Deformable tiles that morph between shapes while preserving tessellation
M.C. Escher (1898-1972) was a master of tessellation art, creating tiles shaped like birds, fish, lizards, and horsemen that interlock perfectly to fill the plane. His technique was systematic: start with a simple tile that tessellates (a square, triangle, or hexagon), then deform its edges in coordinated ways that preserve the tessellation property.
The mathematics behind Escher's art is formalized by Heesch types -- a classification of how tile edges can relate to each other through translations, rotations, and glide reflections. Understanding these types gives you Escher's toolkit for creating your own tessellation art.
Drag the control points on one edge of a square tile to deform it. The opposite edge automatically mirrors your deformation, ensuring the tiles still fit together. Watch the entire tessellation update in real time as your simple square transforms into an organic, Escher-like shape.
Drag the control points on the center tile to deform its edges. Opposite edges deform identically, creating an Escher-like tessellation via translation symmetry.
Cyan dots: top edge · Teal dots: left edge · Opposite edges mirror automatically
Key insight: The key principle is edge correspondence. If the top edge of a tile matches the bottom edge (via translation), then the tile can stack vertically. If the left edge matches the right edge, it stacks horizontally. Any smooth deformation of one edge, copied to its corresponding edge, preserves tessellability.
Escher's most famous works feature gradual metamorphoses where one tessellation smoothly transforms into another. Here, a regular square grid morphs continuously into organic shapes while maintaining perfect tessellation at every frame. Adjust the morph parameter or let it animate automatically.
Watch a regular square tiling smoothly morph into Escher-like creature tessellations. The key insight: opposite edges always deform identically, preserving the tessellation.
Key insight: Metamorphosis works because the set of all edge deformations that preserve tessellability forms a continuous space. You can smoothly interpolate between any two valid deformations and every intermediate state is also a valid tessellation. This is the topological structure underlying Escher's artistic technique.
Heinrich Heesch classified all the ways tile edges can correspond to each other: T (translation -- one edge translates to the opposite), C (rotation -- one edge rotates 180° to become an adjacent edge), and G (glide reflection). Each Heesch type specifies how all four edges of a quadrilateral tile relate, determining the tile's symmetry and tessellation pattern.
Heesch types classify how tile edges relate to each other through symmetry operations. Each type defines a recipe for creating Escher-like tessellations.
Select a type to see how its edge symmetries create different tiling patterns
Key insight: Escher used Heesch types systematically. His notebooks contain careful studies of each type, with sketches showing how the edge correspondences constrain the possible shapes. Type TTTT (all edges translate) is simplest; type CCCC (all edges rotate) produces the most complex interlocking.