Pentagonal antiwedge

The pentagonal antiwedge, or paw, also sometimes called the pentagonal gyrobicupolic ring, is a CRF segmentochoron (designated K-4.133 on Richard Klitzing's list). It consists of 1 pentagonal antiprism, 2 pentagonal cupolas, and 10 square pyramids.

The pentagonal antiwedge can be seen as a wedge of the rectified hexacosichoron]. This is best seen when viewing it as a relative of segmentochoron [[icosahedron atop icosidodecahedron, with the pentagonal antiprism base coming from the icosahedron and the opposite decagon being the central plain of the icosidodecahedron.

Vertex coordinates
The vertices of a pentagonal antiwedge with edge length 1 are given by:


 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\sqrt{\frac{5+\sqrt5}{40}},\,\frac{\sqrt5-1}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,\sqrt{\frac{5+\sqrt5}{40}},\,\frac{\sqrt5-1}{4}\right),$$
 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\sqrt{\frac{5+\sqrt5}{40}},\,\frac{\sqrt5-1}{4}\right),$$
 * $$\left(±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}},\,-\sqrt{\frac{5+\sqrt5}{40}},\,\frac{\sqrt5-1}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}},\,-\sqrt{\frac{5+\sqrt5}{40}},\,\frac{\sqrt5-1}{4}\right),$$
 * $$\left(0,\,-\sqrt{\frac{5+\sqrt5}{10}},\,-\sqrt{\frac{5+\sqrt5}{40}},\,\frac{\sqrt5-1}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,0,\,0\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,0,\,0\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,0,\,0\right).$$

Representations
A pentagonal antiwedge has the following Coxeter diagrams:


 * os2xo10os&#x (full symmetry)
 * xxo5oxx&#x (H2 symmetry only, seen with pentagon atop gyro pentagonal cupola)