Square scalene

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.

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)