Fully Truncated Simplices

Minisymposium: POLYTOPES AND GRAPHS

Content: If you truncate an equilateral triangle $\{3\}$ to its edge midpoints you get another, smaller $\{3\}$. If you truncate a regular tetrahedron $\{3, 3\}$ to its edge midpoints, you get (a little unexpectedly) a regular octahedron $\{3, 4\}$. In higher ranks, the fully truncated n-simplex $\mathcal{T}_n$ is no longer regular, though it is uniform with facets of two types. We want to understand the minimal regular cover $\mathcal{R}_n$ of $\mathcal{T}_n$, which in turn means we need to understand its monodromy group $M_n := \mathrm{Mon}(\mathcal{T}_n)$. For $n\geq 4$, we `know' that this group has Schl\"{a}fli type $\{3,12,3,\ldots,3\}$ and the impressive order $$ \frac{[(n+1)!]^{n-1} \cdot (n-1)!}{2^{n-2}}$$ % I will discuss all this and report on the current state of my confusion amongst all this fun. (How can one call mathematics `work'?)

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