Icosahedral 120-cell

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.

Icosahedral 120-cell

Orthogonal projection
TypeSchläfli-Hess polytope
Cells120 {3,5}
Faces1200 {3}
Edges720
Vertices120
Vertex figure{5,5/2}
Schläfli symbol{3,5,5/2}
Symmetry groupH4, [3,3,5]
Coxeter-Dynkin diagram
DualSmall stellated 120-cell
PropertiesRegular

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

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. With its dual, it can form the compound of icosahedral 120-cell and small stellated 120-cell.

gollark: I guess it depends on exactly what you do, and the resistance of the wires.
gollark: Which is as far as I know more an issue of low voltages than DC itself, but DC means you can't change the voltage very easily.
gollark: There is the problem that low-voltage DC loses power more quickly over longer distances.
gollark: Yes, you're right, let's just replace our lightbulbs with idealized magic visible light emitters.
gollark: If they didn't need that (I think the only practical way to achieve this would just be to stick one larger and more efficient converter somewhere) the bulbs would be individually cheaper and probably more efficient too, as well as safer.

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 .
  • H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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".
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.