# Fully Truncated Simplices

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Barry Monson

University of New Brunswick

####
Leah Berman

U. Alaska- Fairbanks

####
Deborah Oliveros

UNAM - Quer\'{e}taro

####
Gordon Williams

U. Alaska - Fairbanks

PDF

**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
$$ \frac{[(n+1)!]^{n-1} \cdot (n-1)!}{2^{n-2}}$$
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I will discuss all this and report on the current state of my confusion
amongst all this fun. (How can one call mathematics `work'?)