Rank15
TypeRegular
Notation
Coxeter diagramx4o3o3o3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol{4,3,3,3,3,3,3,3,3,3,3,3,3,3}
Elements
Doka3640 dodekeracts
Henda21840 hendekeracts
Daka96096 dekeracts
Xenna320320 enneracts
Yotta823680 octeracts
Zetta1647360 hepteracts
Exa2562560 hexeracts
Peta3075072 penteracts
Tera2795520 tesseracts
Cells1863680 cubes
Faces860160 squares
Edges245760
Vertices32768
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {15}}{2}}\approx 1.93649}$
Inradius${\displaystyle {\frac {1}{2}}=0.5}$
Hypervolume1
Dixennal angle90°
Height1
Central density1
Number of external pieces30
Level of complexitya
Related polytopes
Army*
Regiment*
ConjugateNone
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB15, order 42849873690624000
ConvexYes
NatureTame

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:

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