Notes on tilings, lattices and the shapes that fill a plane. Each one starts from a pattern and tries to find the small piece it is built from.

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  The repeat unit

  How to find the smallest tile that rebuilds a periodic pattern, and why its area never changes.

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  Three regular tilings, and why only three

  Triangles, squares and hexagons are the only regular polygons that tile the plane edge to edge. The proof fits on one line.

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  Truchet tiles

  Two quarter-circle tiles, placed at random, draw endless curves that never cross.

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  Vertex notation and the Archimedean tilings

  Reading names like 4.8.8 and 3.6.3.6, and the eight tilings by regular polygons where every vertex looks the same.

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  The Cairo pentagonal tiling

  A tiling by equal pentagons that are not regular, built from a square grid and a short segment in every square.

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  Basketweave, herringbone and other brick bonds

  A 2:1 rectangle tiles in many ways. Herringbone is the one whose repeat unit is hardest to see.

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  Hexagonal grids and three coordinates

  Cube coordinates make distances, rings and rotations on a hexagon grid as simple as on squares.