Create regular, semi-regular, and custom tessellations. Explore symmetry groups and Escher-style tiling with drag-to-deform.
A tessellation is a pattern of tiles that covers the plane with no gaps and no overlaps, and this tool builds them from regular polygons, the eight semi-regular combinations, or shapes you deform yourself. Choose a starting tile and the tool renders the repeating pattern; in custom mode, dragging an edge reshapes it Escher-style. What surprises most first-time users: an arbitrary drawn shape almost never tiles. Tiling is a constraint, not a property of pretty shapes - regular pentagons, for instance, cannot cover the plane by themselves.
A concrete input and expected output from the current implementation.
Input
Regular hexagon, side length 1
Expected output
A full-plane hexagonal tiling: each tile is a regular hexagon with interior angle 120 degrees, three tiles meet at every vertex (120 x 3 = 360, Schlafli symbol {6,3}), and each tile covers an area of 3*sqrt(3)/2, about 2.598 square units. The hexagon is one of only three regular polygons that tile the plane alone, because its interior angle divides 360 degrees evenly, so three copies meet at each vertex with no gap. The area figure follows from cutting a side-1 regular hexagon into six equilateral triangles of area sqrt(3)/4 each: 6 x sqrt(3)/4 = 3*sqrt(3)/2.
Every vertex of a valid tiling is surrounded by angles totaling 360 degrees. A regular n-gon contributes (n-2) x 180 / n degrees, which divides 360 only for n = 3, 4, 6 - the triangle, square, and hexagon. Mixed tilings combine angles that still close the gap, such as 90 + 135 + 135 for one square and two octagons, and each arrangement is recorded as a vertex figure like 4.8.8.
In custom mode, dragging one edge applies the same deformation to the opposite edge, so neighboring copies interlock along shared edges instead of leaving cracks. This is the Escher method: reshape a tile by editing edges in pairs, matched by translation or rotation. The copies fit because every shared edge exists twice with identical shape; the vertex angles are what change, and they must still sum to 360 degrees.
Eight, and together with the three regular tilings that makes eleven uniform tilings of the plane by convex regular polygons, a count known since Kepler. Their vertex figures are 3.3.3.3.6, 3.3.3.4.4, 3.3.4.3.4, 3.4.6.4, 3.6.3.6, 3.12.12, 4.6.12, and 4.8.8 - each sums to 360 degrees around a vertex.
Their interior angle is 108 degrees, and no whole number fits: 3 x 108 = 324 leaves a gap, 4 x 108 = 432 overlaps. Only angles that divide 360 can tile alone, which is exactly why the single-shape regular tilings are triangle, square, and hexagon. Irregular convex pentagons do tile the plane - 15 families are known, the last found in 2015 - but none is regular.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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