Internal and external duality in abstract polytopes

Gabe Cunningham, Mark Mixer


We define an abstract regular polytope to be internally self-dual if its self-duality can be realized
as one of its symmetries. This property has many interesting implications on the structure of the polytope,
which we present here. Then, we construct many examples of internally self-dual polytopes.
In particular, we show that there are internally self-dual regular polyhedra of each
type $\{p, p\}$ for $p \geq 3$ and that there are both infinitely many internally self-dual and infinitely many externally self-dual polyhedra
of type $\{p, p\}$ for $p$ even. We also show that there are internally self-dual polytopes in each rank,
including a new family of polytopes that we construct here.


abstract polytope; self-dual polytope; regular polytope

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Contributions to Discrete Mathematics. ISSN: 1715-0868