Great duoantiprism

Great duoantiprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Gudap
Coxeter diagram	s10o2s10/3o ()
Elements
Cells	50 tetrahedra, 10 pentagonal antiprisms, 10 pentagrammic retroprisms
Faces	100+100 triangles, 10 pentagons, 10 pentagrams
Edges	50+50+100
Vertices	50
Vertex figure	Semicrossed gyrobifastigium, edge lengths (√5-1)/2, 1, and (√5+1)/2
Measures (edge length 1)
Circumradius	1
Hypervolume	${\frac {55{\sqrt {5}}}{48}}\approx 2.56216$
Dichoral angles	Starp–5/2–starp: 144°
	Starp–3–tet: $\arccos \left({\frac {\sqrt {7-3{\sqrt {5}}}}{4}}\right)\approx 82.23876^{\circ }$
	Pap–5–pap: 72°
	Pap–3–tet: $\arccos \left({\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 22.23876^{\circ }$
Central density	3
Number of external pieces	600
Level of complexity	144
Related polytopes
Army	Pentagonal-pentagonal duoantiprism
Regiment	Gudap
Dual	Pentagonal-pentagrammic concave duoantitegum
Conjugate	Great duoantiprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(10)×I2(10))/2, order 200
Convex	No
Nature	Tame

The great duoantiprism or gudap, also known as the pentagonal-pentagrammic crossed duoantiprism or 5-5/3 duoantiprism, is a nonconvex uniform polychoron that consists of 50 tetrahedra, 10 pentagonal antiprisms, and 10 pentagrammic retroprisms. 4 tetrahedra, 2 pentagonal antiprisms, and 2 pentagrammic retroprisms join at each vertex.

It is one of only two members of the infinite set of duoantiprisms that can be made uniform, the other being the hexadecachoron. It can be obtained through the process of alternating a non-uniform decagonal-decagrammic duoprism where the decagrams have an edge length of ${\frac {1+{\sqrt {5}}}{2}}$ times that of its decagons.

The great duoantiprism contains the vertices of an inscribed pentagonal-pentagrammic duoprism, and in turn can be vertex-inscribed into a small stellated hecatonicosachoron. In fact, it can be derived as a partial faceting of that polychoron, with the pentagrammic retroprisms being facetings of a ring of 10 small stellated dodecahedral cells and the pentagonal antiprisms being facetings of a ring of 10 great dodecahedral cells of the great grand hecatonicosachoron.

Gallery

Convex core

Vertex coordinates

The coordinates of a great duoantiprism, centered at the origin and with unit edge length, are given by:

• $\pm \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0,\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right),$
• $\pm \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),$
• $\pm \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}}\right),$
• $\pm \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0,\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right),$
• $\pm \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),$
• $\pm \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}}\right),$
• $\pm \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right),$
• $\pm \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),$
• $\pm \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}}\right),$

External links

• Bowers, Jonathan. "Category 20: Miscellaneous" (#965).

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

• Klitzing, Richard. "gudap".
• Wikipedia contributors. "Great duoantiprism".