Hexagonal antifastegium

Hexagonal antifastegium
(OFF file)
Rank	4
Type	Segmentotope
Space	Spherical
Notation
Bowers style acronym	Haf
Coxeter diagram	ox xo6ox&#x
Elements
Cells	6 tetrahedra, 6 square pyramids, 1 hexagonal prism, 2 hexagonal antiprisms
Faces	12+12+12 triangles, 6 squares, 1+2 hexagons
Edges	6+6+12+24
Vertices	6+12
Vertex figures	6 wedges, edge lengths √3 (top) and 1 (remaining edges)
	12 isosceles trapezoidal pyramids, base edge lengths 1, 1, 1, √3, side edge lengths 1, 1, √2, √2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{19+6\sqrt3}{23}} ≈ 1.13045$
Hypervolume	$\sqrt{\frac{2+3\sqrt3}8} ≈ 0.94843$
Dichoral angles	Tet–3–squippy: $\arccos\left(\frac{2-3\sqrt3}4\right) ≈ 143.03835°$
	Hip–4–squippy: $\arccos\left(-\sqrt{\frac{7-4\sqrt3}2}\right) ≈ 100.92178°$
	Hap–3–tet: $\arccos\left(-\frac{\sqrt{3\sqrt3-5}}4\right) ≈ 96.35698°$
	Hap–3–squippy: $\arccos\left(\frac{\sqrt{3\sqrt3-5}}4\right) ≈ 83.64302°$
	Hap–6–hap: $\arccos\left(\frac{3-\sqrt3}4\right) ≈ 71.51917°$
	Hip–6–hap: $\arccos\left(\sqrt{\frac{1+\sqrt3}8}\right) ≈ 54.24041°$
Heights	Hig atop hap: $\sqrt{\frac{7-\sqrt3}8} ≈ 0.81148$
	Hig atop gyro hip: $\frac{\sqrt{4\sqrt3-5}}2 ≈ 0.69430$
Central density	1
Related polytopes
Army	Haf
Regiment	Haf
Dual	Hexagonal antitegmatonotch
Conjugate	Hexagonal antifastegium
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	G2×A1×I, order 24
Convex	Yes
Nature	Tame

The hexagonal antifastegium, or haf, is a CRF segmentochoron (designated K-4.46 on Richard Klitzing's list). It consists of 1 hexagonal prism, 2 hexagonal antiprisms, 6 tetrahedra, and 6 square pyramids. It is a member of the infinite family of polygonal antifastegiums.

It is a segmentochoron between a hexagon and a hexagonal antiprism or between a hexagon and a gyro hexagonal prism.

Vertex coordinates

The vertices of a hexagonal antifastegium of edge length 1 are given by:

• $\left(±\frac12,\,±\frac{\sqrt3}2,\,±\frac12,\,0\right),$
• $\left(±1,\,0,\,±\frac12,\,0\right),$
• $\left(±\frac{\sqrt3}2,\,±\frac12,\,0,\,\frac{\sqrt{4\sqrt3-5}}2\right),$
• $\left(0,\,±1,\,0,\,\frac{\sqrt{4\sqrt3-5}}2\right).$

Representations

The hexagonal antifastegium can be represented by the following Coxeter diagrams:

• ox xo6ox&#x
• xoo6oxx&#x

External links

• Klitzing, Richard. "haf".