Square scalene

Square scalene
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Squasc
Coxeter diagram	xo ox4oo&#x
Tapertopic notation	[11]2
Elements
Cells	4 tetrahedra 2 square pyramids
Faces	4+8 triangles 1 square
Edges	1+4+8
Vertices	2+4
Vertex figures	2 square pyramids, edge length 1
	4 digonal disphenoids, edge lengths √2 (1 base) and 1 (remaining edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Hypervolume	${\frac {1}{24}}\approx 0.041667$
Dichoral angles	Tet–3–tet: 120°
	Squippy–4–squippy: 90°
	Tet–3–squippy: 60°
Heights	Point atop squippy: ${\frac {\sqrt {2}}{2}}\approx 0.70711$
	Dyad atop tet: ${\frac {\sqrt {2}}{2}}\approx 0.70711$
	Trig atop inclined trig: ${\frac {\sqrt {2}}{2}}\approx 0.70711$
	Dyad atop perp square: ${\frac {1}{2}}=0.5$
Central density	1
Related polytopes
Army	Squasc
Regiment	Squasc
Dual	Square scalene
Conjugate	None
Abstract & topological properties
Flag count	160
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B2×A1×I, order 16
Flag orbits	10
Convex	Yes
Nature	Tame

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.

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:

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

It can also be given by the integral coordinates:

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

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)

External links

• Klitzing, Richard. "squasc".
• Wikipedia contributors. "Square-pyramidal pyramid".