# Abstract regular polytope

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Just as a regular polytope is a polytope whose symmetry is flag-transitive, an abstract regular polytope is an abstract polytope whose automorphism group is flag-transitive.

As they are not constrained by geometry, abstract regular polytopes are much more diverse than their concrete counterparts.

## Definition

Classically there are two definitions of abstract regular polytopes: one which constrains the definition of abstract polytope, and one which builds them from first principles as new objects objects. Other than the nullitope which only satisfies the first definition, two definitions are equivalent.

### As abstract polytopes

Let π be an abstract polytope. π is regular iff for any two flags x  and y  of π, there is an automorphism ${\displaystyle \gamma \in \Gamma ({\mathcal {P}})}$ such that ${\displaystyle \gamma (x)=y}$.

That is to say, the automorphism group of π acts transitively on its flags.

### Distinguished generators

Let π, be a group G  with set of generators, Ο  indexed by integers 0 to n -1. We call this indexed set the distinguished generators of π. We call the subgroup of G  generated by a subset of Ο  a distinguished subgroup, and we use the following notation:

${\displaystyle \Phi _{I}=\langle \rho _{i}\mid i\in I\rangle }$

Then π is an abstract regular polytope if it satisfies the following properties:

• Each distinguished generator Ο i  is an involution (${\displaystyle \rho _{i}\rho _{i}=1}$).
• Non-adjacent generators commute. i.e. two generators Ο i  and Ο j  commute (${\displaystyle \rho _{i}\rho _{j}=\rho _{j}\rho _{i}}$) if ${\displaystyle |i-j|>1}$.
• For any ${\displaystyle J,K\subseteq \{0,\dots ,n-1\}}$, ${\displaystyle \Phi _{J}\cap \Phi _{K}=\Phi _{J\cap K}}$.

The second property is a version of the diamond condition. This last property is called the intersection property or intersection condition.

From here we can build up the properties normally associated with an abstract polytope:

• The rank of π is the number of distinguished generators.
• The flags of π are the elements of G
• Two flags x  and y  are i -adjacent iff ${\displaystyle \rho _{i}x=y}$
• The proper i -elements of π are the orbits of the group generated by all the distinguished generators of π other than Ο i  acting on G .
• Two elements of π are incident with each other iff they share flags.

## Bibliography

• McMullen, Peter; Schulte, Egon (December 2002). Abstract Regular Polytopes. Cambridge University Press. ISBN 0-521-81496-0.