Pentagonal antiwedge
Rank 4 Type Segmentotope Notation Bowers style acronym Paw Coxeter diagram os2xo10os&#x Elements Cells 10 square pyramids , 1 pentagonal antiprism , 2 pentagonal cupolas Faces 10+10+10 triangles , 10 squares , 2 pentagons , 1 decagon Edges 10+10+10+20 Vertices 10+10 Vertex figures 10 skewed wedges , edge lengths 1 (6), √2 (2), and (1+√5 )/2 (1) 10 sphenoids , edge lengths 1 (3), √2 (2), and √(5+√5 )/2 (1) Measures (edge length 1) Circumradius
5
+
2
5
≈
3.07768
{\displaystyle {\sqrt {5+2{\sqrt {5}}}}\approx 3.07768}
Hypervolume
15
+
13
5
96
≈
0.45905
{\displaystyle {\frac {15+13{\sqrt {5}}}{96}}\approx 0.45905}
Dichoral angles Squippy–3–squippy:
arccos
(
−
1
+
3
5
8
)
≈
164.47751
∘
{\displaystyle \arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }}
Pap–3–squippy:
arccos
(
−
7
+
3
5
4
)
≈
157.76124
∘
{\displaystyle \arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }}
Pecu–10–pecu: 108° Pecu–4–squippy: 45° Pecu–3–squippy:
arccos
(
−
10
4
)
≈
37.76124
∘
{\displaystyle \arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 37.76124^{\circ }}
Pap–5–pecu: 36° Heights Peg atop gyro pecu:
1
2
=
0.5
{\displaystyle {\frac {1}{2}}=0.5}
Pap atop dec:
5
−
1
4
≈
0.30902
{\displaystyle {\frac {{\sqrt {5}}-1}{4}}\approx 0.30902}
Central density 1 Related polytopes Army Paw Regiment Paw Dual Pentagonal gyrocupolanotch Conjugate Retrograde pentagrammic antiwedge Abstract & topological properties Euler characteristic 0 Orientable Yes Properties Symmetry (I2 (10)×A1 /2)×I , order 20Convex Yes Nature Tame
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 plane of the icosidodecahedron.
The vertices of a pentagonal antiwedge with edge length 1 are given by:
(
±
1
2
,
−
5
+
2
5
20
,
5
+
5
40
,
5
−
1
4
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\frac {{\sqrt {5}}-1}{4}}\right),}
(
±
1
+
5
4
,
5
−
5
40
,
5
+
5
40
,
5
−
1
4
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\frac {{\sqrt {5}}-1}{4}}\right),}
(
0
,
5
+
5
10
,
5
+
5
40
,
5
−
1
4
)
,
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\frac {{\sqrt {5}}-1}{4}}\right),}
(
±
1
2
,
5
+
2
5
20
,
−
5
+
5
40
,
5
−
1
4
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\frac {{\sqrt {5}}-1}{4}}\right),}
(
±
1
+
5
4
,
−
5
−
5
40
,
−
5
+
5
40
,
5
−
1
4
)
,
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\frac {{\sqrt {5}}-1}{4}}\right),}
(
0
,
−
5
+
5
10
,
−
5
+
5
40
,
5
−
1
4
)
,
{\displaystyle \left(0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\frac {{\sqrt {5}}-1}{4}}\right),}
(
±
1
2
,
±
5
+
2
5
2
,
0
,
0
)
,
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,0,\,0\right),}
(
±
3
+
5
4
,
±
5
+
5
8
,
0
,
0
)
,
{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,0,\,0\right),}
(
±
1
+
5
2
,
0
,
0
,
0
)
.
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0,\,0\right).}
A pentagonal antiwedge has the following Coxeter diagrams :
os2xo10os&#x (full symmetry)
xxo5oxx&#x (H2 symmetry only, seen with pentagon atop gyro pentagonal cupola) Klitzing, Richard. "paw" . 
