Grand stellated 120-cell

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

Grand stellated 120-cell

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

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

Orthographic projections by Coxeter planes
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 grand stellated 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. 404–408)
  • Klitzing, Richard. "4D uniform polytopes (polychora) x5/2o5o5/2o - gashi".
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