The pieces the atlas is built from, and the mathematics behind them: browsable catalogues of the tiles and their vertex configurations, then background reading with worked examples and interactive previews.
The three regular and eight semiregular tilings of the plane, the angle equation that constrains them, and why there are exactly eleven.
Marek Čtrnáct's edge-subset construction: fix a grid, choose which edges are drawn, and let the tiles fall out. Planar grids and all five Platonic solids.
The design systems behind Islamic star patterns: the tessellations underneath, with interactive previews of the curated tilings.
Brooks, Smith, Stone and Tutte turned a rectangle cut into unequal squares into an electrical circuit, and the circuit into a convex polyhedron. Run backwards over this catalogue's own solids: 667 rectangles across 105 of them, and why symmetry is what stops them being perfect.
Conway's rule needs a grid before it needs a rule. What is known about Life on hexagons, pentagons and Penrose tilings, why a B/S string is undetermined when tiles differ in degree, and what a tiling's periodicity buys the simulator.
The hyperbolic tilings never run out, so the enumeration is bounded by palette, valence and uniformity. What identifies a tiling when its vertex configuration cannot, and where the search hits its limit.