Tessellations in nature and culture

A tessellation is a pattern of shapes that covers a surface with no gaps and no overlaps, and could go on forever. People and nature have been making them for a very long time.

Only three regular shapes work alone

For regular polygons to tessellate on their own, their angles must fit exactly into 360° at every corner. Six triangles fit (6 × 60°), four squares fit (4 × 90°), and three hexagons fit (3 × 120°). A regular pentagon's angle is 108°, and 360 is not a multiple of 108, so pentagons always leave a gap.

Nature's tiles

Honey bees build hexagonal cells. Hexagons hold the most honey for the least wax. Near Dunedin, the Organ Pipes on Mt Cargill are tall rock columns, many with five or six sides. They formed when a thick layer of lava cooled and shrank, cracking into columns much like mud cracks as it dries.

Patterns of Aotearoa and the Pacific

Tukutuku panels line the walls of many wharenui. They are made by lacing coloured strips of pīngao and kiekie around upright toetoe reeds and wooden laths, so the designs are built from straight lines, crosses and steps. Each design has a name and meaning. The stepped poutama is linked to education and progress, and rows of crosses called purapurawhetū are star seeds. Tāniko weavers could not weave curves, so their borders use triangles, diamonds and zig-zags.

Across the Pacific, artists make tapa from beaten bark: ngatu in Tonga, siapo in Samoa, masi in Fiji and hiapo in Niue. Many tapa designs are laid out in grids, with motifs repeated, turned and flipped.

Escher's creatures

The Dutch artist M. C. Escher changed the edges of simple tiles to make birds, fish and lizards that lock together perfectly. His trick was to cut a piece from one edge and move it, by translation or rotation, to another edge. You can use the same trick to design a tile no one has ever seen.