Octagonal ditetragoltriate

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

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.

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:

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