Pentadekeract

Pentadekeract
Rank	15
Type	Regular
Notation
Coxeter diagram	x4o3o3o3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol	{4,3,3,3,3,3,3,3,3,3,3,3,3,3}
Elements
Tedaka	30 tetradekeracts
Tradaka	420 tridekeracts
Doka	3640 dodekeracts
Henda	21840 hendekeracts
Daka	96096 dekeracts
Xenna	320320 enneracts
Yotta	823680 octeracts
Zetta	1647360 hepteracts
Exa	2562560 hexeracts
Peta	3075072 penteracts
Tera	2795520 tesseracts
Cells	1863680 cubes
Faces	860160 squares
Edges	245760
Vertices	32768
Vertex figure	Pentadecatradakon, edge length √2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {15}}{2}}\approx 1.93649$
Inradius	${\frac {1}{2}}=0.5$
Hypervolume	1
Dixennal angle	90°
Height	1
Central density	1
Number of external pieces	30
Level of complexity	a
Related polytopes
Army	*
Regiment	*
Dual	Trismyriadischiliaheptacosihexacontoctatedakon
Conjugate	None
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B15, order 42849873690624000
Convex	Yes
Nature	Tame

The pentadekeract, also called the 15-cube or triacontatedakon, is one of the 3 regular polytedaka. It has 30 tetradekeracts as facets, joining 3 to a dokon and 15 to a vertex.

It is the 15-dimensional hypercube. As such it is a penteract trioprism and cube pentaprism.

It can be alternated into a demipentadekeract, which is uniform.

Vertex coordinates

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

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