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)