The small rhombated pentachoron , or srip , also commonly called the cantellated 5-cell or cantellated pentachoron , is a convex uniform polychoron that consists of 5 regular octahedra , 10 triangular prisms , and 5 cuboctahedra . 1 octahedron, 2 triangular prisms, and 2 cuboctahedra join at each vertex. As one of its names suggests, it can be formed by cantellating the pentachoron . It can also be formed by rectification of the rectified pentachoron .
The vertices of a small rhombated pentachoron of edge length 1 are given by:
(
10
5
,
0
,
0
,
±
1
)
{\displaystyle \left({\frac {\sqrt {10}}{5}},\,0,\,0,\,\pm 1\right)}
,
(
10
5
,
0
,
±
3
2
,
±
1
2
)
{\displaystyle \left({\frac {\sqrt {10}}{5}},\,0,\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right)}
,
(
10
5
,
−
6
3
,
3
3
,
0
)
{\displaystyle \left({\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {6}}{3}},\,{\frac {\sqrt {3}}{3}},\,0\right)}
,
(
10
5
,
6
3
,
−
3
3
,
0
)
{\displaystyle \left({\frac {\sqrt {10}}{5}},\,{\frac {\sqrt {6}}{3}},\,-{\frac {\sqrt {3}}{3}},\,0\right)}
,
(
10
5
,
−
6
3
,
−
3
6
,
±
1
2
)
{\displaystyle \left({\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {6}}{3}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}
,
(
10
5
,
6
3
,
3
6
,
±
1
2
)
{\displaystyle \left({\frac {\sqrt {10}}{5}},\,{\frac {\sqrt {6}}{3}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}
,
(
−
3
10
10
,
−
6
6
,
−
3
3
,
0
)
{\displaystyle \left(-{\frac {3{\sqrt {10}}}{10}},\,-{\frac {\sqrt {6}}{6}},\,-{\frac {\sqrt {3}}{3}},\,0\right)}
,
(
−
3
10
10
,
6
6
,
3
3
,
0
)
{\displaystyle \left(-{\frac {3{\sqrt {10}}}{10}},\,{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{3}},\,0\right)}
,
(
−
10
20
,
−
6
12
,
−
2
3
3
,
0
)
{\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {2{\sqrt {3}}}{3}},\,0\right)}
,
(
−
10
20
,
−
5
6
12
,
−
3
3
,
0
)
{\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,-{\frac {5{\sqrt {6}}}{12}},\,-{\frac {\sqrt {3}}{3}},\,0\right)}
,
(
−
10
20
,
6
4
,
0
,
±
1
)
{\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,{\frac {\sqrt {6}}{4}},\,0,\,\pm 1\right)}
,
(
−
10
20
,
−
6
12
,
3
3
,
±
1
)
{\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {6}}{12}},\,{\frac {\sqrt {3}}{3}},\,\pm 1\right)}
,
(
−
10
20
,
−
5
6
12
,
3
6
,
±
1
2
)
{\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,-{\frac {5{\sqrt {6}}}{12}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}
,
(
−
10
20
,
6
4
,
±
3
2
,
±
1
2
)
{\displaystyle \left(-{\frac {\sqrt {10}}{20}},\,{\frac {\sqrt {6}}{4}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right)}
,
(
−
3
10
10
,
−
6
6
,
3
6
,
±
1
2
)
{\displaystyle \left(-{\frac {3{\sqrt {10}}}{10}},\,-{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}
,
(
−
3
10
10
,
6
6
,
−
3
6
,
±
1
2
)
{\displaystyle \left(-{\frac {3{\sqrt {10}}}{10}},\,{\frac {\sqrt {6}}{6}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}
.
Much simpler coordinates can be given in five dimensions , as all permutations of:
(
2
,
2
2
,
2
2
,
0
,
0
)
{\displaystyle \left({\sqrt {2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,0,\,0\right)}
.
A small rhombated pentachoron has the following Coxeter diagrams :
The small rhombated pentachoron has a semi-uniform variant of the form x3o3y3o that maintains its full symmetry. This variant uses 5 octahedra of size y, 5 rhombitetratetrahedra of form x3o3y, and 10 triangular prisms of form x y3o as cells, with 2 edge lengths.
With edges of length a (surrounds 2 rhombitetratetrahedra) and b (of octahedra), its circumradius is given by
2
a
2
+
3
b
2
+
2
a
b
5
{\displaystyle {\sqrt {\frac {2a^{2}+3b^{2}+2ab}{5}}}}
and its hypervolume is given by
(
a
4
+
12
a
3
b
+
54
a
2
b
2
+
68
a
b
3
+
11
b
4
)
5
96
{\displaystyle (a^{4}+12a^{3}b+54a^{2}b^{2}+68ab^{3}+11b^{4}){\frac {\sqrt {5}}{96}}}
.
The small rhombated pentachoron is the colonel of the largest regiment of uniform polychora with A4 symmetry, which has a total of 7 members. Its facetings include the retrosphenoverted trispentachoron , small rhombic dispentachoron , pseudorhombic prismatopentachoron , grand rhombic prismatopentachoron , prismatopentintercepted dispentachoron , and prismatointercepted prismatodispentachoron .
When viewed in A3 axial symmetry, the small rhombated pentachoron can be cut into 2 segmentochora , namely cuboctahedron atop truncated tetrahedron and octahedron atop truncated tetrahedron , join at the truncated tetrahedral bases.
The triangular pucofastegium occurs as the triangle-first cap of the small rhombated pentachoron.
Uniform polychoron compounds composed of small rhombated pentachora include:
File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png truncationsName OBSA Schläfli symbol CD diagram Image Pentachoron pen {3,3,3} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:Schlegel wireframe 5-cell.png Rectified pentachoron rap r{3,3,3} File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:Schlegel half-solid rectified 5-cell.png Rectified pentachoron rap r{3,3,3} File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Schlegel half-solid rectified 5-cell.png Pentachoron pen {3,3,3} File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:Schlegel wireframe 5-cell.png Truncated pentachoron tip t{3,3,3} File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:Schlegel half-solid truncated pentachoron.png Small rhombated pentachoron srip rr{3,3,3} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Schlegel half-solid cantellated 5-cell.png Small prismatodecachoron spid t0,3 {3,3,3} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:Schlegel half-solid runcinated 5-cell.png Decachoron deca 2t{3,3,3} File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Schlegel half-solid bitruncated 5-cell.png Small rhombated pentachoron srip rr{3,3,3} File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:Schlegel half-solid cantellated 5-cell.png Truncated pentachoron tip t{3,3,3} File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:Schlegel half-solid truncated pentachoron.png Great rhombated pentachoron grip tr{3,3,3} File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Schlegel half-solid cantitruncated 5-cell.png Prismatorhombated pentachoron prip t0,1,3 {3,3,3} File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:Schlegel half-solid runcitruncated 5-cell.png Prismatorhombated pentachoron prip t0,2,3 {3,3,3} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:Schlegel half-solid runcitruncated 5-cell.png Great rhombated pentachoron grip tr{3,3,3} File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:Schlegel half-solid cantitruncated 5-cell.png Great prismatodecachoron gippid t0,1,2,3 {3,3,3} File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:Schlegel half-solid omnitruncated 5-cell.png