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$ \def\T{\mathcal T} % tiling \def\P{\mathcal P} % protoset $

In the theory of tilings a set of prototiles, also called a protoset, is a (finite or infinite) set of tiles that represent (by congruence) all the tiles of a tiling, or of a class of tilings.

A protoset $ \P = \{P_k\} $ admits a tiling if there is a tiling $ \T = \{T_i\} $ such that all its tiles $T_i$ are congruent to some $P_k$.

An $n$-hedral tiling is a tiling in that $n$ (distinct) prototiles occur. For $n=1,2,3,...$ the terms monohedral, dihedral, trihedral, etc. are used.

Tiling problem

The tiling problem, i.e., the problem to determine whether a protoset admits a tiling is (algorithmically) undecidable, both in general and for many special cases. Normally, a protoset will (if at all) admit either a single tiling or uncountably many tilings, but intermediate numbers are also possible. If a protoset admits (up to congruence) only a single tiling then it is called monomorphic, if it admits precisely two distinct tilings it called dimorphic, trimorphic for three tilings, etc., $n$-morphic for $n$ tilings, and $\sigma$-morphic if there are countably many distinct tilings.

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