Tilings and mosaics

Archimedean tilings

All 11 ways to tile the plane with regular polygons so that every vertex looks the same — and their duals, the Laves tilings.

  • vector geometry
  • seamless tile
  • SVG export
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A floor of identical regular polygons can only be laid with triangles, squares or hexagons: a regular pentagon has 108° corners, and no number of them makes 360°. If different polygons are allowed but every vertex must look alike, with the same shapes in the same order, there are exactly eleven ways: the Archimedean tilings.

How it works

A tiling is named by reading around a vertex: 4.8² is a square and two octagons, 3.4.6.4 a triangle, a square, a hexagon and another square. The angles around the point must come to exactly 360°.

angle of an n-gon: 180° · (n − 2) / n 4.8²: 90° + 135° + 135° = 360° 4.6.12: 90° + 120° + 150° = 360° 3.7.42: 60° + 128.57° + 171.43° = 360°, but this vertex cannot be continued

Counting the order of the polygons, the angle equation has 21 solutions, but only eleven extend over the plane with every vertex alike, and some, like 3.7.42, never get past a single vertex. They are three regular and eight semiregular tilings; 3⁴.6 comes in two mirror-image forms. The generator builds each from its exact unit cell, so tiles repeat seamlessly.

“Dual (Laves)” puts a point in the centre of every polygon and joins the centres of neighbours. Each vertex becomes a face, and all tiles of the new tiling are identical: 3.6.3.6 turns into rhombi that the eye stacks into cubes, 3².4.3.4 into the pentagons of the Cairo pavement.

A bit of history

Johannes Kepler first described all eleven in Harmonices Mundi (1619), the same book in which he reconstructed Archimedes' thirteen semiregular polyhedra and stated his third law of planetary motion. Hence the name: like the Archimedean solids, these tilings have regular faces and identical vertices. The duals bear the name of the German crystallographer Fritz Laves. The Cairo tiling really paves Cairo's streets; the tumbling cubes are familiar from parquet and quilts. The hexagonal grid 6³ is also the most economical: of all ways to divide the plane into cells of equal area it has the least perimeter. Thomas Hales proved this “honeycomb conjecture” in 1999.

What to tweak

  • “Dual (Laves)” on 3.6.3.6 and 3².4.3.4 gives the cubes and the Cairo pentagons.
  • “Colouring” “by polygon” gives triangles, squares and hexagons their own colours; “by position” gives every tile of the unit cell its own, and hidden symmetry shows up.
  • “Grout” and “Rounding” turn the tiling into glazed ceramics or a scatter of jelly sweets.
  • “Colouring” “lines only” with a thick “Outline” makes a lattice ready for SVG, a plotter or a laser cutter.
  • Put “Islamic star patterns” with the same “Base tiling” on top to see which edges the stars grow from.

Parameters

Tiling
The polygons around every vertex: 4.8² is a square and two octagons 3⁶ — triangular · 4⁴ — square · 6³ — hexagonal · 3⁴.6 — snub hexagonal · 3³.4² — elongated triangular · 3².4.3.4 — snub square · 3.4.6.4 — rhombitrihexagonal · 3.6.3.6 — trihexagonal · 3.12² — truncated hexagonal · 4.6.12 — truncated trihexagonal · 4.8² — truncated square
Dual (Laves)
A vertex in every polygon, a face around every vertex
Cells
How many repeating cells fit across the short side
Grout
The gap between tiles
Rounding
Round off the tile corners
Outline
Width of the line around every tile
Colouring
By polygon — triangles, squares and hexagons in different colours; by position — every tile of the unit cell its own colour by polygon · by position · random · gradient · lines only
Outline colour
Rotation
Ignored in seamless tile mode