Square scalene

{{Infobox polytope The square scalene, square pyramidal pyramid, or squasc, 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.
 * img=Squasc.png
 * off=Square scalene.off
 * type=Segmentotope
 * dim = 4
 * obsa = squasc
 * cells = 4 tetrahedra, 2 square pyramids
 * faces = 4+8 triangles, 1 square
 * edges = 1+4+8
 * vertices = 2+4
 * verf=2 square pyramids, edge length 1
 * verf2=4 digonal disphenoids, edge lengths $\sqrt{2}$ (1 base) and 1 (remaining edges)
 * coxeter = xo ox4oo&#x
 * army=Squasc
 * reg=Squasc
 * taper=[11]{{sup|2}}
 * symmetry=BC{{sub|2}}×A{{sub|1}}×I, order 16
 * circum = $$\frac{\sqrt2}{2} ≈ 0.70711$$
 * height = Point atop squippy: $$\frac{\sqrt2}{2} ≈ 0.70711$$
 * height2 = Dyad atop tet: $$\frac{\sqrt2}{2} ≈ 0.70711$$
 * height3 = Trig atop inclined trig: $$\frac{\sqrt2]{2} ≈ 0.70711$$
 * height2 = Dyad atop perp square: $$\frac12 = 0.5$$
 * hypervolume = $$\frac{1}{24} ≈ 0.041667$$
 * dich= Tet–3–tet: 120°
 * dich2= Squippy–4–squippy: 90°
 * dich3= Tet–3–squippy: 60°
 * dual=Square scalene
 * conjugate=Square scalene
 * conv = Yes
 * orientable=Yes
 * nat=Tame}}

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: and all permutations, in the first 2 coordinates, of
 * $$\left(0,\,0,\,0,\,\frac{\sqrt2}{2}\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt2}{2},\,0\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0,\,0\right).$$

It can also be given by these coordinates:


 * $$\left(±\frac12,\,±\frac12,\,0,\,0\right),$$
 * $$\left(0,\,0,\,±\frac12,\,\frac12\right).$$

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)