Grand 600-cell

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Grand 600-cell
Ortho solid 015-uniform polychoron 33p-t0.png
Orthogonal projection
Type Regular star 4-polytope
Cells 600 {3,3}
Faces 1200 {3}
Edges 720
Vertices 120
Vertex figure {3,5/2}
Schläfli symbol {3,3,5/2}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
Symmetry group H4, [3,3,5]
Dual Great grand stellated 120-cell
Properties Regular

In geometry, the grand 600-cell or grand polytetrahedron is a regular star 4-polytope with Schläfli symbol {3,3,5/2}. It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600 cells.

It is one of four regular star 4-polytopes discovered by Ludwig Schläfli. It is named by John Horton Conway, extending the naming system by Arthur Cayley for the Kepler-Poinsot solids.

The grand 600-cell can be seen as the four-dimensional analogue of the great icosahedron (which in turn is analogous to the pentagram); both of these are the only regular n-dimensional star polytopes which are derived by performing stellational operations on the pentagonal polytope which has simplectic faces. It can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of said (n-1)-D simplex faces of the core nD polytope (tetrahedra for the grand 600-cell, equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces.

The Grand 600-cell is also dual to the great grand stellated 120-cell, mirroring the great icosahedron's duality with the great stellated dodecahedron (which in turn is also analogous to the pentagram); all of these are the final stellations of the n-dimensional "dodecahedral-type" pentagonal polytope.

Related polytopes[edit]

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

Orthographic projections by Coxeter planes
H3 A2 / B3 / D4 A3 / B2
Schläfli-Hess polychoron-wireframe-4.png Grand 600-cell-ortho-6gon.png Grand 600-cell-ortho-4gon.png

With its dual, it forms the compound of great grand stellated 120-cell and grand 600-cell.

See also[edit]


  • 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-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) x3o3o5/2o - gax".

External links[edit]