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The 11 Archimedean tilings, one by one

1 May 2026

An illustrated tour through the eleven edge-to-edge regular-polygon tilings with one vertex configuration.

The Archimedean tilings are the eleven edge-to-edge tilings of the plane built from regular polygons in which every vertex looks the same. Three of them — the regular tilings — use a single polygon shape. The other eight mix two or three.

This post walks through all eleven, in roughly increasing complexity. If you are unfamiliar with the notation, the primer post covers it in five minutes.

The three regular tilings

3.3.3.3.3.3 — Triangular

3.3.3.3.3.3 triangular tiling.
Six equilateral triangles meet at every vertex.

Six equilateral triangles meet at every vertex. The densest of the regular tilings; the only one whose vertex angle (60°) divides 360° six times.

4.4.4.4 — Square

4.4.4.4 square tiling.
Four squares per vertex — the graph paper of tilings.

The graph paper. Four squares per vertex. Boring on its own; fertile ground for everything that comes later.

6.6.6 — Hexagonal

6.6.6 hexagonal tiling.
Three hexagons per vertex, at 120° each.

Three hexagons per vertex, at 120° each. The dual of the triangular tiling, and (not coincidentally) what bees converged on.

Mixing two polygon shapes

3.6.3.6 — Trihexagonal

3.6.3.6 trihexagonal tiling.
Triangles fill the gaps between hexagons; every vertex sees both, alternating.

The most photogenic of the lot. Hexagons share vertices with each other; triangles fill the kite-shaped gaps in between.

3.12.12 — Truncated hexagonal

3.12.12 truncated hexagonal tiling.
Snip the corners off a hexagonal tiling to get dodecagons; the snipped corners become tiny triangles.

Take 6.6.6, slice each hexagon’s corners off, and you have 3.12.12. The big polygons have twelve sides apiece.

4.8.8 — Truncated square

4.8.8 truncated square tiling.
Octagons sit on a square grid; the leftover corners form small squares.

The same trick on 4.4.4.4. Octagons meet two-on-two-off, with the gaps filled by small squares.

Mixing three polygon shapes

3.4.6.4 — Rhombi-trihexagonal

3.4.6.4 rhombi-trihexagonal tiling.
Hexagons, squares and triangles all share vertices.

4.6.12 — Truncated trihexagonal

4.6.12 truncated trihexagonal tiling.
The most polygon-rich of the Archimedeans: square, hexagon, dodecagon at every vertex.

3.3.3.3.6 — Snub trihexagonal

3.3.3.3.6 snub trihexagonal tiling.
Four triangles plus a hexagon at every vertex. Has a chirality the others lack.

This one is chiral: the tiling is not its own mirror image. There are two distinct snub trihexagonal tilings, one left-handed and one right-handed. Most counts treat them as the same.

3.3.3.4.4 — Elongated triangular

3.3.3.4.4 elongated triangular tiling.
Rows of triangles alternate with rows of squares.

3.3.4.3.4 — Snub square

3.3.4.3.4 snub square tiling.
Squares tilted off-axis, with triangles filling the rotated gaps.

The last two both use only triangles and squares; what differs is the order in which those polygons meet around each vertex.

Open the editor

Every one of these is buildable in the editor in a handful of clicks. Try editor.tessell.art; pick one and see how far you get before the geometry tells you there is only one way it could possibly continue.