Octagonal spinoduoprism

Octagonal spinoduoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Ondip
Elements
Cells	64 tetrahedra, 128 triangular prisms, 64 cubes
Faces	256 triangles, 128+128+128 squares
Edges	128+128+256
Vertices	128
Vertex figure	Blend of two triangular antipodiums, edge lengths 1 (one base each) and √2 (remaining edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {3+{\sqrt {2}}}{2}}}\approx 1.48563$
Hypervolume	0
Dichoral angles	Tet–3–trip: 150°
	Cube–4–trip: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
Number of external pieces	1408
Level of complexity	108
Related polytopes
Army	Oadet
Regiment	Ondip
Conjugate	Great octagonal spinoduoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(8)≀S2, order 512
Convex	No
Nature	Feral

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:

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

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

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

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.

External links

• Bowers, Jonathan. "Category 20: Miscellaneous" (#1846).

• Bowers, Jonathan. "How to Make Ondip".

• Klitzing, Richard. "ondip".