The small disnub dishexacosichoron , or sadsadox , is a nonconvex uniform polychoron that consists of 4800 regular octahedra (falling in pairs into the same hyperplane, thus forming 2400 golden hexagrammic antiprisms ) and 1200 regular icosahedra (also falling in pairs in the same hyperplane, forming 600 snub disoctahedra ). 8 octahedra and 4 icosahedra join at each vertex.
This polychoron can be obtained as the blend of 10 rectified hexacosichora , positioned in a similar way to the compound of 10 hexacosichora known as the snub decahecatonicosachoron . In the process some of the octahedra blend out fully, while the other cells compound as noted above. In addition the vertex figure would in turn be a blend of two pentagonal prismatic vertex figures of the rectified hexacosichoron.
Coordinates for the vertices of a small disnub dishexacosichoron of edge length 1 are given by all permutations of:
(
0
,
0
,
±
2
2
,
±
2
2
+
10
2
)
,
{\displaystyle \left(0,\,0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {2{\sqrt {2}}+{\sqrt {10}}}{2}}\right),}
(
±
2
4
,
±
2
4
,
±
2
2
+
10
4
,
±
3
2
+
2
10
4
)
,
{\displaystyle \left(\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {2{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {3{\sqrt {2}}+2{\sqrt {10}}}{4}}\right),}
(
±
2
+
10
8
,
±
2
+
10
8
,
±
3
2
+
10
8
,
±
7
2
+
3
10
8
)
,
{\displaystyle \left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm 3{\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {7{\sqrt {2}}+3{\sqrt {10}}}{8}}\right),}
(
±
3
2
+
10
8
,
±
3
2
+
10
8
,
±
10
−
2
8
,
±
3
3
2
+
10
8
)
,
{\displaystyle \left(\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm 3{\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}}\right),}
(
±
2
+
10
4
,
±
2
+
10
4
,
±
3
2
+
10
4
,
±
3
2
+
10
4
)
,
{\displaystyle \left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{4}}\right),}
(
±
5
2
+
10
8
,
±
5
2
+
10
8
,
±
2
+
10
8
,
±
7
2
+
3
10
8
)
,
{\displaystyle \left(\pm {\frac {5{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {5{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {7{\sqrt {2}}+3{\sqrt {10}}}{8}}\right),}
(
±
2
2
+
10
4
,
±
2
2
+
10
4
,
±
2
4
,
±
4
2
+
10
4
)
,
{\displaystyle \left(\pm {\frac {2{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {2{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {4{\sqrt {2}}+{\sqrt {10}}}{4}}\right),}
(
±
5
2
+
3
10
8
,
±
5
2
+
3
10
8
,
±
10
−
2
8
,
±
3
2
+
10
8
)
,
{\displaystyle \left(\pm {\frac {5{\sqrt {2}}+3{\sqrt {10}}}{8}},\,\pm {\frac {5{\sqrt {2}}+3{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}}\right),}
together with all even permutations of:
(
0
,
±
2
4
,
±
5
2
+
10
8
,
±
3
3
2
+
10
8
)
,
{\displaystyle \left(0,\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {5{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm 3{\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}}\right),}
(
0
,
±
3
2
+
10
8
,
±
3
2
+
2
10
4
,
±
5
2
+
10
8
)
,
{\displaystyle \left(0,\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}+2{\sqrt {10}}}{4}},\,\pm {\frac {5{\sqrt {2}}+{\sqrt {10}}}{8}}\right),}
(
0
,
±
2
+
10
8
,
±
5
2
+
3
10
8
,
±
4
2
+
10
4
)
,
{\displaystyle \left(0,\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {5{\sqrt {2}}+3{\sqrt {10}}}{8}},\,\pm {\frac {4{\sqrt {2}}+{\sqrt {10}}}{4}}\right),}
(
0
,
±
2
2
+
10
4
,
±
3
2
+
10
8
,
±
5
2
+
3
10
8
)
,
{\displaystyle \left(0,\,\pm {\frac {2{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm 3{\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {5{\sqrt {2}}+3{\sqrt {10}}}{8}}\right),}
(
±
10
−
2
8
,
±
2
4
,
±
2
2
+
10
2
,
±
2
+
10
8
)
,
{\displaystyle \left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {2{\sqrt {2}}+{\sqrt {10}}}{2}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}}\right),}
(
±
10
−
2
8
,
±
2
4
,
±
3
2
+
10
4
,
±
7
2
+
3
10
8
)
,
{\displaystyle \left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {7{\sqrt {2}}+3{\sqrt {10}}}{8}}\right),}
(
±
10
−
2
8
,
±
2
+
10
4
,
±
7
2
+
3
10
8
,
±
2
2
+
10
4
)
,
{\displaystyle \left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {7{\sqrt {2}}+3{\sqrt {10}}}{8}},\,\pm {\frac {2{\sqrt {2}}+{\sqrt {10}}}{4}}\right),}
(
±
2
4
,
±
2
+
10
8
,
±
2
+
10
4
,
±
3
3
2
+
10
8
)
,
{\displaystyle \left(\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm 3{\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}}\right),}
(
±
2
4
,
±
5
2
+
10
8
,
±
3
2
+
10
4
,
±
5
2
+
3
10
8
)
,
{\displaystyle \left(\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {5{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {5{\sqrt {2}}+3{\sqrt {10}}}{8}}\right),}
(
±
2
+
10
8
,
±
3
2
+
10
8
,
±
2
+
10
4
,
±
3
2
+
2
10
4
)
,
{\displaystyle \left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {3{\sqrt {2}}+2{\sqrt {10}}}{4}}\right),}
(
±
2
+
10
8
,
±
3
2
+
10
8
,
±
3
2
+
10
4
,
±
4
2
+
10
4
)
,
{\displaystyle \left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {4{\sqrt {2}}+{\sqrt {10}}}{4}}\right),}
(
±
2
2
,
±
3
2
+
10
8
,
±
2
2
+
10
4
,
±
7
2
+
3
10
8
)
,
{\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {2{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {7{\sqrt {2}}+3{\sqrt {10}}}{8}}\right),}
(
±
3
2
+
10
8
,
±
2
2
+
10
4
,
±
3
2
+
10
4
,
±
3
2
+
10
8
)
,
{\displaystyle \left(\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {2{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm 3{\frac {{\sqrt {2}}+{\sqrt {10}}}{8}}\right),}
(
±
2
+
10
4
,
±
5
2
+
10
8
,
±
5
2
+
3
10
8
,
±
2
2
+
10
4
)
.
{\displaystyle \left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{4}},\,\pm {\frac {5{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {5{\sqrt {2}}+3{\sqrt {10}}}{8}},\,\pm {\frac {2{\sqrt {2}}+{\sqrt {10}}}{4}}\right).}
The regiment of the small disnub dishexacosichoron, known as the "sidtaps", contains 9 uniform members, 11 scaliform members, 3 fissary scaliforms, and a number of uniform compounds.