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:

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. As such, there are 314,248,348 non-uniform Blind polytopes in 4 dimensions and only two in each higher number of dimensions.

References[edit | edit source]

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. Blind, Gerd; Blind, Roswitha (1991). "The semiregular polytopes". Commentarii Mathematici Helvetici. 66: 150–154. doi:10.1007/BF02566640.
  6. 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]