### Carbon Beauty IV

Recall that the genus 0 Euler structure $(V,E,F)=(24,36,14)$ had two interesting models, namely the permutohedron (truncated octahedron) and the truncated cube. Similarly, the Euler structure $(12,18,8)$ has the two models of the truncated tetrahedron and the ternary polytope with four pentagons and four squares. Is there a dual for the truncated icosahedron? Yes, in fact the truncated dodecahedron shares the Euler structure $(60,90,32)$ with its 12 10-sided faces and 20 triangles. In summary, the three pairs of polytopes have the same dual decomposition into two types of face polygon. Note also that the dodecahedron itself is a so called fullerene graph because it has 12 pentagonal faces. Recall that this Platonic trinity is but one of many trinities matching the quaternionic $(\mathbb{R},\mathbb{C},\mathbb{H})$ triple, which appeared for instance in the ribbon graph matrix theory of Mulase et al. Observe that the transformation which takes the truncated tetrahedron to the other 8 sided polytope acts on two edges of a tetrahedron via a string type duality, deforming the hexagons on either side of the edge into pentagons and the triangles into squares. This is the self dual complex number case of flat ribbon graphs. In future M Theory lessons, we will look more carefully at twisted ribbon graphs associated to $\mathbb{R}$ and $\mathbb{H}$ and other triples related to ternary geometry. As Louise Riofrio would say, M Theory can be taught in kindergarten!

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