# 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$\sqrt{\frac{19+6\sqrt3}{23}} ≈ 1.13045$ Hypervolume$\sqrt{\frac{2+3\sqrt3}8} ≈ 0.94843$ Dichoral anglesTet–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: $\arccos\left(\sqrt{\frac{1+\sqrt3}8}\right) ≈ 54.24041°$ HeightsHig atop hap: $\sqrt{\frac{7-\sqrt3}8} ≈ 0.81148$ Hig atop gyro hip: $\frac{\sqrt{4\sqrt3-5}}2 ≈ 0.69430$ 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:

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

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

• ox xo6ox&#x
• xoo6oxx&#x