The great disprismatohexacosihecatonicosachoron , or gidpixhi , also commonly called the omnitruncated 120-cell , is a convex uniform polychoron that consists of 1200 hexagonal prisms , 720 decagonal prisms , 600 truncated octahedra , and 120 great rhombicosidodecahedra . 1 of each type of cell join at each vertex. It is the omnitruncate of the H4 family of uniform polychora, and could also be considered to be the omnitruncated 600-cell. It is therefore the most complex of the non-prismatic convex uniform polychora.
This polychoron can be alternated into a snub hexacosihecatonicosachoron , although it cannot be made uniform.
Vertex coordinates for a great disprismatohexacosihecatonicosachoron of edge length 1 are given by all permutations of:
(
±
1
2
,
±
1
2
,
±
4
+
3
5
2
,
±
12
+
5
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm\frac{4+3\sqrt5}{2},\,\pm\frac{12+5\sqrt5}{2}\right),}
(
±
1
2
,
±
1
2
,
±
7
+
4
5
2
,
±
11
+
4
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm\frac{7+4\sqrt5}{2},\,\pm\frac{11+4\sqrt5}{2}\right),}
(
±
1
2
,
±
1
2
,
±
3
+
2
5
2
,
±
11
+
6
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm\frac{3+2\sqrt5}{2},\,\pm\frac{11+6\sqrt5}{2}\right),}
(
±
1
2
,
±
3
2
,
±
9
+
4
5
2
,
±
9
+
4
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac32,\,\pm\frac{9+4\sqrt5}{2},\,\pm\frac{9+4\sqrt5}{2}\right),}
(
±
1
,
±
1
,
±
2
(
2
+
5
)
,
±
(
5
+
2
5
)
)
,
{\displaystyle \left(\pm1,\,\pm1,\,\pm2(2+\sqrt5),\,\pm(5+2\sqrt5)\right),}
(
±
3
+
5
2
,
±
5
+
5
2
,
±
2
(
2
+
5
)
,
±
2
(
2
+
5
)
)
,
{\displaystyle \left(\pm\frac{3+\sqrt5}{2},\,\pm\frac{5+\sqrt5}{2},\,\pm2(2+\sqrt5),\,\pm2(2+\sqrt5)\right),}
(
±
4
+
5
2
,
±
4
+
5
2
,
±
7
+
4
5
2
,
±
9
+
4
5
2
)
,
{\displaystyle \left(\pm\frac{4+\sqrt5}{2},\,\pm\frac{4+\sqrt5}{2},\,\pm\frac{7+4\sqrt5}{2},\,\pm\frac{9+4\sqrt5}{2}\right),}
(
±
3
+
2
5
2
,
±
5
+
2
5
2
,
±
7
+
4
5
2
,
±
7
+
4
5
2
)
,
{\displaystyle \left(\pm\frac{3+2\sqrt5}{2},\,\pm\frac{5+2\sqrt5}{2},\,\pm\frac{7+4\sqrt5}{2},\,\pm\frac{7+4\sqrt5}{2}\right),}
(
±
(
2
+
5
)
,
±
(
2
+
5
)
,
±
(
3
+
2
5
)
,
±
2
(
2
+
5
)
)
,
{\displaystyle \left(\pm(2+\sqrt5),\,\pm(2+\sqrt5),\,\pm(3+2\sqrt5),\,\pm2(2+\sqrt5)\right),}
plus all even permutations of:
(
±
1
2
,
±
5
3
+
5
4
,
±
15
+
7
5
4
,
±
3
3
+
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm5\frac{3+\sqrt5}{4},\,\pm\frac{15+7\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{2}\right),}
(
±
1
2
,
±
7
+
3
5
2
,
±
17
+
7
5
4
,
±
17
+
5
5
4
