Short answers, pulled from the story.
Tessellation is the covering of a surface using one or more geometric shapes, called tiles, with no overlaps and no gaps. Common rules require that there are no gaps between tiles and that no corner of one tile lies along the edge of another.
Only three shapes can form a regular tessellation: the equilateral triangle, the square, and the regular hexagon. Each of these can be duplicated infinitely to fill a flat plane with no gaps.
An aperiodic tessellation uses a set of tile shapes that can tile the plane but cannot form a repeating pattern. Penrose tilings, which use two different quadrilateral prototiles, are the best-known example of aperiodic tiling.
M. C. Escher visited Spain in 1936 and was inspired by the Moorish tiling of the Alhambra palace. He later made four Circle Limit drawings based on hyperbolic geometry, including Circle Limit IV, completed in 1960.
The einstein tile is a single shape that forces aperiodic tiling of the plane. It was discovered in 2023 by David Smith, a hobbyist mathematician, and was dubbed "the hat." The discovery was under professional review at the time of reporting.
There are exactly 17 wallpaper groups, representing all possible symmetry groups for periodic tilings of the plane. This was proved by the Russian crystallographer Yevgraf Fyodorov in 1891, and it is claimed that all seventeen groups are represented in the Alhambra palace in Granada, though this is disputed.