# Blind polytope > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Blind_polytope > Markdown URL: https://mediated.wiki/source/Blind_polytope.md > Source: https://en.wikipedia.org/wiki/Blind_polytope > Source revision: 1315199722 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Convex polytope composed of regular-polytope facets}} In [geometry](/source/geometry), a '''Blind polytope''' is a [convex polytope](/source/convex_polytope) composed of [regular polytope](/source/regular_polytope) [facets](/source/facet_(mathematics)). The category was named after the German couple Gerd and [Roswitha Blind](/source/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](/source/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](/source/semiregular_polyhedra) and [Johnson solid](/source/Johnson_solid)s 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-polytope](/source/convex_uniform_4-polytope)s (also called semiregular 4-polytopes) are completely known cases, nearly all grouped by their [Wythoff construction](/source/Wythoff_construction)s, sharing symmetries of the [convex regular 4-polytope](/source/convex_regular_4-polytope)s and prismatic forms. Set of convex [uniform 5-polytope](/source/uniform_5-polytope)s, [uniform 6-polytope](/source/uniform_6-polytope)s, [uniform 7-polytope](/source/uniform_7-polytope)s, etc are largely enumerated as Wythoff constructions, but not known to be complete. == Other cases== Pyramidal forms: (4D) # (''Tetrahedral pyramid'', ( ) ∨ {3,3}, a [tetrahedron](/source/tetrahedron) base, and 4 tetrahedral sides, a lower symmetry name of regular [5-cell](/source/5-cell).) # [Octahedral pyramid](/source/Octahedral_pyramid), ( ) ∨ {3,4}, an [octahedron](/source/octahedron) base, and 8 [tetrahedra](/source/tetrahedra) sides meeting at an apex. # [Icosahedral pyramid](/source/Icosahedral_pyramid), ( ) ∨ {3,5}, an [icosahedron](/source/icosahedron) base, and 20 [tetrahedra](/source/tetrahedra) sides. Bipyramid forms: (4D) # [Tetrahedral bipyramid](/source/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](/source/16-cell).) # [Icosahedral bipyramid](/source/Icosahedral_bipyramid), { } + {3,5}, an icosahedron center, and 40 tetrahedral cells on two sides. Augmented forms: (4D) * [Rectified 5-cell](/source/Rectified_5-cell) augmented with one [octahedral pyramid](/source/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 solid](/source/Johnson_solid)s as cells, as well as regular and semiregular polyhedral cells. For example, a [cubic bipyramid](/source/cubic_bipyramid) has 12 [square pyramid](/source/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 '''R'''n (n≥4)|trans-chapter=Convex polytopes with regular facets in '''R'''n (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 '''R'''4, bei denen alle Facetten reguläre Tetraeder sind|trans-title=All convex polytopes in '''R'''4, 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](/source/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 {{geometry-stub}} --- Adapted from the Wikipedia article [Blind polytope](https://en.wikipedia.org/wiki/Blind_polytope) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Blind_polytope?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.