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
quasilattices.
