Icosahedral 120-cell

Icosahedral 120-cell
Orthogonal projection
Type	Schläfli-Hess polytope
Cells	120 {3,5}
Faces	1200 {3}
Edges	720
Vertices	120
Vertex figure	{5,5/2}
Schläfli symbol	{3,5,5/2}
Symmetry group	H4, [3,3,5]
Coxeter-Dynkin diagram
Dual	Small stellated 120-cell
Properties	Regular

In geometry, the icosahedral 120-cell, polyicosahedron, faceted 600-cell or icosaplex is a regular star 4-polytope with Schläfli symbol {3,5,5/2}. It is one of 10 regular Schläfli-Hess polytopes.

It is constructed by 5 icosahedra around each edge in a pentagrammic figure. The vertex figure is a great dodecahedron.

Related polytopes

It has the same edge arrangement as the 600-cell, grand 120-cell and great 120-cell, and shares its vertices with all other Schläfli–Hess 4-polytopes except the great grand stellated 120-cell (another stellation of the 120-cell).

Orthographic projections by Coxeter planes

H4	-	F4
[30]	[20]	[12]
H3	A2 / B3 / D4	A3 / B2
[10]	[6]	[4]

As a faceted 600-cell, replacing the simplicial cells of the 600-cell with icosahedral pentagonal polytope cells, it could be seen as a four-dimensional analogue of the great dodecahedron, which replaces the triangular faces of the icosahedron with pentagonal faces. Indeed, the icosahedral 120-cell is dual to the small stellated 120-cell, which could be taken as a 4D analogue of the small stellated dodecahedron, dual of the great dodecahedron.

See also

• List of regular polytopes
• Convex regular 4-polytope
• Kepler-Poinsot solids - regular star polyhedron
• Star polygon - regular star polygons

References

• Edmund Hess, (1883) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder [1].
• H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26, Regular Star-polytopes, pp. 404–408)
• Klitzing, Richard. "4D uniform polytopes (polychora) x3o5o5/2o - fix".

External links

• Regular polychora Archived 2003-09-06 at the Wayback Machine
• Discussion on names
• Reguläre Polytope
• The Regular Star Polychora