The Six Regular Convex Polychora

In 1852, Swiss mathematician Ludwig Schläfli made a discovery that would define four-dimensional geometry: there are exactly six regular convex polychora—the "Platonic solids" of the fourth dimension.

What Makes Them Special?

A polychoron is regular if:

• All cells are identical regular polyhedra
• All faces are identical regular polygons
• All edges are identical
• All vertices are identical
• The symmetry group acts transitively on flags

This is an extraordinarily restrictive definition. In 2D, infinitely many regular polygons exist. In 3D, only 5 Platonic solids exist. In 4D, exactly 6 regular convex forms exist. In 5D and higher, only 3 exist.

The fourth dimension is a peak of geometric richness.

The Six Regular Convex Polychora

Families and Relationships

The Simplex Family

The Pentachoron {3,3,3} is the 4D simplex—the simplest possible polytope in any dimension. It's self-dual: its dual is another pentachoron.

The Hypercube/Orthoplex Family

The Tesseract {4,3,3} and Hexadecachoron {3,3,4} are duals of each other, extending the cube/octahedron duality to 4D.

The Exceptional 24-Cell

The Icositetrachoron {3,4,3} is unique to four dimensions—it has no analogue in 3D or any other dimension. It's self-dual and possesses F₄ symmetry, a group that exists only in 4D.

The Icosahedral Family

The Hecatonicosachoron {5,3,3} and Hexacosichoron {3,3,5} are duals, extending the dodecahedron/icosahedron relationship to 4D. They are by far the most complex regular forms.

The Geometry of Rarity

Why exactly six? The answer lies in angular geometry.

For a regular polychoron {p,q,r} to exist:

• {p,q} must be a valid regular polyhedron (the cell type)
• r copies of this cell must fit around an edge with total dihedral angle < 360°

This constraint eliminates almost all possibilities. The six survivors represent all ways to tile 4D space around an edge without overlapping or leaving gaps.

Experiencing the Regulars

Each of these six shapes can be explored in the interactive viewer. As they rotate through 4D, watch how:

• The pentachoron turns inside-out—every cell briefly takes the center
• The tesseract reveals the cube-within-cube structure
• The 16-cell shows its spiky, octahedral nature
• The 24-cell flows with exceptional smoothness (F₄ symmetry)
• The 120-cell unfolds its 120 dodecahedral cells
• The 600-cell bristles with 600 tetrahedral spikes

Beyond the Convex Regulars

Schläfli also discovered 4 of the 10 regular star polychora—non-convex forms with self-intersecting cells. Edmund Hess completed the enumeration in 1883. These Schläfli-Hess forms extend the concept of regularity to spectacular star-shaped geometries.

Further Reading

• What is the Fourth Dimension?
• Schläfli Symbols Explained
• History of 4D Geometry