Diacositetracontadiminished 2 41 polytope

Diacositetracontadiminished 2 41 polytope
Rank	8
Type	Scaliform
Elements
Zetta	1920 7-simplices 240 7-demicubes 240 hepteractidiminished 231 polytopes
Exa	15360+15360 6-simplices 6720 tridiminished 221 polytopes 3360 6-demicubes
Peta	53760+107520 5-simplices 53760 hexadecachoric pyramids 20160 5-demicubes
Tera	107520+107520+322560 pentachora 6720+53760 hexadecachora
Cells	26880+107520+107520+430080 tetrahedra
Faces	107520+215040 triangles
Edges	53760
Vertices	1920
Vertex figure	Octadiminished 7-demicube, edge length 1
Measures (edge length 1)
Circumradius	${\sqrt {2}}\approx 1.41421$
Central density	1
Number of external pieces	2400
Related polytopes
Conjugate	None
Abstract & topological properties
Flag count	2136637440
Euler characteristic	0
Orientable	Yes
Properties
Convex	Yes
Nature	Tame

The diacositetracontadiminished 2₄₁ polytope or diacositetracontadiminished diacositetraconta-myriaheptachiliadiacosioctaconta-zetton, abbreviated as 240-diminished 240-17280-zetton, is a convex scaliform 8-polytope. It has 1920 7-simplices, 240 7-demicubes and 240 hepteractidiminished 2₃₁ polytopes as facets. Eight 7-simplices, eight 7-demicubes, and 14 hepteractidiminished 2₃₁ polytopes meet each vertex.

One can create this polyzetton by removing an inscribed 4₂₁ polytope's vertices from the 2₄₁ polytope.

Vertex coordinates

The vertices of a diacositetracontadiminished 2₄₁ polytope of edge length 1, centered at the origin, are given by:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,0,\,0,\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm 1,\,0,\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,\pm 1,\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,0,\,\pm 1\right)$,

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm 1\right)$,

• $\left(\pm 1,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0\right)$,
• $\left(0,\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0\right)$,
• $\left(0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,0\right)$,
• $\left(0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm 1\right)$,

• $\left(\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}},\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm 1\right)$,

• $\left(\pm 1,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}},\,0\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm 1\right)$,

• $\left(\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0\right)$,
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(0 ,\, \pm \frac12 ,\, \pm \frac12 ,\, \pm 1 ,\, 0 ,\, \pm \frac12 ,\, \pm \frac12 ,\, 0\right)} ,
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(0 ,\, \pm \frac12 ,\, \pm \frac12 ,\, 0 ,\, \pm 1 ,\, \pm \frac12 ,\, \pm \frac12 ,\, 0\right)} ,
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(0 ,\, \pm \frac12 ,\, \pm \frac12 ,\, 0 ,\, 0 ,\, \pm \frac12 ,\, \pm \frac12 ,\, \pm 1\right)} ,

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\pm \frac12 ,\, \pm 1 ,\, 0 ,\, \pm \frac12 ,\, 0 ,\, \pm \frac12 ,\, \pm \frac12 ,\, 0\right)} ,
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\pm \frac12 ,\, 0 ,\, \pm 1 ,\, \pm \frac12 ,\, 0 ,\, \pm \frac12 ,\, \pm \frac12 ,\, 0\right)} ,
• $\left(\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0\right)$,
• $\left(\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 1\right)$,

• $\left(\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm 1,\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,\pm 1,\,\pm {\frac {1}{2}}\right)$,

• $\left(\pm {\frac {1}{2}},\,\pm 1,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,0,\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm 1,\,\pm {\frac {1}{2}}\right)$,

• $\left(\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm 1,\,0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,\pm 1,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}}\right)$,

• $\left(\pm 1,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {1}{2}}\right)$,

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,0,\,0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm 1,\,0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,\pm 1,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,0,\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,

• $\left(\pm 1,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,

• $\left(\pm 1,\,0,\,0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,\pm 1,\,0,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,0,\,\pm 1,\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(0,\,0,\,0,\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,

• $\left({\frac {1}{2}},\,{\frac {1}{2}},\,{\frac {1}{2}},\,{\frac {1}{2}},\,{\frac {1}{2}},\,{\frac {1}{2}},\,{\frac {1}{2}},\,{\frac {1}{2}}\right)$ and all odd sign changes.

External links

• Klitzing, Richard. "bay".
• Wikipedia contributors. "2₄₁ polytope".