tessell.art - Blog

The 11 Archimedean tilings, one by one

2026-05-01T00:00:00+00:00

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</em> tilings — use a single polygon shape. The other eight mix two or three.</p>

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

The three regular tilings</h2>
`3.3.3.3.3.3</code> — Triangular</h3>`

Six equilateral triangles meet at every vertex.</figcaption> </figure>
Six equilateral triangles meet at every vertex. The densest of the regular tilings; the only one whose vertex angle (60°) divides 360° six times.</p>
`4.4.4.4</code> — Square</h3>`

Four squares per vertex — the graph paper of tilings.</figcaption> </figure>
The graph paper. Four squares per vertex. Boring on its own; fertile ground for everything that comes later.</p>
`6.6.6</code> — Hexagonal</h3>`

Three hexagons per vertex, at 120° each.</figcaption> </figure>
Three hexagons per vertex, at 120° each. The dual of the triangular tiling, and (not coincidentally) what bees converged on.</p>
Mixing two polygon shapes</h2>
3.6.3.6</code> — Trihexagonal</h3> Triangles fill the gaps between hexagons; every vertex sees both, alternating.</figcaption> </figure> The most photogenic of the lot. Hexagons share vertices with each other; triangles fill the kite-shaped gaps in between.</p> 3.12.12</code> — Truncated hexagonal</h3> Snip the corners off a hexagonal tiling to get dodecagons; the snipped corners become tiny triangles.</figcaption> </figure> Take 6.6.6</code>, slice each hexagon’s corners off, and you have 3.12.12</code>. The big polygons have twelve sides apiece.</p> 4.8.8</code> — Truncated square</h3> Octagons sit on a square grid; the leftover corners form small squares.</figcaption> </figure> The same trick on 4.4.4.4</code>. Octagons meet two-on-two-off, with the gaps filled by small squares.</p> Mixing three polygon shapes</h2> 3.4.6.4</code> — Rhombi-trihexagonal</h3> Hexagons, squares and triangles all share vertices.</figcaption> </figure> 4.6.12</code> — Truncated trihexagonal</h3> The most polygon-rich of the Archimedeans: square, hexagon, dodecagon at every vertex.</figcaption> </figure> 3.3.3.3.6</code> — Snub trihexagonal</h3> Four triangles plus a hexagon at every vertex. Has a chirality the others lack.</figcaption> </figure> This one is chiral</em>: 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.</p> 3.3.3.4.4</code> — Elongated triangular</h3> Rows of triangles alternate with rows of squares.</figcaption> </figure> 3.3.4.3.4</code> — Snub square</h3> Squares tilted off-axis, with triangles filling the rotated gaps.</figcaption> </figure> The last two both use only triangles and squares; what differs is the order in which those polygons meet around each vertex.</p> Open the editor</h2> Every one of these is buildable in the editor in a handful of clicks. Try editor.tessell.art</code></a>; pick one and see how far you get before the geometry tells you there is only one way it could possibly continue.</p>

