Tiling of orientation pinwheels. **a** The variable *orientation preference* has a range between 0 and π, and is mapped by Eq. onto a *circle*, with range 0–2π. The *circle* represents an orientation pinwheel. Orientation pinwheels can be regarded as the basic components of the orientation preference map, as is schematically represented here by their jigsaw puzzle tiling. **b** Results from a simple algorithm for generating orientation pinwheel tiling. Starting from the central pinwheel tile shown in (**a**), *rows* of tiles are added that are reflected about the vertical axis every other row; columns are added that are reflected about the horizontal axis every other column (only the centre nine tiles are shown). This tiling results in a saddle point pattern at each four-way junction of tiles (*left*). At a saddle point, the orientation preference increases with distance from the saddle point along one border direction (e.g. *horizontal*) and decreases with distance along the other border direction (e.g. *vertical*). The saddle-point pattern can be modified by shifting every *second row* of tiles to the left or right by one tile. This operation results in *linear zones* at tile borders (*right*). Here the orientation preference changes continuously along the vertical border. **c** Simulation of fluid tiling of orientation pinwheels (adapted from Wright et al. ). This simulation allows the polar angle of the pinwheel’s coordinate system to vary in phase from tile to tile. A regular spacing of pinwheels is imposed, resulting in a regular spacing of singularities (the pinwheel centres), seen most clearly here in the *right-hand column*. The fluid interactions at tile borders provide for a variety of further topological features, including saddle-points and ‘border’ singularities. This figure shows that a tessellation of orientation pinwheels need not result in a rigidly tiled orientation preference map. The sign of the singularity (direction of change of orientation preference about the singularity) in each tile is shown by a ‘+’ or ‘−’ sign. Analysis of the signs of singularities shows that saddle-points and linear zones generally arise at the borders of tiles containing two odd and two even singularities. ‘Border’ singularities are marked with *circles*. In each case it can be seen that these arise at tile borders where three or more tiles have singularities of the same sign. ‘Border’ singularities arise where local map borders cannot be resolved as a smooth transition in the orientation preference gradient