# Octagonal spinoduoprism

(Redirected from Ondip)
Octagonal spinoduoprism
Rank4
TypeUniform
Notation
Bowers style acronymOndip
Elements
Cells64 tetrahedra, 128 triangular prisms, 64 cubes
Faces256 triangles, 128+128+128 squares
Edges128+128+256
Vertices128
Vertex figureBlend of two triangular antipodiums, edge lengths 1 (one base each) and 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {3+{\sqrt {2}}}{2}}}\approx 1.48563}$
Hypervolume0
Dichoral anglesTet–3–trip: 150°
Cube–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Number of external pieces1408
Level of complexity108
Related polytopes
RegimentOndip
ConjugateGreat octagonal spinoduoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(8)≀S2, order 512
ConvexNo
NatureFeral

The octagonal spinoduoprism, or ondip, is a nonconvex uniform polychoron that consists of 64 regular tetrahedra, 128 triangular prisms, and 64 cubes. 2 tetrahedra, 6 triangular prisms, and 4 cubes join at each vertex.

It was discovered in March 2006, constructed as a blend of 4 small disprismatotesseractihexadecachora. Its vertex figure is in turn a blend of two vertex figures of the small disprismatotesseractihexadecachoron. It has the same symmetry as the octagonal duoprism.

## Vertex coordinates

The vertices of an octagonal spinoduoprism of edge length 1 are given by all permutations of:

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}$

along with all permutations of the first two and/or last two coordinates of:

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right).}$

The first set of vertices are identical to the vertices of an inscribed small disprismatotesseractihexadecachoron.

## Related polychora

The regiment of the octagonal spinoduoprism contains two other uniform members (the small ditetragonal spinoduoprism and small altersquare duoantiprismoid), a fissary uniform member (the small ditetragonal fissary duoprism, and 10 scaliform members.