# Square cupofastegium

Square cupofastegium Rank4
TypeSegmentotope
SpaceSpherical
Notation
Bowers style acronymSquicuf
Coxeter diagramox xx4xo&#x
Elements
Cells4 tetrahedra, 4 triangular prisms, 1 cube, 2 square cupolas
Faces8+8 triangles, 2+4+8 squares, 1 octagon
Edges4+4+4+8+16
Vertices8+8
Vertex figures8 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$\sqrt{\frac{3+\sqrt2}{2}} \approx 1.48563$ Hypervolume$\frac{7+4\sqrt2}{24} \approx 0.52737$ Dichoral anglesTet–3–trip: 150°
Cube–4–trip: $\arccos\left(-\frac{\sqrt6}{3}\right) \approx 144.73561°$ Squacu–8–squacu: 90°
Squacu–3–tet: 60°
Squacu–4–trip: $\arccos\left(\frac{\sqrt3}{3}\right) \approx 54.73561°$ Cube–4–squacu: 45°
HeightsSquare atop squacu: $\frac{\sqrt2}{2} \approx 0.70711$ Cube atop oc: $\frac12 = 0.5$ Central density1
Related polytopes
ArmySquicuf
RegimentSquicuf
DualSquare cupolanotch
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB2×A1×I, order 16
ConvexYes
NatureTame

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:

• $\left(\pm\frac12,\,\pm\frac12,\,\pm\frac12,\,\frac12\right)$ ,
• $\left(\pm\frac12,\,\pm\frac{1+\sqrt2}{2},\,0,\,0\right)$ ,
• $\left(\pm\frac{1+\sqrt2}{2},\,\pm\frac12,\,0,\,0\right)$ .

## 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)