Small stellated 120-cell

In geometry, the small stellated 120-cell or stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol {5/2,5,3}. It is one of 10 regular Schläfli-Hess polytopes.

Small stellated 120-cell

Orthogonal projection
TypeSchläfli-Hess polytope
Cells120 {5/2,5}
Faces720 {5/2}
Edges1200
Vertices120
Vertex figure{5,3}
Schläfli symbol{5/2,5,3}
Coxeter-Dynkin diagram
Symmetry groupH4, [3,3,5]
DualIcosahedral 120-cell
PropertiesRegular

It has the same edge arrangement as the great grand 120-cell, and also shares its 120 vertices with the 600-cell and eight other regular star 4-polytope. It may also be seen as the first stellation of the 120-cell. In this sense it could be seen as analogous to the three-dimensional small stellated dodecahedron, which is the first stellation of the dodecahedron. Indeed, the small stellated 120-cell is dual to the icosahedral 120-cell, which could be taken as a 4D analogue of the great dodecahedron, dual of the small stellated dodecahedron. With its dual, it forms the compound of icosahedral 120-cell and small stellated 120-cell.

The edges of the small stellated 120-cell are τ2 as long as those of the 120-cell core inside the 4-polytope.

Orthographic projections by Coxeter planes
H3 A2 / B3 / D4 A3 / B2

See also
  • List of regular polytopes
  • Convex regular 4-polytope - Set of 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) o3o5o5/2x - sishi".
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