# Distinguished generators

The distinguished generators are a generating set for a regular polytope's symmetry group that are useful for several definitions and operations, primarily for regular skew polytopes.

## Definition

### Flag changes

Given an regular polytope, we choose an arbitrary flag f  to be the base flag. Then the n th distinguished generator ${\displaystyle \rho _{n}}$ is the (unique) automorphism that maps f  to its n -adjacent flag.[1] The number of generators ${\displaystyle \rho _{n}}$ for a given base flag will be the same as the polytope's rank, and the group formed from these generators is the monodromy group which is equal to symmetry group when the polytope is regular.

Since all flags of a regular polytope are identical the choice of a base flag makes no difference to the ultimate structure of the group.

### Algebraic

The distinguished generators of an abstract regular polytope are indexed generators of the polytope's automorphism group, ${\displaystyle \left\langle \rho _{0},\dots ,\rho _{n-1}\right\rangle }$ such that:

• Each generator ${\displaystyle \rho _{i}}$ is an involution (${\displaystyle \rho _{i}\rho _{i}=1}$).
• Two generators ${\displaystyle \rho _{i}}$ and ${\displaystyle \rho _{j}}$ commute (${\displaystyle \rho _{i}\rho _{j}=\rho _{j}\rho _{i}}$) if ${\displaystyle 0\leq i\leq j-2\leq n-3}$.
• No generator ${\displaystyle \rho _{i}}$ can be expressed as a product of the other generators.

This last property is the intersection property. It may also be stated in other ways. Generalizations of abstract polytopes, like maniplexes, may require a weaker version of the intersection property.

In fact if an indexed set of generators satisfies the above properties they are the distinguished generators of some abstract regular polytope.

## Polytope realizations

A symmetric realization of a regular polytope also has distinguished generators for its symmetry group. These distinguished generators are generating mirrors, which satisfy the abstract requirements of distinguished generators.

Using the characterization of isometries in Euclidean space, it can be shown that any involutionary symmetry fixing the origin is a reflection over some subspace.

## Constructing a polytope

Given a set of distinguished generators we can reconstruct the regular polytope.

## Operations

### Operations on regular polyhedra

The following operations can be defined in terms of distinguished generators of the form ${\displaystyle \left\langle \rho _{0},\rho _{1},\rho _{2}\right\rangle }$:

• Dual (δ ): ${\displaystyle \left\langle \rho _{2},\rho _{1},\rho _{0}\right\rangle }$
• Petrial (π ): ${\displaystyle \left\langle \rho _{0}\rho _{2},\rho _{1},\rho _{2}\right\rangle }$
• Halving (η ): ${\displaystyle \left\langle \rho _{0}\rho _{1}\rho _{0},\rho _{2},\rho _{1}\right\rangle }$
• Skewing (σ ): ${\displaystyle \left\langle \rho _{1},\rho _{0}\rho _{2},(\rho _{1}\rho _{2})^{2}\right\rangle }$

The results of some of these operations (halving and skewing) may not always be an abstract polytope, these operations are often presented with additional restrictions on the input that ensure the result is an abstract polytope.

## References

1. McMullen & Schulte (2002:33)

## Bibliography

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