Schläfli Symbols: The Language of Regular Polytopes

The Schläfli symbol is a compact notation that completely describes the local structure of any regular polytope. Invented by Ludwig Schläfli in the 1850s, it remains the primary way mathematicians denote regular shapes in any dimension.

The Basic Idea

A Schläfli symbol is a sequence of numbers in curly braces that recursively defines a polytope's structure:

{p} describes a regular polygon with p sides
{p, q} describes a regular polyhedron
{p, q, r} describes a regular polychoron (4D)
{p, q, r, s} describes a regular 5-polytope

Each number tells you about the local arrangement of elements.

Two-Dimensional: Polygons

For polygons, the Schläfli symbol is simply the number of sides:

Symbol	Polygon	Sides
{3}	Triangle	3
{4}	Square	4
{5}	Pentagon	5
{6}	Hexagon	6
{n}	n-gon	n

Star Polygons

For star polygons (those that self-intersect), we use fractions:

Symbol	Star Polygon	Description
{5/2}	Pentagram	5 points, skip 2 vertices
{7/2}	Heptagram	7 points, skip 2 vertices
{7/3}	Heptagram	7 points, skip 3 vertices

The notation {p/q} means: connect every q-th vertex of a regular p-gon.

Three-Dimensional: Polyhedra

For polyhedra, {p, q} means:

p: The face type is a regular {p} polygon
q: Exactly q faces meet at each vertex

The Platonic Solids

Symbol	Polyhedron	Face	Faces at Vertex
{3, 3}	Tetrahedron	Triangle	3
{4, 3}	Cube	Square	3
{3, 4}	Octahedron	Triangle	4
{5, 3}	Dodecahedron	Pentagon	3
{3, 5}	Icosahedron	Triangle	5

Reading the Symbol

Take the cube {4, 3}:

{4} = Faces are squares (4-sided)
3 = Three squares meet at each vertex

Take the icosahedron {3, 5}:

{3} = Faces are triangles
5 = Five triangles meet at each vertex

Four-Dimensional: Polychora

For polychora, {p, q, r} means:

{p, q}: The cell type (a regular polyhedron)
r: Exactly r cells meet around each edge

The Six Convex Regular Polychora

Symbol	Polychoron	Cell	Cells per Edge
{3, 3, 3}	Pentachoron	Tetrahedron	3
{4, 3, 3}	Tesseract	Cube	3
{3, 3, 4}	Hexadecachoron	Tetrahedron	4
{3, 4, 3}	Icositetrachoron	Octahedron	3
{5, 3, 3}	Hecatonicosachoron	Dodecahedron	3
{3, 3, 5}	Hexacosichoron	Tetrahedron	5

Reading a 4D Symbol

Take the tesseract {4, 3, 3}:

{4, 3} = Cells are cubes
The final 3 = Three cubes meet around each edge

Take the 600-cell {3, 3, 5}:

{3, 3} = Cells are tetrahedra
The final 5 = Five tetrahedra meet around each edge

The Vertex Figure

The Schläfli symbol also encodes the vertex figure—the shape you see if you slice the polytope near a vertex:

For {p, q}: The vertex figure is {q}
For {p, q, r}: The vertex figure is {q, r}

Example: The tesseract {4, 3, 3} has vertex figure {3, 3}, which is a tetrahedron. If you slice a tesseract near any vertex, the cross-section is a tetrahedron.

Duality

A remarkable property: reversing a Schläfli symbol gives the dual polytope!

Symbol	Polytope	Dual Symbol	Dual Polytope
{4, 3}	Cube	{3, 4}	Octahedron
{4, 3, 3}	Tesseract	{3, 3, 4}	16-cell
{5, 3, 3}	120-cell	{3, 3, 5}	600-cell

Self-dual polytopes have palindromic symbols:

{3, 3} Tetrahedron (self-dual)
{3, 3, 3} Pentachoron (self-dual)
{3, 4, 3} 24-cell (self-dual)

Constraints on Valid Symbols

Not every combination of numbers produces a valid polytope. The constraints relate to angular defect:

For Polyhedra {p, q}

The angles around a vertex must sum to less than 360°. This limits us to exactly 5 convex regular polyhedra (Platonic solids).

For Polychora {p, q, r}

The cell's dihedral angles must allow r of them to fit around an edge without exceeding 360°. This analysis yields exactly 6 convex regular polychora.

Summary

The Schläfli symbol packs enormous information into a compact form:

It completely specifies local topology
It works in any dimension
Reversing it gives the dual
It reveals impossibility through angular constraints

When you see {4, 3, 3}, you know immediately: cubic cells, three per edge—a tesseract. This elegant notation remains central to polytope theory 170 years after its invention.