# Hexagonal antifastegium

Hexagonal antifastegium
Rank4
TypeSegmentotope
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${\displaystyle {\sqrt {\frac {19+6{\sqrt {3}}}{23}}}\approx 1.13045}$
Hypervolume${\displaystyle {\sqrt {\frac {2+3{\sqrt {3}}}{8}}}\approx 0.94843}$
Dichoral anglesTet–3–squippy: ${\displaystyle \arccos \left({\frac {2-3{\sqrt {3}}}{4}}\right)\approx 143.03835^{\circ }}$
Hip–4–squippy: ${\displaystyle \arccos \left(-{\sqrt {\frac {7-4{\sqrt {3}}}{2}}}\right)\approx 100.92178^{\circ }}$
Hap–3–tet: ${\displaystyle \arccos \left(-{\frac {\sqrt {3{\sqrt {3}}-5}}{4}}\right)\approx 96.35698^{\circ }}$
Hap–3–squippy: ${\displaystyle \arccos \left({\frac {\sqrt {3{\sqrt {3}}-5}}{4}}\right)\approx 83.64302^{\circ }}$
Hap–6–hap: ${\displaystyle \arccos \left({\frac {3-{\sqrt {3}}}{4}}\right)\approx 71.51917^{\circ }}$
Hip–6–hap: ${\displaystyle \arccos \left({\sqrt {\frac {1+{\sqrt {3}}}{8}}}\right)\approx 54.24041^{\circ }}$
HeightsHig atop hap: ${\displaystyle {\sqrt {\frac {7-{\sqrt {3}}}{8}}}\approx 0.81148}$
Hig atop gyro hip: ${\displaystyle {\frac {\sqrt {4{\sqrt {3}}-5}}{2}}\approx 0.69430}$
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:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,0\right),}$
• ${\displaystyle \left(\pm 1,\,0,\,\pm {\frac {1}{2}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,0,\,{\frac {\sqrt {4{\sqrt {3}}-5}}{2}}\right),}$
• ${\displaystyle \left(0,\,\pm 1,\,0,\,{\frac {\sqrt {4{\sqrt {3}}-5}}{2}}\right).}$

## Representations

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

• ox xo6ox&#x
• xoo6oxx&#x