Distinguished generators

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

Definition

Flag changes

The distinguished generators acting as flag changes of the flag of a cube.

Given an abstract regular polytope the nth distinguished generator $\rho_n$ is the automorphism that maps each flag to the unique n-adjacent flag.

Algebraic

The distinguished generators of an abstract regular polytope are indexed generators of the polytope's automorphism group, $\left\langle\rho_0,\dots,\rho_{n-1}\right\rangle$ such that:

• Each generator $\rho_i$ is an involution ($\rho_i\rho_i = 1$).
• Two generators $\rho_i$ and $\rho_j$ commute ($\rho_i\rho_j=\rho_j\rho_i$) if $0\leq i\leq j-2\leq n-3$.
• For any $J,K \subseteq \{0,\dots,n-1\}$, $\left\langle\rho_i\mid i\in J\right\rangle \cap \left\langle\rho_i\mid i\in K\right\rangle = \left\langle\rho_i\mid i\in J\cap K\right\rangle$.

This last property is called 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.

Concrete polytopes

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

Unlike abstract distinguished generators, concrete distinguished generators can be ambiguous, however most operations yield the same result irrespective of the choice of distinguished generators.

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 $\left\langle\rho_0,\dots,\rho_{n-1}\right\rangle$:

• Dual (δ): $\left\langle\rho_2,\rho_1,\rho_0\right\rangle$
• Petrial (π): $\left\langle\rho_0\rho_2,\rho_1,\rho_2\right\rangle$
• Halving (η): $\left\langle\rho_0\rho_1\rho_0,\rho_2,\rho_1\right\rangle$
• Skewing (σ): $\left\langle\rho_1,\rho_0\rho_2,(\rho_1\rho_2)^2\right\rangle$

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.

See also

• Complex