Blind polytope

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.^[1] It generalizes the set of semiregular polyhedra and Johnson solids to higher dimensions.^[2]

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)

1. (Tetrahedral pyramid, ( ) ∨ {3,3}, a tetrahedron base, and 4 tetrahedral sides, a lower symmetry name of regular 5-cell.)
2. Octahedral pyramid, ( ) ∨ {3,4}, an octahedron base, and 8 tetrahedra sides meeting at an apex.
3. Icosahedral pyramid, ( ) ∨ {3,5}, an icosahedron base, and 20 tetrahedra sides.

Bipyramid forms: (4D)

1. Tetrahedral bipyramid, { } + {3,3}, a tetrahedron center, and 8 tetrahedral cells on two side.
2. (Octahedral bipyramid, { } + {3,4}, an octahedron center, and 8 tetrahedral cells on two side, a lower symmetry name of regular 16-cell.)
3. 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 13 total. It retains 5 tetrahedral cells, reduced to 4 octahedral cells and adds 8 new tetrahedral cells.^[3]

Convex Regular-Faced Polytopes

Blind polytopes are a subset of convex regular-faced polytopes (CRF).^[4] 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

1. Blind, R. (1979), "Konvexe Polytope mit kongruenten regulären ${\displaystyle (n-1)}$ -Seiten im ${\displaystyle \mathbb {R} ^{n}}$  (${\displaystyle n\geq 4}$ )", Commentarii Mathematici Helvetici (in German), 54 (2): 304–308, doi:10.1007/BF02566273, MR 0535060, S2CID 121754486
2. Klitzing, Richard, "Johnson solids, Blind polytopes, and CRFs", Polytopes, retrieved 2022-11-14
3. "aurap". bendwavy.org. Retrieved 10 April 2023.
4. "Johnson solids et al". bendwavy.org. Retrieved 10 April 2023.

• 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