Solutions to seven and a half problems on tilings (Q6170508)

A closed topological disk \(T\) in the Euclidean plane \(\mathbb{E}^2\) is said to be a tiling (or tessellation) of the plane if there is a collection \(\mathcal{T}\) of congruent copies of \(T\) such that every two elements of \(\mathcal{T}\) have disjoint interiors and their union is \(\mathbb{E}^2\). The Heesch number of \(T\) is the nonnegative integer, possible \(\infty\), showing how many times \(T\) can be completely surrounded by its congruent copies. If \(m\) is a natural number then the Heesch's problem is the question whether there exists a figure whose Heesch number is \(m\). The isohedral number of a tiling \(\mathcal{T}\) is defined as the number of equivalence classes of the tiles in \(\mathcal{T}\), where two tiles are equivalent if and only if they are identified by a symmetry of the tiling. Figure \(T\) is called an \(m\)-anisohedral figure if its isohedral number of \(T\) equals \(m\). Figure \(T\) is said to be \(m\)-morphic if it tiles the plane in exactly \(m\) non-congruent ways. Figure \(T\) is called \(\sigma\)-morphic if it tiles the plane in infinitely many ways, but only countably many. For every natural number \(m\) the question of existence of an \(m\)-anisohedral figure, of an \(m\)-morphic figure arises, as well as the existence of a \(\sigma\)-morphic figure. From mathematical point of view there are twelve possibilities of combining these four problems, that are discussed in this paper. The solution of five of these problems are mentioned from the previous literature and the remaining seven combinations are solved in this paper.