Notation primer

Vertex notation

Vertex notation describes a tiling by walking around a single vertex and listing the polygons you meet, by their number of sides, in order. The notation 3.6.3.6 reads as: triangle, hexagon, triangle, hexagon — four polygons meeting at one vertex, alternating triangle and hexagon.

Read the cycle in either direction; rotations and reflections of the same string describe the same vertex. So 3.6.3.6, 6.3.6.3, 3.6.3.6 (reversed) are all the same configuration. The canonical form is the lexicographically smallest rotation — which is one of the things the ring-seq library exists to compute.

When a tiling has more than one kind of vertex, the notation lists each vertex configuration, separated by semicolons: 3.3.4.3.4 ; 3.3.4.12 describes a 2-uniform tiling with two distinct vertex types.

The 11 Archimedean tilings

The Archimedean tilings are the eleven edge-to-edge tilings of the plane built from regular polygons in which every vertex configuration is the same (1-uniform). Three of them are the regular tilings — triangles only, squares only, hexagons only. The other eight mix two or three polygon types.

Notation	Name	Thumbnail
3.3.3.3.3.3	Triangular
4.4.4.4	Square
6.6.6	Hexagonal
3.6.3.6	Trihexagonal
3.4.6.4	Rhombi-trihexagonal
3.12.12	Truncated hexagonal
4.6.12	Truncated trihexagonal
4.8.8	Truncated square
3.3.3.3.6	Snub trihexagonal
3.3.3.4.4	Elongated triangular
3.3.4.3.4	Snub square

k-uniform tilings

A tiling is k-uniform if there are exactly k distinct vertex configurations under the tiling’s symmetry group. The Archimedean tilings are 1-uniform. There are 20 known 2-uniform tilings, and the count grows quickly with k.

Canonical sources for the full classification:

• B. Grünbaum and G. C. Shephard, Tilings and Patterns (1987).
• The Wikipedia page on Euclidean tilings by convex regular polygons is a serviceable secondary reference.

Glossary

• Edge-to-edge. A tiling is edge-to-edge when every shared border between two tiles is a complete edge of both — no T-junctions.
• Vertex configuration. The cyclic sequence of polygons around a vertex, written in vertex notation.
• Uniformity. The number of distinct vertex configurations under the tiling’s symmetry group. 1-uniform = Archimedean.
• Symmetry group. The group of rigid motions (translations, rotations, reflections, glide reflections) that map the tiling onto itself.
• Demiregular. A non-standard term sometimes used for 2- or 3-uniform tilings; ambiguous enough that we avoid it on this site.