The Hexacosichoron: The 600-Cell
The densest regular polychoron—600 tetrahedral cells forming the 4D analogue of the icosahedron.
Understanding the 600-Cell
The hexacosichoron, commonly called the 600-cell, is the densest of the regular convex polychora. Bounded by 600 tetrahedral cells—with 5 meeting around every edge—it represents the four-dimensional analogue of the icosahedron. Its 120 vertices form the binary icosahedral group, connecting it to deep mathematics.
600 Tetrahedral Cells
The 600-cell is bounded by 600 regular tetrahedra. This is:
- 100× more cells than the pentachoron (5)
- 75× more cells than the tesseract (8)
- 5× more cells than the 120-cell (120)
Five tetrahedra meet around each edge—the maximum possible for any convex regular polychoron. This creates an extremely dense, tightly-packed structure.
Comparison with the 120-Cell
The 600-cell and 120-cell are duals with reciprocal counts:
| Property | 600-cell | 120-cell |
|---|---|---|
| Vertices | 120 | 600 |
| Edges | 720 | 1,200 |
| Faces | 1,200 | 720 |
| Cells | 600 | 120 |
| Cell type | Tetrahedra | Dodecahedra |
The 600-cell is "spikier" (more cells, smaller cells); the 120-cell is "rounder" (fewer cells, larger cells).
Mathematical Structure
The Schläfli Symbol {3,3,5}
The symbol {3,3,5} encodes the 600-cell:
- {3} = Faces are triangles
- {3,3} = Cells are tetrahedra (3 triangles at each vertex)
- {3,3,5} = Five tetrahedra meet around each edge
The ending 5 signals icosahedral (H₄) symmetry.
Symmetry: The H₄ Group
Like the 120-cell, the 600-cell has H₄ symmetry with 14,400 elements. The two shapes share this exceptional group because dual polytopes always share symmetry groups, and H₄ is the unique exceptional group governing pentagonal/icosahedral symmetry in 4D.
The 120 Vertices
The 120 vertices of a 600-cell form the binary icosahedral group (times 2). These can be expressed as unit quaternions involving the golden ratio φ = (1+√5)/2.
Measurements
For a 600-cell with edge length 1:
| Measure | Value |
|---|---|
| Circumradius | φ = (1+√5)/2 ≈ 1.618 |
| Inradius | φ²/2 ≈ 1.309 |
| Hypervolume | ≈ 55.68 |
| Dihedral angle | ≈ 164.48° |
The circumradius equals the golden ratio—another manifestation of φ in this icosahedral structure.
Unique Properties
Maximum Density
The 600-cell is the "densest" regular polychoron:
- 5 cells meet at each edge (maximum among regulars)
- 20 cells meet at each vertex
- The dihedral angle (164.48°) is the largest—almost flat
The Binary Icosahedral Group
The 600-cell's 120 vertices form the binary icosahedral group 2I, which has 120 elements. This group is:
- The double cover of the icosahedral rotation group (60 elements)
- Isomorphic to SL(2,5), the 2×2 matrices over the field with 5 elements
- Central to the classification of finite subgroups of SU(2)
Decomposition into 24-Cells
The 600-cell can be partitioned into five 24-cells! Its 120 vertices divide perfectly into 5 groups of 24, each forming a 24-cell. This decomposition reveals deep connections between the H₄ and F₄ symmetry groups.
The Grand Antiprism
In 1965, John Conway and Michael Guy discovered the Grand Antiprism—the only convex uniform polychoron that cannot be constructed by Wythoff's method. It's obtained by removing a specific ring of vertices from the 600-cell.
Visualizing the 600-Cell
Stereographic Projection
In stereographic projection:
- The 600 tetrahedral cells appear as a dense, intricate mesh
- The structure looks bristly, with tetrahedra pointing outward
- 4D rotation creates rapid, complex motion
The "Exploding" Appearance
During 4D rotation, the 600-cell can appear to:
- Explode outward (as cells move toward the viewer)
- Collapse inward (as cells recede)
- Form and reform icosahedral patterns
Cross-Sections
The 600-cell's 3D cross-sections include icosahedra (through vertex centers), dodecahedra (related to dual structure), and various Archimedean solids.
The E8 Connection
Two 600-cells can be combined to form a 240-vertex structure related to the E8 root system—one of the most important structures in mathematics and theoretical physics. This connects the humble 600-cell to:
- The largest exceptional Lie group
- Heterotic string theory
- Proposed theories of everything
Citation: The hexacosichoron (600-cell), denoted {3,3,5}, is the densest regular convex polychoron, bounded by 600 tetrahedral cells with 5 meeting at each edge. Discovered by Ludwig Schläfli in 1852, it has 1,200 triangular faces, 720 edges, and 120 vertices that form the binary icosahedral group. The 600-cell possesses H₄ symmetry (14,400 elements), shares this with its dual the 120-cell, and can be decomposed into five 24-cells. Its circumradius equals the golden ratio φ.
Related Polychora
- Dual: Hecatonicosachoron (120-cell)
- Same symmetry: 120-cell (H₄ group)
- Contains: Five 24-cells
- Same cell type: Pentachoron, 16-cell