructing octahedrons from tetrahedrons as a child. Lots of us were inspred by geometry.

Thanks, Tony! The golden ratio comes up in so many places. This is particularly interesting because it links together some of Arnold's trinities.

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