Square antiwedge

Square antiwedge
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Squaw
Coxeter diagram	os2xo8os&#x
Elements
Cells	8 square pyramids, 1 square antiprism, 2 square cupolas
Faces	8+8+8 triangles, 2+8 squares, 1 octagon
Edges	8+8+8+16
Vertices	8+8
Vertex figures	8 skewed wedges, edge lengths 1 (6) and √2 (3)
	8 sphenoids, edge lengths 1 (3), √2 (2), and √2+√2 (1)
Measures (edge length 1)
Circumradius	${\frac {{\sqrt {7}}+2{\sqrt {14}}}{7}}\approx 1.44701$
Hypervolume	${\sqrt {\frac {99+72{\sqrt {2}}}{448}}}\approx 0.66953$
Dichoral angles	Squippy–3–squippy: $\arccos \left({\frac {1-3{\sqrt {2}}}{4}}\right)\approx 144.16048^{\circ }$
	Squap–3–squippy: $\arccos \left(-{\frac {\sqrt {4+3{\sqrt {2}}}}{4}}\right)\approx 135.86903^{\circ }$
	Squacu–8–squacu: $\arccos \left({\frac {2-{\sqrt {2}}}{2}}\right)\approx 72.96875^{\circ }$
	Squacu–4–squippy: $\arccos \left({\frac {2-{\sqrt {2}}}{2}}\right)\approx 72.96875^{\circ }$
	Squacu–3–squippy: $\arccos \left({\frac {3{\sqrt {2}}-2}{4}}\right)\approx 55.89853^{\circ }$
	Squap–4–squacu: $\arccos \left({\frac {\sqrt[{4}]{2}}{2}}\right)\approx 53.51562^{\circ }$
Heights	Square atop gyro squacu: ${\frac {\sqrt {2{\sqrt {2}}-1}}{2}}\approx 0.67610$
	Squap atop oc: ${\sqrt {\frac {4-{\sqrt {2}}}{8}}}\approx 0.56853$
Central density	1
Related polytopes
Army	Squaw
Regiment	Squaw
Dual	Square gyrocupolanotch
Conjugate	Square antiwedge
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(8)×A1)/2×, order 16
Convex	Yes
Nature	Tame

The square antiwedge, or squaw, also sometimes called the square gyrobicupolic ring, is a CRF segmentochoron (designated K-4.64 on Richard Klitzing's list). It consists of 1 square antiprism, 2 square cupolas, and 8 square pyramids.

The square antiwedge can be thought of as a piece of the larger segmentochoron cuboctahedron atop small rhombicuboctahedron, with one base square being a face of the cuboctahedron, and the opposite square cupola being part of the small rhombicuboctahedron.

Vertex coordinates

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

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt[{4}]{8}}{4}},\,{\sqrt {\frac {4-{\sqrt {2}}}{8}}}\right),$
• $\left(0,\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt[{4}]{8}}{4}},\,{\sqrt {\frac {4-{\sqrt {2}}}{8}}}\right),$
• $\left(\pm {\frac {\sqrt {2}}{2}},\,0,\,-{\frac {\sqrt[{4}]{8}}{4}},\,{\sqrt {\frac {4-{\sqrt {2}}}{8}}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,0,\,0\right),$
• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,0,\,0\right).$

Representations

A square antiwedge has the following Coxeter diagrams:

• os2xo8os&#x (full symmetry)
• xxo4oxx&#x (BC2 symmetry only, seen with square atop gyro square cupola)

External links

• Klitzing, Richard. "squaw".