A non-involutory selfduality

• Published in 1989
• Added on 2026-04-27

Two polyhedra \(P_1\) and \(P_2\) are said to be duals of each other provided there exists a bijection \(\delta\) from the family of vertices and faces of \(P_1\) to the family of vertices and faces of \(P_2\) which reverses inclusion. IF there exists a duality map \(\delta\) from a polyhedron \(P\) to itself, we say that \(P\) is selfdual. In this case the selfduality map is a permutation on the set of vertices and faces of \(P\). The rank \(r(\delta)\) of a selfduality map \(\delta\) is defined as the smallest positive integer \(n\) such that \(\delta^n\) is the identity. The rank \(r(P)\) of a selfdual polyhedron \(P\) is the minimum value of \(r(\delta)\) over all selfduality maps \(\delta\) of \(P\). Grünbaum and Shephard have asked whether every selfdual convex polyhedron \(P\) has rank 2 or, equivalently, whether every selfdual \(P\) admits an involutary selfduality map. In this note we give an example of a polyhedron which provides the negative answer to the above problem.

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• https://www.sciencedirect.com/science/article/pii/0012365X89901441

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BibTeX entry

@article{Anoninvolutoryselfduality,
	key = {Anoninvolutoryselfduality},
	type = {article},
	title = {A non-involutory selfduality},
	author = {Stanislav Jendrol'},
	abstract = {Two polyhedra \(P{\_}1\) and \(P{\_}2\) are said to be duals of each other provided there exists a bijection \(\delta\) from the family of vertices and faces of \(P{\_}1\) to the family of vertices and faces of \(P{\_}2\) which reverses inclusion. IF there exists a duality map \(\delta\) from a polyhedron \(P\) to itself, we say that \(P\) is selfdual. In this case the selfduality map is a permutation on the set of vertices and faces of \(P\). The rank \(r(\delta)\) of a selfduality map \(\delta\) is defined as the smallest positive integer \(n\) such that \(\delta^n\) is the identity. The rank \(r(P)\) of a selfdual polyhedron \(P\) is the minimum value of \(r(\delta)\) over all selfduality maps \(\delta\) of \(P\).
Gr{\"{u}}nbaum and Shephard have asked whether every selfdual convex polyhedron \(P\) has rank 2 or, equivalently, whether every selfdual \(P\) admits an involutary selfduality map.
In this note we give an example of a polyhedron which provides the negative answer to the above problem.},
	comment = {},
	date_added = {2026-04-27},
	date_published = {1989-04-27},
	urls = {https://www.sciencedirect.com/science/article/pii/0012365X89901441},
	collections = {fun-maths-facts,geometry,things-to-make-and-do},
	url = {https://www.sciencedirect.com/science/article/pii/0012365X89901441},
	year = 1989,
	urldate = {2026-04-27}
}