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,\,±\frac{\sqrt2}{2},\,±\frac{2\sqrt2+\sqrt{10}}{2}\right),}
(
±
2
4
,
±
2
4
,
±
2
2
+
10
4
,
±
3
2
+
2
10
4
)
,
{\displaystyle \left(±\frac{\sqrt2}{4},\,±\frac{\sqrt2}{4},\,±\frac{2\sqrt2+\sqrt{10}}{4},\,±\frac{3\sqrt2+2\sqrt{10}}{4}\right),}
(
±
2
+
10
8
,
±
2
+
10
8
,
±
3
2
+
10
8
,
±
7
2
+
3
10
8
)
,
{\displaystyle \left(±\frac{\sqrt2+\sqrt{10}}{8},\,±\frac{\sqrt2+\sqrt{10}}{8},\,±3\frac{\sqrt2+\sqrt{10}}{8},\,±\frac{7\sqrt2+3\sqrt{10}}{8}\right),}
(
±
3
2
+
10
8
,
±
3
2
+
10
8
,
±
10
−
2
8
,
±
3
3
2
+
10
8
)
,
{\displaystyle \left(±\frac{3\sqrt2+\sqrt{10}}{8},\,±\frac{3\sqrt2+\sqrt{10}}{8},\,±\frac{\sqrt{10}-\sqrt2}{8},\,±3\frac{3\sqrt2+\sqrt{10}}{8}\right),}
(
±
2
+
10
4
,
±
2
+
10
4
,
±
3
2
+
10
4
,
±
3
2
+
10
4
)
,
{\displaystyle \left(±\frac{\sqrt2+\sqrt{10}}{4},\,±\frac{\sqrt2+\sqrt{10}}{4},\,±\frac{3\sqrt2+\sqrt{10}}{4},\,±\frac{3\sqrt2+\sqrt{10}}{4}\right),}
(
±
5
2
+
10
8
,
±
5
2
+
10
8
,
±
2
+
10
8
,
±
7
2
+
3
10
8
)
,
{\displaystyle \left(±\frac{5\sqrt2+\sqrt{10}}{8},\,±\frac{5\sqrt2+\sqrt{10}}{8},\,±\frac{\sqrt2+\sqrt{10}}{8},\,±\frac{7\sqrt2+3\sqrt{10}}{8}\right),}
(
±
2
2
+
10
4
,
±
2
2
+
10
4
,
±
2
4
,
±
4
2
+
10
4
)
,
{\displaystyle \left(±\frac{2\sqrt2+\sqrt{10}}{4},\,±\frac{2\sqrt2+\sqrt{10}}{4},\,±\frac{\sqrt2}{4},\,±\frac{4\sqrt2+\sqrt{10}}{4}\right),}
(
±
5
2
+
3
10
8
,
±
5
2
+
3
10
8
,
±
10
−
2
8
,
±
3
2
+
10
8
)
,
{\displaystyle \left(±\frac{5\sqrt2+3\sqrt{10}}{8},\,±\frac{5\sqrt2+3\sqrt{10}}{8},\,±\frac{\sqrt{10}-\sqrt2}{8},\,±\frac{3\sqrt2+\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,\,±\frac{\sqrt2}{4},\,±\frac{5\sqrt2+\sqrt{10}}{8},\,±3\frac{3\sqrt2+\sqrt{10}}{8}\right),}
(
0
,
±
3
2
+
10
8
,
±
3
2
+
2
10
4
,
±
5
2
+
10
8
)
,
{\displaystyle \left(0,\,±\frac{3\sqrt2+\sqrt{10}}{8},\,±\frac{3\sqrt2+2\sqrt{10}}{4},\,±\frac{5\sqrt2+\sqrt{10}}{8}\right),}
(
0
,
±
2
+
10
8
,
±
5
2
+
3
10
8
,
±
4
2
+
10
4
)
,
{\displaystyle \left(0,\,±\frac{\sqrt2+\sqrt{10}}{8},\,±\frac{5\sqrt2+3\sqrt{10}}{8},\,±\frac{4\sqrt2+\sqrt{10}}{4}\right),}
(
0
,
±
2
2
+
10
4
,
±
3
2
+
10
8
,
±
5
2
+
3
10
8
)
,
