The small rhombated hexacosichoron , or srix , also commonly called the cantellated 600-cell , is a convex uniform polychoron that consists of 600 cuboctahedra , 720 pentagonal prisms , and 120 icosidodecahedra . 1 icosidodecahedron, 2 pentagonal prisms, and 2 cuboctahedra join at each vertex. As one of its names suggests, it can be obtained by cantellating the hexacosichoron .
Coordinates for the vertices of a small rhombated hexacosichoron of edge length 1 are given by all permutations of:
(
0
,
0
,
±
1
+
5
2
,
±
5
+
3
5
2
)
{\displaystyle \left(0,\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{2}}\right)}
,
(
0
,
±
1
,
±
(
2
+
5
)
,
±
(
2
+
5
)
)
{\displaystyle \left(0,\,\pm 1,\,\pm (2+{\sqrt {5}}),\,\pm (2+{\sqrt {5}})\right)}
,
(
±
1
2
,
±
1
2
,
±
3
+
2
5
2
,
±
5
+
2
5
2
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}
,
(
±
2
+
5
2
,
±
2
+
5
2
,
±
3
+
2
5
2
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±
3
+
2
5
2
)
{\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right)}
,
together with all even permutations of:
(
0
,
±
1
2
,
±
3
1
+
5
4
,
±
11
+
5
5
4
)
{\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}
,
(
0
,
±
1
+
5
2
,
±
(
3
+
5
)
,
±
3
+
5
2
)
{\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm (3+{\sqrt {5}}),\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)}
,
(
0
,
±
4
+
5
2
,
±
7
+
3
5
4
,
±
3
3
+
5
4
)
{\displaystyle \left(0,\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}
,
(
±
1
2
,
±
1
+
5
4
,
±
5
+
3
5
2
,
±
3
+
5
4
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}
,
(
±
1
2
,
±
1
+
5
4
,
±
(
3
+
5
)
,
±
5
+
3
5
4
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm (3+{\sqrt {5}}),\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)}
,
(
±
1
2
,
±
1
+
5
2
,
±
11
+
5
5
4
,
±
5
+
5
4
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}
,
(
±
1
2
,
±
5
+
5
4
,
±
(
2
+
5
)
,
±
3
3
+
5
4
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}
,
(
±
1
2
,
±
2
+
5
2
,
±
5
+
2
5
2
,
±
4
+
5
2
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)}
,
(
±
1
+
5
4
,
±
1
,
±
2
+
5
2
,
±
11
+
5
5
4
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}
,
(
±
1
+
5
4
,
±
3
+
5
2
,
±
3
+
2
5
2
,
±
3
3
+
5
4
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}
,
(
±
1
+
5
4
,
±
5
+
3
5
4
,
±
(
2
+
5
)
,
±
4
+
5
2
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)}
,
(
±
1
,
±
3
+
5
4
,
±
5
+
2
5
2
,
±
7
+
3
5
4
)
{\displaystyle \left(\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}
,
(
±
3
+
5
4
,
±
5
+
5
4
,
±
2
+
5
2
,
±
(
3
+
5
)
)
{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm (3+{\sqrt {5}})\right)}
,
(
±
3
+
5
4
,
±
5
+
5
4
,
±
3
+
2
5
2
,
±
(
2
+
5
)
)
{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}})\right)}
,
(
±
3
+
5
4
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±
3
1
+
5
4
,
±
5
+
2
5
2
,
±
3
+
5
2
)
{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)}
,
(
±
1
+
5
2
,
±
5
+
5
4
,
±
5
+
3
5
4
,
±
5
+
2
5
2
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}
,
(
±
1
+
5
2
,
±
5
+
3
5
4
,
±
3
+
2
5
2
,
±
7
+
3
5
4
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}
,
(
±
2
+
5
2
,
±
3
1
+
5
4
,
±
(
2
+
5
)
,
±
5
+
3
5
4
)
{\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)}
.
The small rhombated hexacosichoron has a semi-uniform variant of the form o5y3o3x that maintains its full symmetry. This variant uses 120 icosidodecahedra of size y, 600 rhombitetratetrahedra of form x3o3y, and 720 pentagonal prisms of form x y5o as cells, with 2 edge lengths.
With edges of length a (surrounds 2 rhombitetratetrahedra) and b (of icosidodecahedra), its circumradius is given by
3
a
2
+
21
b
2
+
14
a
b
+
(
a
2
+
9
b
2
+
6
a
b
)
5
2
{\displaystyle {\sqrt {\frac {3a^{2}+21b^{2}+14ab+(a^{2}+9b^{2}+6ab){\sqrt {5}}}{2}}}}
.
The small rhombated hexacosichoron is the colonel of a seven-member regiment. Its other members include the small retrosphenoverted hecatonicosihexacosihecatonicosachoron , rhombic small hexacosihecatonicosachoron , pseudorhombic small dishecatonicosachoron , grand rhombic small hexacosihecatonicosachoron , small dishecatonicosintercepted hexacosihecatonicosachoron , and hecatonicosintercepted prismatohexacosihecatonicosachoron .
The segmentochoron icosidodecahedron atop truncated icosahedron can be obtained as a cap of the small rhombated hexacosichoron.
truncationsName OBSA Schläfli symbol CD diagram Image Hecatonicosachoron hi {5,3,3} Rectified hecatonicosachoron rahi r{5,3,3} Rectified hexacosichoron rox r{3,3,5} Hexacosichoron ex {3,3,5} Truncated hecatonicosachoron thi t{5,3,3} Small rhombated hecatonicosachoron srahi rr{5,3,3} Small disprismatohexacosihecatonicosachoron sidpixhi t0,3 {5,3,3} Hexacosihecatonicosachoron xhi 2t{5,3,3} Small rhombated hexacosichoron srix rr{3,3,5} Truncated hexacosichoron tex t{3,3,5} Great rhombated hecatonicosachoron grahi tr{5,3,3} Prismatorhombated hexacosichoron prix t0,1,3 {5,3,3} Prismatorhombated hecatonicosachoron prahi t0,2,3 {5,3,3} Great rhombated hexacosichoron grix tr{3,3,5} Great disprismatohexacosihecatonicosachoron gidpixhi t0,1,2,3 {5,3,3}