Octagonal ditetragoltriate

The octagonal ditetragoltriate or odet is a convex isogonal polychoron and the sixth member of the ditetragoltriate family. It consists of 16 octagonal prisms and 64 rectangular trapezoprisms. 2 octagonal prisms and 4 rectangular trapezoprisms join at each vertex. However, it cannot be made uniform. It is the first in an infinite family of isogonal octagonal prismatic swirlchora.

Octagonal ditetragoltriate
File:Octagonal ditetragoltriate.png (OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Odet
Elements
Cells	64 rectangular trapezoprisms, 16 octagonal prisms
Faces	128 isosceles trapezoids, 128 rectangles, 16 octagons
Edges	64+128+128
Vertices	128
Vertex figure	Notch
Measures (based on variant with trapezoids with 3 unit edges)
Edge lengths	Edges of smaller octagon (128): 1
	Lacing edges (64): 1
	Edges of larger octagon (128): ${\frac {2+{\sqrt {4-2{\sqrt {2}}}}}{2}}\approx 1.54120$
Circumradius	${\sqrt {\frac {5+2{\sqrt {2}}+2{\sqrt {2+{\sqrt {2}}}}}{2}}}\approx 2.40041$
Central density	1
Related polytopes
Army	Odet
Regiment	Odet
Dual	Octagonal tetrambitriate
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(8)≀S2, order 512
Convex	Yes
Nature	Tame

This polychoron can be alternated into a square double antiprismoid, which is also nonuniform.

It can be obtained as the convex hull of 2 similarly oriented semi-uniform octagonal duoprisms, one with a larger xy octagon and the other with a larger zw octagon.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {2+{\sqrt {4-2{\sqrt {2}}}}}{2}}$ ≈ 1:1.54120. This value is also the ratio between the two sides of the two semi-uniform duoprisms.

Vertex coordinates

The vertices of an octagonal ditetragoltriate, assuming that the trapezoids have three equal edges of length 1, centered at the origin, are given by:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {2+{\sqrt {4-2{\sqrt {2}}}}}{4}},\,\pm {\frac {2+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {2+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{4}},\,\pm {\frac {2+{\sqrt {4-2{\sqrt {2}}}}}{4}}\right),$
• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {4-2{\sqrt {2}}}}}{4}},\,\pm {\frac {2+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{4}}\right),$
• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{4}},\,\pm {\frac {2+{\sqrt {4-2{\sqrt {2}}}}}{4}}\right),$
• $\left(\pm {\frac {2+{\sqrt {4-2{\sqrt {2}}}}}{4}},\,\pm {\frac {2+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}}\right),$
• $\left(\pm {\frac {2+{\sqrt {4-2{\sqrt {2}}}}}{4}},\,\pm {\frac {2+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{4}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {2+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{4}},\,\pm {\frac {2+{\sqrt {4-2{\sqrt {2}}}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}}\right),$
• $\left(\pm {\frac {2+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}{4}},\,\pm {\frac {2+{\sqrt {4-2{\sqrt {2}}}}}{4}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right).$