Square antifastegium

Square antifastegium
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Squaf
Coxeter diagram	ox xo4ox&#x
Elements
Cells	4 tetrahedra, 4 square pyramids, 1 cube, 2 square antiprisms
Faces	8+8+8 triangles, 1+2+4 squares
Edges	4+4+8+16
Vertices	4+8
Vertex figures	4 wedges, edge lengths √2 (top) and 1 (remaining edges)
	8 isosceles trapezoidal pyramids, base edge lengths 1, 1, 1, √2, side edge lengths 1, 1, √2, √2
Measures (edge length 1)
Circumradius	${\sqrt {\frac {4+{\sqrt {2}}}{7}}}\approx 0.87947$
Hypervolume	${\frac {\sqrt {10+8{\sqrt {2}}}}{12}}\approx 0.38472$
Dichoral angles	Squap–3–tet: $\arccos \left(-{\frac {\sqrt {4+3{\sqrt {2}}}}{4}}\right)\approx 135.86903^{\circ }$
	Tet–3–squippy: $\arccos \left(-{\frac {3{\sqrt {2}}-2}{4}}\right)\approx 124.10147^{\circ }$
	Cube–4–squippy: $\arccos \left(-{\frac {2-{\sqrt {2}}}{2}}\right)\approx 107.03125^{\circ }$
	Squap–4–squap: $\arccos \left({\frac {2-{\sqrt {2}}}{2}}\right)\approx 72.96875^{\circ }$
	Cube–4–squap: $\arccos \left({\frac {\sqrt[{4}]{2}}{2}}\right)\approx 53.51562^{\circ }$
	Squap–3–squippy: $\arccos \left({\frac {\sqrt {4+3{\sqrt {2}}}}{4}}\right)\approx 44.13097^{\circ }$
Heights	Square atop squap: ${\frac {\sqrt {4-{\sqrt {2}}}}{2}}\approx 0.80402$
	Square atop gyro cube: ${\frac {\sqrt {2{\sqrt {2}}-1}}{2}}\approx 0.67610$
Central density	1
Related polytopes
Army	Squaf
Regiment	Squaf
Dual	Square antitegmatonotch
Conjugate	Square antifastegium
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B2×A1×I, order 16
Convex	Yes
Nature	Tame

The square antifastegium, or squaf, is a CRF segmentochoron (designated K-4.14 on Richard Klitzing's list). It consists of 1 cube, 2 square antiprisms, 4 tetrahedra, and 4 square pyramids. It is a member of the infinite family of polygonal antifastegiums.

It is a segmentochoron between a square and a square antiprism or between a square and a gyro cube.

It can be obtained as a diminishing of the segmentochoron octahedron atop cube by removing two opposite vertices from the top octahedron, cutting off two square antiprismatic pyramids.

Vertex coordinates

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

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

Representations

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

• ox xo4ox&#x (square atop gyro cube)
• xoo4oxx&#x (square atop gyro square atop gyro square)

External links

• Klitzing, Richard. "squaf".