Pentagonal double antitegmoid

Pentagonal double antitegmoid
Rank	4
Type	Uniform dual
Elements
Cells	100 order-5 truncated bi-apiculated tetrahedra
Faces	200 kites, 100 isosceles trapezoids, 200 pentagons
Edges	20+100+200+400
Vertices	20+100+200
Vertex figure	100+200 tetrahedra, 20 pentagonal antitegums
Measures (edge length 1)
Central density	1
Related polytopes
Dual	Grand antiprism
Abstract & topological properties
Flag count	8800
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(10)≀(S2/2), order 400
Convex	Yes
Nature	Tame

The pentagonal double antitegmoid or pentaantitegmatoswirlic hectochoron is a convex isotopic polychoron and member of the double antitegmoid family with 100 order-5 truncated bi-apiculated tetrahedra as cells. It can be obtained as the dual of the uniform grand antiprism. It is the first in an infinite family of isochoric pentagonal antitegmatic swirlchora.

It can be constructed by raising tall pyramids on 20 of the cells of the hecatonicosachoron corresponding to the vertices of a decagonal duotegum such that adjacent cells merge into order-5 truncated bi-apiculated tetrahedra.

Being the dual of the grand antiprism, this shape is sometimes called the grand antitegum, in analogy to how antitegums are the duals of antiprisms. Despite this name, this shape is neither a stellation nor an antitegum in any common sense of the word.

The cells of this polychoron can be constructed by augmenting tall pyramids onto two of the faces of a regular dodecahedron. As such they each have 4 identical geometrically regular pentagonal faces, with 2 isosceles trapezoids and 4 kites as well.

The ratio between the longest and shortest edges is $1:{\tfrac {3+{\sqrt {5}}}{2}}\approx 1:2.61803$.

Variations

The pentagonal double antitegmoid is part of a continuum of more general isochoric variations with full symmetry. It has two degrees of freedom. These polychora have mirror-symmetric pentagons instead of fully regular pentagons.

External links

• Klitzing, Richard. "gap dual".