Blind polytope explained

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.It generalizes the set of semiregular polyhedra and Johnson solids to higher dimensions.

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, ( ) ∨, a tetrahedron base, and 4 tetrahedral sides, a lower symmetry name of regular 5-cell.)
2. Octahedral pyramid, ( ) ∨, an octahedron base, and 8 tetrahedra sides meeting at an apex.
3. Icosahedral pyramid, ( ) ∨, an icosahedron base, and 20 tetrahedra sides.

Bipyramid forms: (4D)

1. Tetrahedral bipyramid, +, a tetrahedron center, and 8 tetrahedral cells on two side.
2. (Octahedral bipyramid, +, an octahedron center, and 8 tetrahedral cells on two side, a lower symmetry name of regular 16-cell.)
3. Icosahedral bipyramid, +, 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.^[1]

Convex Regular-Faced Polytopes

Blind polytopes are a subset of convex regular-faced polytopes (CRF).^[2] 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

• Book: Blind, Roswitha. Contributions to Geometry: Proceedings of the Geometry-Symposium held in Siegen June 28, 1978 to July 1, 1978. 1979. Jürgen. Tölke. Jörg. M.. Wills. Konvexe Polytope mit regulären Facetten im Rⁿ (n≥4). Convex polytopes with regular facets in Rⁿ (n≥4). Birkhäuser, Basel. 248–254. de. 10.1007/978-3-0348-5765-9_10.
• de. 10.1007/BF01476586. Gerd. Blind. Roswitha. Blind. 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. 89. 87–93. 1980. 2 . 117654776 .
• de. 10.1007/BF01308665. Über die Symmetriegruppen von regulärseitigen Polytopen. On the symmetry groups of regular-faced polytopes. Gerd. Blind. Roswitha. Blind. Monatshefte für Mathematik. 108. 103–114. 1989. 2–3 . 118720486 .
• 10.1007/BF02566640. The semiregular polytopes. Gerd. Blind. Roswitha. Blind. Commentarii Mathematici Helvetici. 66. 150–154. 1991. 119695696 .

External links

• Blind polytope
• Convex regular-faced polytopes

Notes and References

1. Web site: aurap. bendwavy.org. 10 April 2023.
2. Web site: Johnson solids et al.. bendwavy.org. 10 April 2023.