# Square pucofastegium

Square pucofastegium
Rank4
TypeSegmentotope
Notation
Bowers style acronymSquipuf
Coxeter diagramox ox4xx&#x
Elements
Cells4 square pyramids, 4 triangular prisms, 2 square cupolas, 1 octagonal prism
Faces8+8 triangles, 1+4+4+8 squares, 2 octagons
Edges4+8+8+8+16
Vertices4+16
Vertex figures4 wedges, edge lengths 1 (base square) and 2 (top and side edges)
16 irregular tetrahedra, edge lengths 1 (2), 2 (3), and 2+2 (1)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {2+{\sqrt {2}}}}\approx 1.84776}$
Hypervolume${\displaystyle {\frac {9+8{\sqrt {2}}}{24}}\approx 0.84640}$
Dichoral anglesSquippy–3–trip: 150º
Squacu–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Squippy–3–squacu: 120º
Squacu–4–squacu: 90º
Op–8–squacu: 45º
Op–4–squippy: 45º
Trip–4–op: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{3}}\right)\approx 35.26439^{\circ }}$
HeightsOc atop squacu: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Square atop op: ${\displaystyle {\frac {1}{2}}=0.5}$
Central density1
Related polytopes
ArmySquipuf
RegimentSquipuf
DualSquare pucolanotch
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB2×A1×I, order 16
ConvexYes
NatureTame

The square pucofastegium, or squipuf, also sometimes called the square magnabicupolic ring, is a CRF segmentochoron (designated K-4.105 on Richard Klitzing's list). It consists of 4 triangular prisms, 4 square pyramids, 2 square cupolas, and 1 octagonal prism.

The square pucofastegium occurs as the square-first cap of the small rhombated tesseract. In fact diminishing 8 such caps from the small rhombated tesseract results in the octagonal duoprism.

## Vertex coordinates

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

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

## Representations

A square pucofastegium has the following Coxeter diagrams:

• ox ox4xx&#x (full symmetry)
• xxx4oxx&#x (BC2 symmetry only, seen with octagon atop square cupola)