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{7+3\sqrt5}{2},\,\pm\frac{17+7\sqrt5}{4},\,\pm\frac{17+5\sqrt5}{4}\right),}
(
±
1
2
,
±
1
,
±
7
+
5
5
4
,
±
23
+
11
5
4
)
,
{\displaystyle \left(\pm\frac12,\,\pm1,\,\pm\frac{7+5\sqrt5}{4},\,\pm\frac{23+11\sqrt5}{4}\right),}
(
±
1
2
,
±
3
+
5
4
,
±
3
7
+
3
5
4
,
±
(
3
+
2
5
)
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{3+\sqrt5}{4},\,\pm3\frac{7+3\sqrt5}{4},\,\pm(3+2\sqrt5)\right),}
(
±
1
2
,
±
3
+
5
4
,
±
25
+
9
5
4
,
±
5
+
3
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{3+\sqrt5}{4},\,\pm\frac{25+9\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{2}\right),}
(
±
1
2
,
±
1
+
5
2
,
±
23
+
9
5
4
,
±
11
+
7
5
4
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{1+\sqrt5}{2},\,\pm\frac{23+9\sqrt5}{4},\,\pm\frac{11+7\sqrt5}{4}\right),}
(
±
1
2
,
±
2
+
5
2
,
±
11
+
6
5
2
,
±
4
+
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{2+\sqrt5}{2},\,\pm\frac{11+6\sqrt5}{2},\,\pm\frac{4+\sqrt5}{2}\right),}
(
±
1
2
,
±
7
+
5
4
,
±
17
+
9
5
4
,
±
2
(
2
+
5
)
)
,
{\displaystyle \left(\pm\frac12,\, \pm\frac{7+\sqrt5}{4},\,\pm\frac{17+9\sqrt5}{4},\,\pm2(2+\sqrt5)\right),}
(
±
1
2
,
±
5
+
3
5
4
,
±
25
+
9
5
4
,
±
(
3
+
5
)
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{25+9\sqrt5}{4},\,\pm(3+\sqrt5)\right),}
(
±
1
2
,
±
5
+
3
5
4
,
±
23
+
11
5
4
,
±
5
+
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{23+11\sqrt5}{4},\,\pm\frac{5+\sqrt5}{2}\right),}
(
±
1
2
,
±
(
1
+
5
)
,
±
23
+
9
5
4
,
±
13
+
5
5
4
)
,
{\displaystyle \left(\pm\frac12,\,\pm(1+\sqrt5),\,\pm\frac{23+9\sqrt5}{4},\,\pm\frac{13+5\sqrt5}{4}\right),}
(
±
1
2
,
±
3
3
+
5
4
,
±
17
+
9
5
4
,
±
3
3
+
5
2
)
,
{\displaystyle \left(\pm\frac12,\,\pm3\frac{3+\sqrt5}{4},\,\pm\frac{17+9\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{2}\right),}
(
±
1
2
,
±
(
2
+
5
)
,
±
19
+
9
5
4
,
±
17
+
5
5
4
)
,
{\displaystyle \left(\pm\frac12,\,\pm(2+\sqrt5),\,\pm\frac{19+9\sqrt5}{4},\,\pm\frac{17+5\sqrt5}{4}\right),}
(
±
1
,
±
3
+
5
4
,
±
11
+
6
5
2
,
±
7
+
3
5
4
)
,
{\displaystyle \left(\pm1,\,\pm\frac{3+\sqrt5}{4},\,\pm\frac{11+6\sqrt5}{2},\,\pm\frac{7+3\sqrt5}{4}\right),}
(
±
1
,
±
5
+
5
4
,
±
19
+
9
5
4
,
±
7
+
4
5
2
)
,
{\displaystyle \left(\pm1,\,\pm\frac{5+\sqrt5}{4},\,\pm\frac{19+9\sqrt5}{4},\,\pm\frac{7+4\sqrt5}{2}\right),}
(
±
1
,
±
2
+
5
2
,
±
25
+
9
5
4
,
±
11
+
5
5
4
)
,
{\displaystyle \left(\pm1,\,\pm\frac{2+\sqrt5}{2},\,\pm\frac{25+9\sqrt5}{4},\,\pm\frac{11+5\sqrt5}{4}\right),}
(
±
1
,
±
3
1
+
5
4
,
±
23
+
9
5
4
,
±
3
2
+
5
2
)
,
