Blind polytope
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The Blind polytopes are the strictly convex polytopes whose facets are all regular. As such, they are a subclass of the convex regular-faced polytopes, and the non-uniform Blind polytopes generalize the Johnson solids. Blind polytopes are named after the researching German couple Gerd and Roswitha Blind, who listed all such polytopes in a series of papers during the 1980s.[1][2][3][4][5]
The uniform Blind polytopes are precisely the convex semiregular polytopes. The non-uniform ones are:
- the 92 Johnson solids (3-dimensional),
- the simplicial bipyramids (generalizing the triangular bipyramid, one in each dimension greater than 3),
- the orthoplecial pyramids (generalizing the square pyramid, one in each dimension greater than 3),
- the icosahedral pyramid (4-dimensional),
- the icosahedral bipyramid (4-dimensional),
- the augmented rectified pentachoron (4-dimensional),
- and the special cuts of the hexacosichoron (4-dimensional), minus the semiregular snub disicositetrachoron.
In 2008 Mathieu Dutour Sikirić and Wendy Myrvold finally managed to provide the number of polytopes in the last class to be 314,248,344, including the snub disicositetrachoron.[6] The only asymmetrical Blind polytopes are found in the special cuts.
References[edit | edit source]
- ↑ Blind, Roswitha (1979). "Konvexe Polytope mit kongruenten regulären (n–1)-Seiten im ℝⁿ (n≥4)" [Convex polytopes with congruent regular (n–1)-faces in ℝⁿ (n≥4)]. Commentarii Mathematici Helvetici (in German). 54: 304–308.
- ↑ 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.
- ↑ 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: 87–93. doi:10.1007/BF01476586.
- ↑ 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: 103–114. doi:10.1007/BF01308665.
- ↑ Blind, Gerd; Blind, Roswitha (1991). "The semiregular polytopes". Commentarii Mathematici Helvetici. 66: 150–154. doi:10.1007/BF02566640.
- ↑ Sikirić, Mathieu Dutour; Myrvold, Wendy (2008). "The Special Cuts of the 600-cell". Contributions to Algebra and Geometry. 49 (1): 269–275.
External links[edit | edit source]
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