# Hexagonal antifastegium

Hexagonal antifastegium
Rank4
TypeSegmentotope
SpaceSpherical
Notation
Bowers style acronymHaf
Coxeter diagramox xo6ox&#x
Elements
Cells6 tetrahedra, 6 square pyramids, 1 hexagonal prism, 2 hexagonal antiprisms
Faces12+12+12 triangles, 6 squares, 1+2 hexagons
Edges6+6+12+24
Vertices6+12
Vertex figures6 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${\displaystyle \sqrt{\frac{19+6\sqrt3}{23}} ≈ 1.13045}$
Hypervolume${\displaystyle \sqrt{\frac{2+3\sqrt3}8} ≈ 0.94843}$
Dichoral anglesTet–3–squippy: ${\displaystyle \arccos\left(\frac{2-3\sqrt3}4\right) ≈ 143.03835°}$
Hip–4–squippy: ${\displaystyle \arccos\left(-\sqrt{\frac{7-4\sqrt3}2}\right) ≈ 100.92178°}$
Hap–3–tet: ${\displaystyle \arccos\left(-\frac{\sqrt{3\sqrt3-5}}4\right) ≈ 96.35698°}$
Hap–3–squippy: ${\displaystyle \arccos\left(\frac{\sqrt{3\sqrt3-5}}4\right) ≈ 83.64302°}$
Hap–6–hap: ${\displaystyle \arccos\left(\frac{3-\sqrt3}4\right) ≈ 71.51917°}$
Hip–6–hap: ${\displaystyle \arccos\left(\sqrt{\frac{1+\sqrt3}8}\right) ≈ 54.24041°}$
HeightsHig atop hap: ${\displaystyle \sqrt{\frac{7-\sqrt3}8} ≈ 0.81148}$
Hig atop gyro hip: ${\displaystyle \frac{\sqrt{4\sqrt3-5}}2 ≈ 0.69430}$
Central density1
Related polytopes
ArmyHaf
RegimentHaf
DualHexagonal antitegmatonotch
ConjugateHexagonal antifastegium
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryG2×A1×I, order 24
ConvexYes
NatureTame

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:

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

## Representations

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

• ox xo6ox&#x
• xoo6oxx&#x