{\displaystyle \left(\pm1,\,\pm3\frac{1+\sqrt5}{4},\,\pm\frac{23+9\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2}\right),}
(
±
1
,
±
5
+
3
5
4
,
±
12
+
5
5
2
,
±
11
+
3
5
4
)
,
{\displaystyle \left(\pm1,\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{12+5\sqrt5}{2},\,\pm\frac{11+3\sqrt5}{4}\right),}
(
±
1
,
±
4
+
5
2
,
±
17
+
9
5
4
,
±
17
+
7
5
4
)
,
{\displaystyle \left(\pm1,\,\pm\frac{4+\sqrt5}{2},\,\pm\frac{17+9\sqrt5}{4},\,\pm\frac{17+7\sqrt5}{4}\right),}
(
±
1
,
±
3
+
2
5
2
,
±
3
7
+
3
5
4
,
±
5
3
+
5
4
)
,
{\displaystyle \left(\pm1,\,\pm\frac{3+2\sqrt5}{2},\,\pm3\frac{7+3\sqrt5}{4},\,\pm5\frac{3+\sqrt5}{4}\right),}
(
±
3
+
5
4
,
±
13
+
5
5
4
,
±
7
+
4
5
2
,
±
3
3
+
5
2
)
,
{\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{13+5\sqrt5}{4},\,\pm\frac{7+4\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{2}\right),}
(
±
3
+
5
4
,
±
3
2
+
5
2
,
±
2
(
2
+
5
)
,
±
17
+
5
5
4
)
,
{\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2},\,\pm2(2+\sqrt5),\,\pm\frac{17+5\sqrt5}{4}\right),}
(
±
3
+
5
4
,
±
3
2
,
±
(
2
+
5
)
,
±
23
+
11
5
4
)
,
{\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac32,\,\pm(2+\sqrt5),\,\pm\frac{23+11\sqrt5}{4}\right),}
(
±
3
+
5
4
,
±
3
1
+
5
4
,
±
11
+
6
5
2
,
±
3
+
5
2
)
,
{\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm3\frac{1+\sqrt5}{4},\,\pm\frac{11+6\sqrt5}{2},\,\pm\frac{3+\sqrt5}{2}\right),}
(
±
3
+
5
4
,
±
4
+
5
2
,
±
(
1
+
5
)
,
±
23
+
11
5
4
)
,
{\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{4+\sqrt5}{2},\,\pm(1+\sqrt5),\,\pm\frac{23+11\sqrt5}{4}\right),}
(
±
3
+
5
4
,
±
11
+
3
5
4
,
±
9
+
4
5
2
,
±
3
3
+
5
2
)
,
{\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{11+3\sqrt5}{4},\,\pm\frac{9+4\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{2}\right),}
(
±
3
+
5
4
,
±
5
+
2
5
2
,
±
(
5
+
2
5
)
,
±
17
+
5
5
4
)
,
{\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{5+2\sqrt5}{2},\,\pm(5+2\sqrt5),\,\pm\frac{17+5\sqrt5}{4}\right),}
(
±
3
2
,
±
2
+
5
2
,
±
12
+
5
5
2
,
±
5
+
2
5
2
)
,
{\displaystyle \left(\pm\frac32,\,\pm\frac{2+\sqrt5}{2},\,\pm\frac{12+5\sqrt5}{2},\,\pm\frac{5+2\sqrt5}{2}\right),}
(
±
3
2
,
±
3
+
5
2
,
±
19
+
9
5
4
,
±
15
+
7
5
4
)
,
{\displaystyle \left(\pm\frac32,\,\pm\frac{3+\sqrt5}{2},\,\pm\frac{19+9\sqrt5}{4},\,\pm\frac{15+7\sqrt5}{4}\right),}
(
±
3
2
,
±
5
+
3
5
4
,
±
3
7
+
3
5
4
,
±
7
+
3
5
2
)
,
{\displaystyle \left(\pm\frac32,\,\pm\frac{5+3\sqrt5}{4},\,\pm3\frac{7+3\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{2}\right),}
(
±
1
+
5
2
,
±
(
3
+
5
)
,
±
2
(
2
+
5
)
,
±
3
3
+
5
