Square cupofastegium

Square cupofastegium
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Squicuf
Coxeter diagram	ox xx4xo&#x
Elements
Cells	4 tetrahedra, 4 triangular prisms, 1 cube, 2 square cupolas
Faces	8+8 triangles, 2+4+8 squares, 1 octagon
Edges	4+4+4+8+16
Vertices	8+8
Vertex figures	8 isosceles trapezoidal pyramids, base edge lengths 1, √2, √2, √2, side edge lengths 1, 1, √2. √2
	8 sphenoids, edge lengths 1 (3), √2 (2), and √2+√2 (1)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {3+{\sqrt {2}}}{2}}}\approx 1.48563$
Hypervolume	${\frac {7+4{\sqrt {2}}}{24}}\approx 0.52737$
Dichoral angles	Tet–3–trip: 150°
	Cube–4–trip: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Squacu–8–squacu: 90°
	Squacu–3–tet: 60°
	Squacu–4–trip: $\arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 54.73561^{\circ }$
	Cube–4–squacu: 45°
Heights	Square atop squacu: ${\frac {\sqrt {2}}{2}}\approx 0.70711$
	Cube atop oc: ${\frac {1}{2}}=0.5$
Central density	1
Related polytopes
Army	Squicuf
Regiment	Squicuf
Dual	Square cupolanotch
Conjugate	Retrograde square cupofastegium
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B2×A1×I, order 16
Convex	Yes
Nature	Tame

The square cupofastegium, or squicuf, also sometimes called the square orthobicupolic ring, is a CRF segmentochoron (designated K-4.73 on Richard Klitzing's list). It consists of 1 cube, 4 tetrahedra, 4 triangular prisms, and 2 square cupolas.

It has two representations as a segmentochoron: square atop square cupola or cube atop octagon.

The square cupofastegium can be obtained as a cap of the small disprismatotesseractihexadecachoron from one of the 24 cubes with square prism symmetry.

Vertex coordinates

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

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

Representations

A square cupofastegium has the following Coxeter diagrams:

• ox xx4xo&#x (full symmetry)
• xxx4oxo&#x (B₂ symmetry only, seen with square atop square cupola)

External links

• Klitzing, Richard. "squicuf".