We study 1D quasilattices, especially self-similar ones that can be used to
    generate two-, three- and higher-dimensional quasicrystalline tessellations
    that have matching rules and invertible self-similar substitution rules (also
    known as inflation rules) analogous to the rules for generating Penrose
    tilings. The lattice positions can be expressed in a closed-form expression we
    call {\it floor form}: $x_{n}=S(n-\alpha)+(L-S)\lfloor \kappa(n-\beta)\rfloor$,
    where $L >S>0$ and $0<\kappa<1$ is an irrational number. We describe two
    equivalent geometric constructions of these quasilattices and show how they can
    be subdivided into various types of equivalence classes: (i) {\it lattice
    equivalent}, where any two quasilattices in the same lattice equivalence class
    may be derived from one another by a local decoration/gluing rule; (ii) {\it
    self-similar}, a proper subset of lattice equivalent where, in addition, the
    two quasilattices are locally isomorphic; and (iii) {\it self-same}, a proper
    subset of self-similar where, in addition, the two quasilattices are globally
    isomorphic (i.e.) identical up to rescaling). For all three types of
    equivalence class, we obtain the explicit transformation law between the floor
    form expression for two quasilattices in the same class. We tabulate (in Table
    I and Figure 5) the ten special self-similar 1D quasilattices relevant for
    constructing Ammann patterns and Penrose-like tilings in two dimensions and
    higher, and we explicitly construct and catalog the corresponding self-same

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