{\displaystyle \left(0,\,±\frac{2\sqrt2+\sqrt{10}}{4},\,±3\frac{\sqrt2+\sqrt{10}}{8},\,±\frac{5\sqrt2+3\sqrt{10}}{8}\right),}
(
±
10
−
2
8
,
±
2
4
,
±
2
2
+
10
2
,
±
2
+
10
8
)
,
{\displaystyle \left(±\frac{\sqrt{10}-\sqrt2}{8},\,±\frac{\sqrt2}{4},\,±\frac{2\sqrt2+\sqrt{10}}{2},\,±\frac{\sqrt2+\sqrt{10}}{8}\right),}
(
±
10
−
2
8
,
±
2
4
,
±
3
2
+
10
4
,
±
7
2
+
3
10
8
)
,
{\displaystyle \left(±\frac{\sqrt{10}-\sqrt2}{8},\,±\frac{\sqrt2}{4},\,±\frac{3\sqrt2+\sqrt{10}}{4},\,±\frac{7\sqrt2+3\sqrt{10}}{8}\right),}
(
±
10
−
2
8
,
±
2
+
10
4
,
±
7
2
+
3
10
8
,
±
2
2
+
10
4
)
,
{\displaystyle \left(±\frac{\sqrt{10}-\sqrt2}{8},\,±\frac{\sqrt2+\sqrt{10}}{4},\,±\frac{7\sqrt2+3\sqrt{10}}{8},\,±\frac{2\sqrt2+\sqrt{10}}{4}\right),}
(
±
2
4
,
±
2
+
10
8
,
±
2
+
10
4
,
±
3
3
2
+
10
8
)
,
{\displaystyle \left(±\frac{\sqrt2}{4},\,±\frac{\sqrt2+\sqrt{10}}{8},\,±\frac{\sqrt2+\sqrt{10}}{4},\,±3\frac{3\sqrt2+\sqrt{10}}{8}\right),}
(
±
2
4
,
±
5
2
+
10
8
,
±
3
2
+
10
4
,
±
5
2
+
3
10
8
)
,
{\displaystyle \left(±\frac{\sqrt2}{4},\,±\frac{5\sqrt2+\sqrt{10}}{8},\,±\frac{3\sqrt2+\sqrt{10}}{4},\,±\frac{5\sqrt2+3\sqrt{10}}{8}\right),}
(
±
2
+
10
8
,
±
3
2
+
10
8
,
±
2
+
10
4
,
±
3
2
+
2
10
4
)
,
{\displaystyle \left(±\frac{\sqrt2+\sqrt{10}}{8},\,±\frac{3\sqrt2+\sqrt{10}}{8},\,±\frac{\sqrt2+\sqrt{10}}{4},\,±\frac{3\sqrt2+2\sqrt{10}}{4}\right),}
(
±
2
+
10
8
,
±
3
2
+
10
8
,
±
3
2
+
10
4
,
±
4
2
+
10
4
)
,
{\displaystyle \left(±\frac{\sqrt2+\sqrt{10}}{8},\,±\frac{3\sqrt2+\sqrt{10}}{8},\,±\frac{3\sqrt2+\sqrt{10}}{4},\,±\frac{4\sqrt2+\sqrt{10}}{4}\right),}
(
±
2
2
,
±
3
2
+
10
8
,
±
2
2
+
10
4
,
±
7
2
+
3
10
8
)
,
{\displaystyle \left(±\frac{\sqrt2}{2},\,±\frac{3\sqrt2+\sqrt{10}}{8},\,±\frac{2\sqrt2+\sqrt{10}}{4},\,±\frac{7\sqrt2+3\sqrt{10}}{8}\right),}
(
±
3
2
+
10
8
,
±
2
2
+
10
4
,
±
3
2
+
10
4
,
±
3
2
+
10
8
)
,
{\displaystyle \left(±\frac{3\sqrt2+\sqrt{10}}{8},\,±\frac{2\sqrt2+\sqrt{10}}{4},\,±\frac{3\sqrt2+\sqrt{10}}{4},\,±3\frac{\sqrt2+\sqrt{10}}{8}\right),}
(
±
2
+
10
4
,
±
5
2
+
10
8
,
±
5
2
+
3
10
8
,
±
2
2
+
10
4
)
.
{\displaystyle \left(±\frac{\sqrt2+\sqrt{10}}{4},\,±\frac{5\sqrt2+\sqrt{10}}{8},\,±\frac{5\sqrt2+3\sqrt{10}}{8},\,±\frac{2\sqrt2+\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.
