An example of a two-dimensional subshift is the set of arrays $\{x_{(i,j)}:(i,j)\in\mathbb{Z}^2\}$ of zeros and ones for which $x_{(i,j)}+x_{(i+1,j)}+x_{(i,j+1)}=0$ for each $i,j\in\mathbb{Z}$, where addition is taken modulo 2. There are two shift operators, the horizontal shift and the vertical shift. This subshift is called the Ledrappier example or the three dot dynamical system. In this paper we give an internal description of all transitive subshifts for which one of the shift operators is sofic in the usual one-dimensional sense.

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A finite collection P of finite sets tiles the integers iff the integers can be expressed as a disjoint union of translates of members of P. We associate with such a tiling a doubly infinite sequence with entries from P. The set of all such sequences is a sofic system, called a tiling system. We show that, up to powers of the shift, every shift of finite type can be realized as a tiling system.

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A query, about the orbit P W in real 3-space of a point P under an isometry group W generated by edge rotations of a tetrahedron, leads to contrasting notions, W versus S, of "rotation group". The set R = {r A 1 , r A 2 } of rotations r A i about axes Ai generates two manifestations of an isometry group on ℜ 3 :(1). In the stationary group S := S(R), all axes B are fixed under a rotation r A about A.(2). In the peripatetic group W := W(R), each r A transforms every rotational axis B = A.Theorem. If the line A1 is skew to A2, if each r A i is of infinite order, and if P ∈ ℜ 3 , then both of the orbits P S and P W are dense in ℜ 3 .