Are all convex polyhedra tetragonalizable?
type: Item
instance of: scholarly article
zbMATH Open document ID: 1366.52009
author: Branko Grünbaum
publication date: 2016-10-13T00:00:00Z
Mathematics Subject Classification ID: 52B10
zbMATH DE Number: 6638579
zbMATH Keywords: tetragonalization
zbMATH Keywords: Platonic solids
zbMATH Keywords: semiregular polyhedra
zbMATH Keywords: circumscribed polyhedra
zbMATH Keywords: inscribe polyhedra
MaRDI profile type: Publication
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title: Are all convex polyhedra tetragonalizable?
published in: Geombinatorics
review text: A convex polyhedron in 3-space is called tetragonizable by a family of pyramids, all of them erected on the facets of that polyhedron, if the two pyramidal faces containing one of the edges of the original polyhedron are coplanar, thus forming a convex quadrangle which is then a facet of the obtained tetragonalization. E.g., the cube is a tetragonalization of the regular tetrahedron, the rhombic dodecahedron of the cube, and the rhombitriacontahedron of the regular icosahedron. Based on the nice (and unsettled) conjecture that each convex polyhedron is tetragonizable, and supporting this conjecture, the author proves that all simplicial polyhedra (having only triangular facets) are tetragonizable, and that all midscribable polyhedra (all whose edges are tangent to a sphere) have the same property.NEWLINENEWLINE Further parts of this nice paper refer to in-, mid- and circumscribable polyhedra and the conjecture that every tetragonizable polyhedron admits a continuum of tetragonalizations.
reviewed by: Horst Martini
description: scientific article; zbMATH DE number 6638579