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)
- (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 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
- ↑ Blind, R. (1979), "Konvexe Polytope mit kongruenten regulären [math]\displaystyle{ (n-1) }[/math]-Seiten im [math]\displaystyle{ \R^n }[/math] ([math]\displaystyle{ n\ge 4 }[/math])" (in de), Commentarii Mathematici Helvetici 54 (2): 304–308, doi:10.1007/BF02566273
- ↑ Klitzing, Richard, "Johnson solids, Blind polytopes, and CRFs", Polytopes, https://bendwavy.org/klitzing/explain/johnson.htm#blind, retrieved 2022-11-14
- ↑ "aurap". https://bendwavy.org/klitzing/incmats/aurap.htm. Retrieved 10 April 2023.
- ↑ "Johnson solids et al.". https://bendwavy.org/klitzing/explain/johnson.htm#crf. Retrieved 10 April 2023.
- Blind, Roswitha (1979). "Konvexe Polytope mit regulären Facetten im Rn (n≥4)". in Tölke, Jürgen (in de). Contributions to Geometry: Proceedings of the Geometry-Symposium held in Siegen June 28, 1978 to July 1, 1978. Birkhäuser, Basel. pp. 248–254. doi:10.1007/978-3-0348-5765-9_10.
- Blind, Gerd; Blind, Roswitha (1980). "Die konvexen Polytope im R4, bei denen alle Facetten reguläre Tetraeder sind" (in de). Monatshefte für Mathematik 89 (2): 87–93. doi:10.1007/BF01476586.
- Blind, Gerd; Blind, Roswitha (1989). "Über die Symmetriegruppen von regulärseitigen Polytopen" (in de). Monatshefte für Mathematik 108 (2–3): 103–114. doi:10.1007/BF01308665.
- Blind, Gerd; Blind, Roswitha (1991). "The semiregular polytopes". Commentarii Mathematici Helvetici 66: 150–154. doi:10.1007/BF02566640.
External links
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