# Maniplex

A maniplex is a combinatorial model of polytopes.

Maniplexes can be thought of as complexes with a version of dyadicity.

## Definition

### Permutation sequence

An n -maniplex is tuple ${\displaystyle (\Omega ,A)}$, where Ω  is a set, and A  is a sequence of n  permutations on Ω , ${\displaystyle A:\{1,2,\dots ,n\}\rightarrow \operatorname {Aut} (\Omega )}$:

1. ${\displaystyle (\Omega ,A)}$ is a complex.
2. For any element, a , of the complex, the image of a  under ${\displaystyle A(n)}$ is also an element of the complex.
3. Every facet of the complex is a maniplex.

The second requirement can be seen as a form of dyadicity, however because maniplexes retain more information than just incidence the same facet of a maniplex can meet itself at a ridge.

## Bibliography

• Garza-Vargas, Jorge; Hubard, Isabel (7 July 2018). "Polytopality of Maniplexes" (PDF). arXiv:1604.01164.
• Wilson, Steve (2012). "Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators". Symmetry. doi:10.3390/sym4020265.