In geometry, a Blind polytope is a convex polytope composed of regular polytope facets. The category was named after the German couple Gerd and Roswitha Blind, who described them in a series of papers beginning in 1979. It generalizes the set of semiregular polyhedra and Johnson solids to higher dimensions.
Uniform cases The set of convex uniform 4-polytopes (also called semiregular 4-polytopes) are completely known cases, nearly all grouped by their Wythoff constructions, sharing symmetries of the convex regular 4-polytopes and prismatic forms. Set of convex uniform 5-polytopes, uniform 6-polytopes, uniform 7-polytopes, etc are largely enumerated as Wythoff constructions, but not known to be complete.
Other cases Pyramidal forms: (4D)
(Tetrahedral pyramid, ( ) ∨ {3,3}, a tetrahedron base, and 4 tetrahedral sides, a lower symmetry name of regular 5-cell.) Octahedral pyramid, ( ) ∨ {3,4}, an octahedron base, and 8 tetrahedra sides meeting at an apex. Icosahedral pyramid, ( ) ∨ {3,5}, an icosahedron base, and 20 tetrahedra sides. Bipyramid forms: (4D)
Tetrahedral bipyramid, { } + {3,3}, a tetrahedron center, and 8 tetrahedral cells on two side. (Octahedral bipyramid, { } + {3,4}, an octahedron center, and 8 tetrahedral cells on two side, a lower symmetry name of regular 16-cell.) Icosahedral bipyramid, { } + {3,5}, an icosahedron center, and 40 tetrahedral cells on two sides. Augmented forms: (4D)
Rectified 5-cell augmented with one octahedral pyramid, adding one vertex for 11 total. It retains 5 tetrahedral cells, reduced to 4 octahedral cells and adds 8 new tetrahedral cells. Diminished forms: (4D)
Precisely 314,248,344 "special cuts" of the 600-cell, all but one of which (the icositetradiminished 600-cell) are non-uniform. They are created by diminishing a set of non-adjacent vertices and replacing 20 tetrahedra with an icosahedron for each selected vertex.
Convex regular-faced polytopes Blind polytopes are a subset of convex regular-faced polytopes (CRF). This much larger set allows CRF 4-polytopes to have Johnson solids as cells, as well as regular and semiregular polyhedral cells. For example, a cubic bipyramid has 12 square pyramid cells.
References
Blind, Roswitha (1979). "Konvexe Polytope mit regulären Facetten im Rn (n≥4)" [Convex polytopes with regular facets in Rn (n≥4)]. In Tölke, Jürgen; Wills, Jörg. M. (eds.). Contributions to Geometry: Proceedings of the Geometry-Symposium held in Siegen June 28, 1978 to July 1, 1978 (in German). Birkhäuser, Basel. pp. 248–254. doi:10.1007/978-3-0348-5765-9_10.{{cite book}}: CS1 maint: location missing publisher (link) Blind, Gerd; Blind, Roswitha (1980). "Die konvexen Polytope im R4, bei denen alle Facetten reguläre Tetraeder sind" [All convex polytopes in R4, the facets of which are regular tetrahedra]. Monatshefte für Mathematik (in German). 89 (2): 87–93. doi:10.1007/BF01476586. S2CID 117654776. Blind, Gerd; Blind, Roswitha (1989). "Über die Symmetriegruppen von regulärseitigen Polytopen" [On the symmetry groups of regular-faced polytopes]. Monatshefte für Mathematik (in German). 108 (2–3): 103–114. doi:10.1007/BF01308665. S2CID 118720486. Blind, Gerd; Blind, Roswitha (1991). "The semiregular polytopes". Commentarii Mathematici Helvetici. 66: 150–154. doi:10.1007/BF02566640. S2CID 119695696.
External links Blind polytope Convex regular-faced polytopes