# Square scalene

The square scalene (OBSA: squasc), or square pyramidal pyramid, is a CRF segmentochoron (designated K-4.4 on Richard Klitzing's list). It consists of 2 square pyramids and 4 tetrahedra. It can be thought of as a pyramid based on the square pyramid.

Square scalene
Rank4
TypeSegmentotope
Notation
Bowers style acronymSquasc
Coxeter diagramxo ox4oo&#x
Tapertopic notation[11]2
Elements
Cells
Faces
Edges1+4+8
Vertices2+4
Vertex figures2 square pyramids, edge length 1
4 digonal disphenoids, edge lengths 2 (1 base) and 1 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Hypervolume${\displaystyle {\frac {1}{24}}\approx 0.041667}$
Dichoral anglesTet–3–tet: 120°
Squippy–4–squippy: 90°
Tet–3–squippy: 60°
HeightsPoint atop squippy: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Dyad atop tet: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Trig atop inclined trig: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Dyad atop perp square: ${\displaystyle {\frac {1}{2}}=0.5}$
Central density1
Related polytopes
ArmySquasc
RegimentSquasc
DualSquare scalene
ConjugateNone
Abstract & topological properties
Flag count160
Euler characteristic0
OrientableYes
Properties
SymmetryB2×A1×I, order 16
Flag orbits10
ConvexYes
NatureTame

Apart from being a point atop square pyramid, it has alternate segmentochoron representations as a dyad atop tetrahedron, dyad atop perpendicular square and triangle atop inclined triangle.

It can be viewed as a quarter of the hexadecachoron or a half of the octahedral pyramid.

## Vertex coordinates

The vertices of a square scalene with unit edge length are given by:

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

It can also be given by the integral coordinates:

• ${\displaystyle \left(\pm 1,\,\pm 1,\,0,\,0\right)}$ ,
• ${\displaystyle \left(0,\,0,\,\pm 1,\,1\right)}$ .

## Representations

The square scalene has the following Coxeter diagrams:

• xo ox4oo&#x (full symmetry, dyad atop fully orthogonal square)
• xo ox ox&#x (A1×A1×A1 symmetry, rectangle scalene)
• oox4ooo&#x (BC2 symmetry, square pyramidal pyramid)
• oox oox&#x (A1×A1 symmetry, rectangle pyramid pyramid)
• xoo oxx&#x (A1×A1 symmetry, trapezoid scalene)
• xoox&#x (bilateral symmetry only)