Octagonal antifastegium

The octagonal antifastegium, or oaf, is a CRF segmentochoron (designated K-4.58 on Richard Klitzing's list). It consists of 1 octagonal prism, 2 octagonal antiprisms, 8 tetrahedra, and 8 square pyramids. It is a member of the infinite family of polygonal antifastegiums.

It is a segmentochoron between an octagon and an octagonal antiprism or between an octagon and a gyro octagonal prism.

Vertex coordinates
The vertices of an octagonal antifastegium of edge length 1 are given by:
 * $$\left(±\frac12,\,±\frac{1+\sqrt2}2,\,±\frac12,\,0\right),$$
 * $$\left(±\frac{1+\sqrt2}2,\,±\frac12,\,±\frac12,\,0\right),$$
 * $$\left(0,\,±\sqrt{\frac{2+\sqrt2}2},\,0,\,\frac{\sqrt{2\sqrt{20+14\sqrt2}-4\sqrt2-5}}2\right),$$
 * $$\left(±\sqrt{\frac{2+\sqrt2}2},\,0,\,0,\,\frac{\sqrt{2\sqrt{20+14\sqrt2}-4\sqrt2-5}}2\right),$$
 * $$\left(±\frac{\sqrt{2+\sqrt2}}2,\,±\frac{\sqrt{2+\sqrt2}}2,\,0,\,\frac{\sqrt{2\sqrt{20+14\sqrt2}-4\sqrt2-5}}2\right).$$

Representations
The octagonal antifastegium can be represented by the following Coxeter diagram s:
 * ox xo8ox&#x
 * xoo8oxx&#x