Great 120-cell

In geometry, the great 120-cell or great polydodecahedron is a regular star 4-polytope with Schläfli symbol {5,5/2,5}. It is one of 10 regular Schläfli-Hess polytopes. It is one of the two such polytopes that is self-dual.

Great 120-cell

Orthogonal projection
TypeSchläfli-Hess polytope
Cells120 {5,5/2}
Faces720 {5}
Vertex figure{5/2,5}
Schläfli symbol{5,5/2,5}
Coxeter-Dynkin diagram
Symmetry groupH4, [3,3,5]

It has the same edge arrangement as the 600-cell, icosahedral 120-cell as well as the same face arrangement as the grand 120-cell.

Orthographic projections by Coxeter planes
H4 - F4



H3 A2 / B3 / D4 A3 / B2




Due to its self-duality, it does not have a good three-dimensional analogue, but (like all other star polyhedra and polychora) is analogous to the two-dimensional pentagram. With itself, it can form the compound of two great 120-cells.

See also


  • 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. 404408)
  • Klitzing, Richard. "4D uniform polytopes (polychora) o5o5/2o5x - gohi".
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