Decagrammic duoprism

Decagrammic duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Stadidip
Coxeter diagram	x10/3o x10/3o ()
Elements
Cells	20 decagrammic prisms
Faces	100 squares, 20 decagrams
Edges	200
Vertices	100
Vertex figure	Tetragonal disphenoid, edge lengths √(5–√5)/2 (bases) and √2 (sides)
Measures (edge length 1)
Circumradius	${\frac {{\sqrt {10}}-{\sqrt {2}}}{2}}\approx 0.87403$
Inradius	${\frac {\sqrt {5-2{\sqrt {5}}}}{2}}\approx 0.36327$
Hypervolume	$25{\frac {5-2{\sqrt {5}}}{4}}\approx 3.29915$
Dichoral angles	Stiddip–4–stiddip: 90°
	Stiddip–10/3–stiddip: 72°
Central density	9
Number of external pieces	40
Level of complexity	12
Related polytopes
Army	Dedip
Regiment	Stadidip
Dual	Decagrammic duotegum
Conjugate	Decagonal duoprism
Abstract & topological properties
Flag count	2400
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(10)≀S2, order 800
Convex	No
Nature	Tame

The decagrammic duoprism or stadidip, also known as the decagrammic-decagrammic duoprism, the 10/3 duoprism or the 10/3-10/3 duoprism, is a noble uniform duoprism that consists of 20 decagrammic prisms, with 4 meeting at each vertex.

A unit decagrammic duoprism can be vertex-inscribed into a grand ditetrahedronary hexacosidishecatonicosachoron.

Vertex coordinates

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

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5-2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5-2{\sqrt {5}}}}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5-2{\sqrt {5}}}}{2}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5-2{\sqrt {5}}}}{2}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,0\right)$,
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{8}}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5-2{\sqrt {5}}}}{2}}\right)$,
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{8}}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\right)$,
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{8}}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,0\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5-2{\sqrt {5}}}}{2}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,0,\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,0,\,\pm {\frac {{\sqrt {5}}-1}{2}},\,0\right)$.

Representations

A decagrammic duoprism duoprism has the following Coxeter diagrams:

• x10/3o x10/3o () (full symmetry)
• x5/3x x10/3o () (H₂×I₂(10) symmetry, some decagons as dipentagons)
• x5/3x x5/3x () (H₂≀S₂ symmetry, all decagons as dipentagons)

External links

• Bowers, Jonathan. "Category A: Duoprisms".

• Klitzing, Richard. "stadidip".