Dodecagonal ditetragoltriate

Dodecagonal ditetragoltriate
File:Dodecagonal ditetragoltriate.png (OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Twaddet
Elements
Cells	144 rectangular trapezoprisms, 24 dodecagonal prisms
Faces	288 isosceles trapezoids, 288 rectangles, 24 dodecagons
Edges	144+288+288
Vertices	288
Vertex figure	Notch
Measures (based on variant with trapezoids with 3 unit edges)
Edge lengths	Edges of smaller dodecagon (288): 1
	Lacing edges (144): 1
	Edges of larger dodecagon (288): ${\frac {1+{\sqrt {3}}}{2}}\approx 1.36603$
Circumradius	${\sqrt {\frac {11+6{\sqrt {3}}}{2}}}\approx 3.27050$
Central density	1
Related polytopes
Army	Twaddet
Regiment	Twaddet
Dual	Dodecagonal tetrambitriate
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(12)≀S2, order 1152
Convex	Yes
Nature	Tame

The dodecagonal ditetragoltriate or twaddet is a convex isogonal polychoron and the tenth member of the ditetragoltriate family. It consists of 24 dodecagonal prisms and 144 rectangular trapezoprisms. 2 dodecagonal 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 dodecagonal prismatic swirlchora.

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

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

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {1+{\sqrt {3}}}{2}}$ ≈ 1:1.36603. This value is also the ratio between the two sides of the two semi-uniform duoprisms.

Vertex coordinates

The vertices of a dodecagonal ditetragoltriate, assuming that the trapezoids have three equal edges of length 1, centered at the origin, are given by all permutations of the first and second, as well as the third and fourth coordinates of:

• $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{4}},\,\pm {\frac {5+3{\sqrt {3}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{4}},\,\pm {\frac {5+3{\sqrt {3}}}{4}}\right),$
• $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}}\right),$
• $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {3}}}{4}},\,\pm {\frac {5+3{\sqrt {3}}}{4}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {3}}}{4}},\,\pm {\frac {5+3{\sqrt {3}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}}\right).$