Blind polytope

In

facets

. The category was named after the German couple Gerd and Roswitha Blind, who described them in a series of papers beginning in 1979.^[1] It generalizes the set of

semiregular polyhedra and Johnson solids to higher dimensions.^[2]

Uniform cases

The set of

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)

(Tetrahedral pyramid, ( ) ∨ {3,3}, a tetrahedron base, and 4 tetrahedral sides, a lower symmetry name of regular 5-cell.)
tetrahedra
sides meeting at an apex.
tetrahedra
sides.

Bipyramid forms: (4D)

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.)
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 13 total. It retains 5 tetrahedral cells, reduced to 4 octahedral cells and adds 8 new tetrahedral cells.^[3]

Convex Regular-Faced Polytopes

Blind polytopes are a subset of convex regular-faced polytopes (CRF).^[4] 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

^ Blind, R. (1979), "Konvexe Polytope mit kongruenten regulären $(n-1)$ -Seiten im $\mathbb {R} ^{n}$ ( $n\geq 4$ )", Commentarii Mathematici Helvetici (in German), 54 (2): 304–308,
S2CID 121754486

^ Klitzing, Richard, "Johnson solids, Blind polytopes, and CRFs", Polytopes, retrieved 2022-11-14
^ "aurap". bendwavy.org. Retrieved 10 April 2023.
^ "Johnson solids et al". bendwavy.org. Retrieved 10 April 2023.