May 2026 - Planar Tilings by Polyominoes, Polyhexes and Polyiamonds

This is another computational maths paper that considers tiling problems, though it finds application beyond tetris tilings. The tilings considered in the paper are monohedral, meaning that each tile is congruent to a single fixed tile. In tiling, symmetry is an isometry, mapping each tile onto another. Such associated tiles are considered equivalent, forming a transitivity class. Tilings with only one transitivity class are isohedral. A tiling with exactly k > 1 transitivity classes is k-isohedral. If a prototile admits isohedral tiling, it's itself considered isohedral. A patch is considered a collection of nonoverlapping tiles without topological holes. Translational patches are translations of a patch. A fundamental domain is a translational patch of minimal size. A tiling is periodic if the group of symmetries contains at least two linearly independent translations. These are not unique (Section 1). A closed topological disk satisfies Conway's criterion if it can be split into six segments labeled clockwise A through F, so that A and D are translations of each other and the remaining ones are symmetric wrt. a 180 degree rotation about their center point. Any prototile satisfying Conway's criterion admits periodic tiling of the plane (Theorem 1). A closed topological disk satisfies the translation criterion if the boundary can be split into segments labeled clockwise A through F, so that the three pairs of A-D, B-E, C-F are translations of each other. Prototiles that are closed topological disks admit tilings of the plane by a lattice of translations iff it satisfies this translation criterion (Section 3).

Modelling the two criteria can be achieved by tracing the shape of the polyomino through its connected edges. Internal holes aren't explicitly tracked through this representation, and can't be filtered for using this. For this edge string representation a portion of the boundary is symmetric with respect to 180° rotation iff the substring is palindrome. Two portions of the boundary are translations iff their corresponding substrings are inverses. With these, the criteria can be checked for (Section 4). From this, a table of polyominoes can be created, tracking their properties as listed by the criterion (Section 5). A translation of the same method can be applied to other types of tilings (Sections 6 - 7).

To avoid the "trial & error" method, one can introduce a limitation stating that new tile placements need to be close to the prototile. Given a finite set of prototiles T of closed topological disks, all its admitted tilings must include a tile with no more than 6 neighbors (Theorem 3). This can introduce a hard search limit of 6. Adding tiles during this search, the current tile placement needs to be matched with the previous one. If the tile matches, it combines with it into a patch. A heuristic continuation will place tiles as closely to that patch as possible. It begins with a check for unfillable concavities, which, if failed, will remove the previously placed tile (Section 9). There is a case in which the patch needs to control for holes that fit exactly another patch. This complication arises from the placement order of the tiles (Section 10).