2
)
,
{\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm(3+\sqrt5),\,\pm2(2+\sqrt5),\,\pm3\frac{3+\sqrt5}{2}\right),}
(
±
1
+
5
2
,
±
11
+
5
5
4
,
±
9
+
4
5
2
,
±
17
+
5
5
4
)
,
{\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm\frac{11+5\sqrt5}{4},\,\pm\frac{9+4\sqrt5}{2},\,\pm\frac{17+5\sqrt5}{4}\right),}
(
±
1
+
5
2
,
±
5
+
5
4
,
±
5
+
3
5
4
,
±
11
+
6
5
2
)
,
{\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm\frac{5+\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{11+6\sqrt5}{2}\right),}
(
±
1
+
5
2
,
±
7
+
5
4
,
±
3
+
2
5
2
,
±
23
+
11
5
4
)
,
{\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm\frac{7+\sqrt5}{4},\,\pm\frac{3+2\sqrt5}{2},\,\pm\frac{23+11\sqrt5}{4}\right),}
(
±
5
+
5
4
,
±
(
1
+
5
)
,
±
12
+
5
5
2
,
±
3
3
+
5
4
)
,
{\displaystyle \left(\pm\frac{5+\sqrt5}{4},\,\pm(1+\sqrt5),\,\pm\frac{12+5\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{4}\right),}
(
±
5
+
5
4
,
±
5
+
5
2
,
±
9
+
4
5
2
,
±
17
+
7
5
4
)
,
{\displaystyle \left(\pm\frac{5+\sqrt5}{4},\,\pm\frac{5+\sqrt5}{2},\,\pm\frac{9+4\sqrt5}{2},\,\pm\frac{17+7\sqrt5}{4}\right),}
(
±
5
+
5
4
,
±
(
2
+
5
)
,
±
11
+
4
5
2
,
±
5
3
+
5
4
)
,
{\displaystyle \left(\pm\frac{5+\sqrt5}{4},\,\pm(2+\sqrt5),\,\pm\frac{11+4\sqrt5}{2},\,\pm5\frac{3+\sqrt5}{4}\right),}
(
±
2
+
5
2
,
±
13
+
5
5
4
,
±
(
3
+
2
5
)
,
±
17
+
7
5
4
)
,
{\displaystyle \left(\pm\frac{2+\sqrt5}{2},\,\pm\frac{13+5\sqrt5}{4},\,\pm(3+2\sqrt5),\,\pm\frac{17+7\sqrt5}{4}\right),}
(
±
2
+
5
2
,
±
5
3
+
5
4
,
±
11
+
7
5
4
,
±
2
(
2
+
5
)
)
,
{\displaystyle \left(\pm\frac{2+\sqrt5}{2},\,\pm5\frac{3+\sqrt5}{4},\,\pm\frac{11+7\sqrt5}{4},\,\pm2(2+\sqrt5)\right),}
(
±
7
+
5
4
,
±
3
1
+
5
4
,
±
12
+
5
5
2
,
±
(
2
+
5
)
)
,
{\displaystyle \left(\pm\frac{7+\sqrt5}{4},\,\pm3\frac{1+\sqrt5}{4},\,\pm\frac{12+5\sqrt5}{2},\,\pm(2+\sqrt5)\right),}
(
±
7
+
5
4
,
±
4
+
5
2
,
±
(
5
+
2
5
)
,
±
15
+
7
5
4
)
,
{\displaystyle \left(\pm\frac{7+\sqrt5}{4},\,\pm\frac{4+\sqrt5}{2},\,\pm(5+2\sqrt5),\,\pm\frac{15+7\sqrt5}{4}\right),}
(
±
7
+
5
4
,
±
7
+
3
5
4
,
±
11
+
4
5
2
,
±
7
+
3
5
2
)
,
{\displaystyle \left(\pm\frac{7+\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{4},\,\pm\frac{11+4\sqrt5}{2},\,\pm\frac{7+3\sqrt5}{2}\right),}
(
±
3
1
+
5
4
,
±
(
3
+
5
)
,
±
7
+
4
5
2
,
±
17
+
7
5
4
)
,
{\displaystyle \left(\pm3\frac{1+\sqrt5}{4},\,\pm(3+\sqrt5),\,\pm\frac{7+4\sqrt5}{2},\,\pm\frac{17+7\sqrt5}{4}\right),}
(
±
3
1
+
5
4
,
±
5
+
3
5
2
,
±
9
+
4
5
2
,
±
5
3
+
5
4
)
,
