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Blind polytope#Convex Regular-Faced Polytopes

{{Short description|Convex polytope composed of regular-polytope facets}}

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.{{citation | last = Blind | first = R. | doi = 10.1007/BF02566273 | issue = 2 | journal = Commentarii Mathematici Helvetici | language = de | mr = 535060 | pages = 304–308 | title = Konvexe Polytope mit kongruenten regulären (n-1)-Seiten im \R^n (n\ge 4) | volume = 54 | year = 1979| s2cid = 121754486 }}

It generalizes the set of semiregular polyhedra and Johnson solids to higher dimensions.{{citation|url=https://bendwavy.org/klitzing/explain/johnson.htm#blind|title=Johnson solids, Blind polytopes, and CRFs|work=Polytopes|first=Richard|last=Klitzing|access-date=2022-11-14}}

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 11 total. It retains 5 tetrahedral cells, reduced to 4 octahedral cells and adds 8 new tetrahedral cells.{{Cite web|url=https://bendwavy.org/klitzing/incmats/aurap.htm|title=aurap|website=bendwavy.org|accessdate=10 April 2023}}

Convex Regular-Faced Polytopes

Blind polytopes are a subset of convex regular-faced polytopes (CRF).{{Cite web|url=https://bendwavy.org/klitzing/explain/johnson.htm#crf|title=Johnson solids et al.|website=bendwavy.org|accessdate=10 April 2023}}

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

{{reflist}}

  • {{cite book|title=Contributions to Geometry: Proceedings of the Geometry-Symposium held in Siegen June 28, 1978 to July 1, 1978|year=1979|editor-first=Jürgen|editor-last=Tölke|editor-first2=Jörg. M.|editor-last2=Wills|first=Roswitha|last=Blind|author-link=Roswitha Blind|chapter=Konvexe Polytope mit regulären Facetten im Rn (n≥4)|trans-chapter=Convex polytopes with regular facets in Rn (n≥4)|place=Birkhäuser, Basel|pages=248–254|language=de|doi=10.1007/978-3-0348-5765-9_10}}
  • {{cite journal|language=de|doi=10.1007/BF01476586|first1=Gerd|last1=Blind|first2=Roswitha|last2=Blind|author-link2=Roswitha Blind|title=Die konvexen Polytope im R4, bei denen alle Facetten reguläre Tetraeder sind|trans-title=All convex polytopes in R4, the facets of which are regular tetrahedra|journal=Monatshefte für Mathematik|volume=89|pages=87–93|year=1980|issue=2 |s2cid=117654776 }}
  • {{cite journal|language=de|doi=10.1007/BF01308665|title=Über die Symmetriegruppen von regulärseitigen Polytopen|trans-title=On the symmetry groups of regular-faced polytopes|first1=Gerd|last1=Blind|first2=Roswitha|last2=Blind|author-link2=Roswitha Blind|journal=Monatshefte für Mathematik|volume=108|pages=103–114|year=1989|issue=2–3 |s2cid=118720486 }}
  • {{cite journal|doi=10.1007/BF02566640|title=The semiregular polytopes|first1=Gerd|last1=Blind|first2=Roswitha|last2=Blind|author-link2=Roswitha Blind|journal=Commentarii Mathematici Helvetici|volume=66|pages=150–154|year=1991|s2cid=119695696 }}

External links

  • [https://polytope.miraheze.org/wiki/Blind_polytope Blind polytope]
  • [https://polytope.miraheze.org/wiki/Convex_regular-faced_polytopes Convex regular-faced polytopes]

Category:Polytopes

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