Distinguished generators

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

Definition[edit | edit source]

Flag changes[edit | edit source]

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

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

Algebraic[edit | edit source]

The distinguished generators of an abstract regular polytope are indexed generators of the polytope's automorphism group, such that:

  • Each generator is an involution ().
  • Two generators and commute () if .
  • For any , .

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[edit | edit source]

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[edit | edit source]

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

Operations[edit | edit source]

Operations on regular polyhedra[edit | edit source]

The following operations can be defined in terms of distinguished generators of the form :

  • Dual (δ):
  • Petrial (π):
  • Halving (η):
  • Skewing (σ):

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[edit | edit source]