{\displaystyle \left(\pm3\frac{1+\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{2},\,\pm\frac{9+4\sqrt5}{2},\,\pm5\frac{3+\sqrt5}{4}\right),}
(
±
3
+
5
2
,
±
3
+
2
5
2
,
±
25
+
9
5
4
,
±
3
3
+
5
4
)
,
{\displaystyle \left(\pm\frac{3+\sqrt5}{2},\,\pm\frac{3+2\sqrt5}{2},\,\pm\frac{25+9\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{4}\right),}
(
±
3
+
5
2
,
±
7
+
5
5
4
,
±
11
+
4
5
2
,
±
13
+
5
5
4
)
,
{\displaystyle \left(\pm\frac{3+\sqrt5}{2},\,\pm\frac{7+5\sqrt5}{4},\,\pm\frac{11+4\sqrt5}{2},\,\pm\frac{13+5\sqrt5}{4}\right),}
(
±
5
+
3
5
4
,
±
4
+
3
5
2
,
±
(
5
+
2
5
)
,
±
13
+
5
5
4
)
,
{\displaystyle \left(\pm\frac{5+3\sqrt5}{4},\,\pm\frac{4+3\sqrt5}{2},\,\pm(5+2\sqrt5),\,\pm\frac{13+5\sqrt5}{4}\right),}
(
±
5
+
3
5
4
,
±
3
2
+
5
2
,
±
(
3
+
2
5
)
,
±
15
+
7
5
4
)
,
{\displaystyle \left(\pm\frac{5+3\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2},\,\pm(3+2\sqrt5),\,\pm\frac{15+7\sqrt5}{4}\right),}
(
±
5
+
3
5
4
,
±
11
+
7
5
4
,
±
7
+
4
5
2
,
±
7
+
3
5
2
)
,
{\displaystyle \left(\pm\frac{5+3\sqrt5}{4},\,\pm\frac{11+7\sqrt5}{4},\,\pm\frac{7+4\sqrt5}{2},\,\pm\frac{7+3\sqrt5}{2}\right),}
(
±
5
+
3
5
4
,
±
4
+
5
2
,
±
(
2
+
5
)
,
±
25
+
9
5
4
)
,
{\displaystyle \left(\pm\frac{5+3\sqrt5}{4},\,\pm\frac{4+\sqrt5}{2},\,\pm(2+\sqrt5),\,\pm\frac{25+9\sqrt5}{4}\right),}
(
±
5
+
3
5
4
,
±
11
+
3
5
4
,
±
7
+
4
5
2
,
±
2
(
2
+
5
)
)
,
{\displaystyle \left(\pm\frac{5+3\sqrt5}{4},\,\pm\frac{11+3\sqrt5}{4},\,\pm\frac{7+4\sqrt5}{2},\,\pm2(2+\sqrt5)\right),}
(
±
4
+
5
2
,
±
3
+
2
5
2
,
±
11
+
4
5
2
,
±
3
2
+
5
2
)
,
{\displaystyle \left(\pm\frac{4+\sqrt5}{2},\,\pm\frac{3+2\sqrt5}{2},\,\pm\frac{11+4\sqrt5}{2},\,\pm3\frac{2+\sqrt5}{2}\right),}
(
±
4
+
5
2
,
±
(
2
+
5
)
,
±
23
+
9
5
4
,
±
11
+
3
5
4
)
,
{\displaystyle \left(\pm\frac{4+\sqrt5}{2},\,\pm(2+\sqrt5),\,\pm\frac{23+9\sqrt5}{4},\,\pm\frac{11+3\sqrt5}{4}\right),}
(
±
4
+
5
2
,
±
7
+
5
5
4
,
±
3
7
+
3
5
4
,
±
(
3
+
5
)
)
,
{\displaystyle \left(\pm\frac{4+\sqrt5}{2},\,\pm\frac{7+5\sqrt5}{4},\,\pm3\frac{7+3\sqrt5}{4},\,\pm(3+\sqrt5)\right),}
(
±
(
1
+
5
)
,
±
11
+
5
5
4
,
±
7
+
4
5
2
,
±
15
+
7
5
4
)
,
{\displaystyle \left(\pm(1+\sqrt5),\,\pm\frac{11+5\sqrt5}{4},\,\pm\frac{7+4\sqrt5}{2},\,\pm\frac{15+7\sqrt5}{4}\right),}
(
±
(
1
+
5
)
,
±
5
+
3
5
2
,
±
2
(
2
+
5
)
,
±
7
+
3
5
2
)
,
{\displaystyle \left(\pm(1+\sqrt5),\,\pm\frac{5+3\sqrt5}{2},\,\pm2(2+\sqrt5),\,\pm\frac{7+3\sqrt5}{2}\right),}
(
±
7
+
3
5
4
,
±
(
3
+
5
)
,
±
4
+
3
5
2
,
±
19
+
9
5
4
)
,
{\displaystyle \left(\pm\frac{7+3\sqrt5}{4},\,\pm(3+\sqrt5),\,\pm\frac{4+3\sqrt5}{2},\,\pm\frac{19+9\sqrt5}{4}\right),}
