- The discovery of the 'hat' tile in March 2023 solved a long-standing problem of forced aperiodicity with a single shape, but it required both handednesses (reflections).
- Two months later, the same authors announced the 'Spectre' tile, which can tile the plane with only one handedness, addressing public criticism.
- The Spectre tile is similar to the hat but has all edges the same length (with angles of 90° or 120°) and is a 'weakly chiral' aperiodic monotile; edge constraints make it 'strongly chiral'.
- A metatile substitution system using nine hexagon types (all the same shape) generates the Spectre tiling, with expansion rules that reverse handedness.
- The combinatorial coordinates method tracks each Spectre's position in the infinite hierarchy of metatiles, allowing efficient generation of tiling patches via graph search or raster traversal.
- Coordinates are represented exactly using complex numbers (based on 30° rotations) to avoid floating-point errors.
- A simple four-colouring scheme exists for both the Hats and Spectres tilings: special tiles (reflected hats or odd-orientation Spectres) are given a fourth colour, while the rest form a three-colourable hexagonal tiling.
- The algorithm maintains colouring by tracking permutations or hexagon colour assignments during coordinate transitions.
- The combinatorial coordinates approach is advantageous because it avoids dealing with the complex geometry of the metatile expansions directly.