How to read vertex notation 2026-04-22T00:00:00+00:00 If you have spent any time around tilings you have probably seen strings of digits separated by full stops — 3.6.3.6</code>, 4.8.8</code>, 3.3.3.3.6</code>. They look like cryptic file extensions; they are in fact one of the densest pieces of notation in mathematics.</p> Here is the rule. Stand at a vertex. Look at the polygons that meet there. Walk around the vertex, in either direction, and write down the number of sides of each polygon you pass.</p> That is it.</p> At every vertex of 3.6.3.6</code>: triangle, hexagon, triangle, hexagon. Four polygons, alternating.</figcaption> </figure> Same vertex, many spellings</h2> 3.6.3.6</code> and 6.3.6.3</code> describe the same vertex — you just started walking from a different polygon. So do 3.6.3.6</code> reversed (reading anticlockwise instead of clockwise) and any rotation of those.</p> So a vertex configuration is really an equivalence class</em> of strings: anything related to the others by rotation or reflection is the same vertex. We pick a canonical representative — usually the lexicographically smallest rotation — when we need a unique spelling for, say, comparing two tilings programmatically.</p> This is exactly what the ring-seq</code></a> library is for. Treat the string as a ring; the smallest rotation is the canonical form.</p> More than one kind of vertex</h2> Some tilings have more than one kind</em> of vertex. The notation extends naturally: list each vertex configuration, separated by semicolons. 3.3.4.3.4 ; 3.3.4.12</code> is a 2-uniform tiling — two distinct vertex types, each appearing repeatedly.</p> The number of distinct vertex types is the tiling’s uniformity</em>. The 11 Archimedean tilings are 1-uniform; there are 20 known 2-uniform tilings; the count grows fast.</p> For a fuller treatment — k</em>-uniform tilings, the 11 Archimedean tilings, and a short glossary — see the notation primer</a>.</p> Introducing tessell.art 2026-04-15T00:00:00+00:00 Tessell.art is a place to design polygon tessellations — squares, triangles, hexagons and friends, arranged so every edge meets edge-to-edge. It is two things at once: an editor for making tilings, and a small set of foundation libraries we extracted from the editor along the way.</p> The editor</h2> The editor runs in your browser at editor.tessell.art</a> and as a native build for macOS, Windows and Linux. Pick a polygon, click an existing edge to grow the tiling, fan polygons around a vertex, mirror a region. Export to SVG when you are happy.</p> The editor is bilingual: English and Spanish. The site is English only for now, but the editor itself is not.</p> The libraries</h2> Building Tessella surfaced one piece of code that was not really about tessellations: cyclic-sequence operations. Vertex configurations are sequences-as-rings, so we extracted that behaviour into ring-seq</code></a> — and ported it to Scala, Rust and Python so it does not pull anyone into the JVM who does not want to be there.</p> dcel</code> (a doubly-connected edge list, the data structure that powers the editor’s geometric model) is queued behind it. That one stays Scala-only, at least for the foreseeable future.</p> What is not</em> here</h2> This site does not teach tessellation theory from scratch. The notation primer</a> explains how to read 3.6.3.6</code> and how the 11 Archimedean tilings work, but it assumes you arrived already curious.</p> What is next</h2> A gallery of curated tilings, deep-link from gallery into the editor, and more… None of those are here yet; they are on the v2 list. The blog is the place we will say so when they land.</p>

Mixing two polygon shapes</h2>

Open the editor</h2>
Every one of these is buildable in the editor in a handful of clicks. Try `editor.tessell.art</code></a>; pick one and see how far you get before the geometry tells you there is only one way it could possibly continue.</p>`

`What is next</h2> A gallery of curated tilings, deep-link from gallery into the editor, and more… None of those are here yet; they are on the v2 list. The blog is the place we will say so when they land.</p>`

tessell.art - Blog

The 11 Archimedean tilings, one by one

How to read vertex notation

Introducing tessell.art

tessell.art - Blog

The 11 Archimedean tilings, one by one

The three regular tilings</h2> 3.3.3.3.3.3</code> — Triangular</h3> Six equilateral triangles meet at every vertex.</figcaption> </figure> Six equilateral triangles meet at every vertex. The densest of the regular tilings; the only one whose vertex angle (60°) divides 360° six times.</p>

3.3.3.3.3.3</code> — Triangular</h3> Six equilateral triangles meet at every vertex.</figcaption> </figure> Six equilateral triangles meet at every vertex. The densest of the regular tilings; the only one whose vertex angle (60°) divides 360° six times.</p>

4.4.4.4</code> — Square</h3> Four squares per vertex — the graph paper of tilings.</figcaption> </figure> The graph paper. Four squares per vertex. Boring on its own; fertile ground for everything that comes later.</p>

Mixing two polygon shapes</h2>

Open the editor</h2> Every one of these is buildable in the editor in a handful of clicks. Try editor.tessell.art</code></a>; pick one and see how far you get before the geometry tells you there is only one way it could possibly continue.</p>

How to read vertex notation

Introducing tessell.art

What is next</h2> A gallery of curated tilings, deep-link from gallery into the editor, and more… None of those are here yet; they are on the v2 list. The blog is the place we will say so when they land.</p>

Open the editor</h2>
Every one of these is buildable in the editor in a handful of clicks. Try `editor.tessell.art</code></a>; pick one and see how far you get before the geometry tells you there is only one way it could possibly continue.</p>`

`What is next</h2> A gallery of curated tilings, deep-link from gallery into the editor, and more… None of those are here yet; they are on the v2 list. The blog is the place we will say so when they land.</p>`