(
±
7
+
3
5
4
,
±
5
+
5
2
,
±
5
+
2
5
2
,
±
23
+
9
5
4
)
,
{\displaystyle \left(\pm\frac{7+3\sqrt5}{4},\,\pm\frac{5+\sqrt5}{2},\,\pm\frac{5+2\sqrt5}{2},\,\pm\frac{23+9\sqrt5}{4}\right),}
(
±
7
+
3
5
4
,
±
3
3
+
5
4
,
±
(
3
+
2
5
)
,
±
9
+
4
5
2
)
,
{\displaystyle \left(\pm\frac{7+3\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{4},\,\pm(3+2\sqrt5),\,\pm\frac{9+4\sqrt5}{2}\right),}
(
±
5
+
5
2
,
±
3
+
2
5
2
,
±
3
7
+
3
5
4
,
±
11
+
5
5
4
)
,
{\displaystyle \left(\pm\frac{5+\sqrt5}{2},\,\pm\frac{3+2\sqrt5}{2},\,\pm3\frac{7+3\sqrt5}{4},\,\pm\frac{11+5\sqrt5}{4}\right),}
(
±
3
+
2
5
2
,
±
4
+
3
5
2
,
±
9
+
4
5
2
,
±
3
2
+
5
2
)
,
{\displaystyle \left(\pm\frac{3+2\sqrt5}{2},\,\pm\frac{4+3\sqrt5}{2},\,\pm\frac{9+4\sqrt5}{2},\,\pm3\frac{2+\sqrt5}{2}\right),}
(
±
3
+
2
5
2
,
±
3
3
+
5
4
,
±
11
+
7
5
4
,
±
(
5
+
2
5
)
)
,
{\displaystyle \left(\pm\frac{3+2\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{4},\,\pm\frac{11+7\sqrt5}{4},\,\pm(5+2\sqrt5)\right),}
(
±
3
+
2
5
2
,
±
11
+
3
5
4
,
±
5
+
3
5
2
,
±
19
+
9
5
4
)
,
{\displaystyle \left(\pm\frac{3+2\sqrt5}{2},\,\pm\frac{11+3\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{2},\,\pm\frac{19+9\sqrt5}{4}\right),}
(
±
(
2
+
5
)
,
±
4
+
3
5
2
,
±
17
+
9
5
4
,
±
11
+
5
5
4
)
,
{\displaystyle \left(\pm(2+\sqrt5),\,\pm\frac{4+3\sqrt5}{2},\,\pm\frac{17+9\sqrt5}{4},\,\pm\frac{11+5\sqrt5}{4}\right),}
(
±
(
2
+
5
)
,
±
7
+
5
5
4
,
±
9
+
4
5
2
,
±
11
+
7
5
4
)
,
{\displaystyle \left(\pm(2+\sqrt5),\,\pm\frac{7+5\sqrt5}{4},\,\pm\frac{9+4\sqrt5}{2},\,\pm\frac{11+7\sqrt5}{4}\right),}
(
±
7
+
5
5
4
,
±
5
+
2
5
2
,
±
5
+
3
5
2
,
±
17
+
9
5
4
)
.
{\displaystyle \left(\pm\frac{7+5\sqrt5}{4},\,\pm\frac{5+2\sqrt5}{2},\,\pm\frac{5+3\sqrt5}{2},\,\pm\frac{17+9\sqrt5}{4}\right).}
The great disprismatohexacosihecatonicosachoron has a semi-uniform variant of the form a5b3c3d that maintains its full symmetry. This variant uses 120 great rhombicosidodecahedra of form a5b3c, 600 great rhombitetratetrahedra of form b3c3d, 720 dipentagonal prisms of form d a5b, and 1200 ditrigonal prisms of form a c3d as cells, with 4 edge lengths.
With edges of length a, b, c, and d (such that it forms a5b3c3d), its circumradius is given by
14
a
2
+
21
b
2
+
10
c
2
+
3
d
2
+
33
a
b
+
22
a
c
+
11
a
d
+
28
b
c
+
14
b
d
+
10
c
d
+
(
6
a
2
+
9
b
2
+
4
c
2
+
d
2
+
15
a
b
+
10
a
c
+
5
a
d
+
12
b
c
+
6
b
d
+
4
c
d
)
5
2
{\displaystyle \sqrt{\frac{14a^2+21b^2+10c^2+3d^2+33ab+22ac+11ad+28bc+14bd+10cd+(6a^2+9b^2+4c^2+d^2+15ab+10ac+5ad+12bc+6bd+4cd)\sqrt5}{2}}}
