Digonal-hexagonal duoantiprism

Digonal-hexagonal duoantiprism
File:Digonal-hexagonal duoantiprism.png (OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Dihidap
Coxeter diagram	s4o2s12o ()
Elements
Cells	24 digonal disphenoids, 12 tetragonal disphenoids, 4 hexagonal antiprisms
Faces	48+48 isosceles triangles, 4 hexagons
Edges	12+24+48
Vertices	24
Vertex figure	Augmented triangular prism
Measures (based on polygons of edge length 1)
Edge lengths	Lacing (48): ${\sqrt {\frac {5-2{\sqrt {3}}}{2}}}\approx 0.87633$
	Digons (12): 1
	Edges of hexagons (24): 1
Circumradius	${\frac {\sqrt {5}}{2}}\approx 1.11803$
Central density	1
Related polytopes
Army	Dihidap
Regiment	Dihidap
Dual	Digonal-hexagonal duoantitegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(B2×I2(12))/2, order 96
Convex	Yes
Nature	Tame

The digonal-hexagonal duoantiprism or dihidap, also known as the 2-6 duoantiprism, is a convex isogonal polychoron that consists of 4 hexagonal antiprisms, 12 tetragonal disphenoids, and 24 digonal disphenoids. 2 hexagonal antiprisms, 2 tetragonal disphenoids, and 4 digonal disphenoids join at each vertex. It can be obtained through the process of alternating the square-dodecagonal duoprism. However, it cannot be made uniform, as it generally has 3 edge lengths, which can be minimized to no fewer than 2 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\sqrt {\frac {10+4{\sqrt {3}}}{13}}}$ ≈ 1:1.14113.

Vertex coordinates

The vertices of a digonal-hexagonal duoantiprism, assuming that the hexagonal antiprisms are uniform of edge length 1, centered at the origin, are given by:

• $\left(0,\,\pm 1,\,{\sqrt {\frac {{\sqrt {3}}-1}{2}}},\,{\sqrt {\frac {{\sqrt {3}}-1}{2}}}\right),$
• $\left(0,\,\pm 1,\,-{\sqrt {\frac {{\sqrt {3}}-1}{2}}},\,-{\sqrt {\frac {{\sqrt {3}}-1}{2}}}\right),$
• $\left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,{\sqrt {\frac {{\sqrt {3}}-1}{2}}},\,{\sqrt {\frac {{\sqrt {3}}-1}{2}}}\right),$
• $\left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {{\sqrt {3}}-1}{2}}},\,-{\sqrt {\frac {{\sqrt {3}}-1}{2}}}\right),$
• $\left(\pm 1,\,0,\,{\sqrt {\frac {{\sqrt {3}}-1}{2}}},\,-{\sqrt {\frac {{\sqrt {3}}-1}{2}}}\right),$
• $\left(\pm 1,\,0,\,-{\sqrt {\frac {{\sqrt {3}}-1}{2}}},\,{\sqrt {\frac {{\sqrt {3}}-1}{2}}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,{\sqrt {\frac {{\sqrt {3}}-1}{2}}},\,-{\sqrt {\frac {{\sqrt {3}}-1}{2}}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,-{\sqrt {\frac {{\sqrt {3}}-1}{2}}},\,{\sqrt {\frac {{\sqrt {3}}-1}{2}}}\right).$

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:

• $\left(0,\,\pm 1,\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right),$
• $\left(0,\,\pm 1,\,-{\frac {\sqrt {2}}{4}},\,-{\frac {\sqrt {2}}{4}}\right),$
• $\left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right),$
• $\left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,-{\frac {\sqrt {2}}{4}},\,-{\frac {\sqrt {2}}{4}}\right),$
• $\left(\pm 1,\,0,\,{\frac {\sqrt {2}}{4}},\,-{\frac {\sqrt {2}}{4}}\right),$
• $\left(\pm 1,\,0,\,-{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,{\frac {\sqrt {2}}{4}},\,-{\frac {\sqrt {2}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,-{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right).$