The hexacosihecatonicosachoron , or xhi , also commonly called the bitruncated 120-cell , is a convex uniform polychoron that consists of 600 truncated tetrahedra and 120 truncated icosahedra . 2 truncated tetrahedra and 2 truncated icosahedra join at each vertex. It is the medial stage of the truncation series between a hecatonicosachoron and its dual hexacosichoron . As such, it could also be called a bitruncated 600-cell .
Coordinates for the vertices of a hexacosihecatonicosachoron of edge length 1 are given by all permutations of:
(
0
,
0
,
±
(
1
+
5
)
,
±
7
+
3
5
2
)
{\displaystyle \left(0,\,0,\,\pm (1+{\sqrt {5}}),\,\pm {\frac {7+3{\sqrt {5}}}{2}}\right)}
,
(
±
5
+
5
4
,
±
5
+
5
4
,
±
9
+
5
5
4
,
±
9
+
5
5
4
)
{\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}}\right)}
,
together with all even permutations of:
(
0
,
±
1
2
,
±
13
+
5
5
4
,
±
7
+
5
5
4
)
{\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}}\right)}
,
(
0
,
±
1
2
,
±
3
1
+
5
4
,
±
13
+
7
5
4
)
{\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {13+7{\sqrt {5}}}{4}}\right)}
,
(
0
,
±
1
+
5
4
,
±
5
3
+
5
4
,
±
3
+
2
5
2
)
{\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 5{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right)}
,
(
0
,
±
3
3
+
5
4
,
±
5
+
2
5
2
,
±
11
+
3
5
4
)
{\displaystyle \left(0,\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}},\,\pm {\frac {11+3{\sqrt {5}}}{4}}\right)}
,
(
±
1
2
,
±
1
+
5
2
,
±
13
+
7
5
4
,
±
5
+
5
4
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {13+7{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}
,
(
±
1
2
,
±
1
+
5
2
,
±
5
3
+
5
4
,
±
7
+
3
5
4
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 5{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}
,
(
±
1
2
,
±
1
,
±
11
+
5
5
4
,
±
9
+
5
5
4
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}}\right)}
,
(
±
1
2
,
±
3
+
5
2
,
±
11
+
5
5
4
,
±
11
+
3
5
4
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {11+3{\sqrt {5}}}{4}}\right)}
,
(
±
1
+
5
4
,
±
1
,
±
2
+
5
2
,
±
13
+
7
5
4
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {13+7{\sqrt {5}}}{4}}\right)}
,
(
±
1
+
5
4
,
±
7
+
3
5
4
,
±
9
+
5
5
4
,
±
11
+
3
5
4
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {11+3{\sqrt {5}}}{4}}\right)}
,
(
±
1
+
5
2
,
±
3
+
2
5
2
,
±
9
+
5
5
4
,
±
3
3
+
5
4
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}
,
(
±
1
+
5
2
,
±
7
+
3
5
4
,
±
7
+
5
5
4
,
±
5
+
2
5
2
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}
,
(
±
1
,
±
1
+
5
2
,
±
7
+
3
5
2
,
±
3
+
5
2
)
{\displaystyle \left(\pm 1,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)}
,
(
±
1
,
±
2
+
5
2
,
±
13
+
5
5
4
,
±
3
3
+
5
4
)
{\displaystyle \left(\pm 1,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}
,
(
±
1
,
±
5
+
5
4
,
±
11
+
5
5
4
,
±
5
+
2
5
2
)
{\displaystyle \left(\pm 1,\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}
,
(
±
2
+
5
2
,
±
(
1
+
5
)
,
±
11
+
5
5
4
,
±
7
+
3
5
4
)
{\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm (1+{\sqrt {5}}),\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}
,
(
±
2
+
5
2
,
±
3
+
5
2
,
±
7
+
5
5
4
,
±
9
+
5
5
4
)
{\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}}\right)}
,
(
±
3
1
+
5
4
,
±
3
+
5
2
,
±
3
+
2
5
2
,
±
11
+
5
5
4
)
{\displaystyle \left(\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}
,
(
±
5
+
5
4
,
±
2
+
5
2
,
±
5
3
+
5
4
,
±
3
+
5
2
)
{\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 5{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)}
,
(
±
5
+
5
4
,
±
3
1
+
5
4
,
±
13
+
5
5
4
,
±
7
+
3
5
4
)
{\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}
.
The hexacosihecatonicosachoron has a semi-uniform variant of the form o5x3y3o that maintains its full symmetry. This variant uses 600 semi-uniform truncated tetrahedra of form x3y3o and 120 semi-uniform truncated icosahedra of form o5x3y as cells, with 2 edge lengths.
With edges of length a (of pentagonal faces) and b (of triangular faces), its circumradius is given by
21
a
2
+
10
b
2
+
28
a
b
+
(
9
a
2
+
4
b
2
+
12
a
b
)
5
2
{\displaystyle {\sqrt {\frac {21a^{2}+10b^{2}+28ab+(9a^{2}+4b^{2}+12ab){\sqrt {5}}}{2}}}}
.
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}