  We exhibit two distinct renormalization scenarios in many-parameter families
of piecewise isometries (PWI) of a rhombus. The rotational component, defined
over the quadratic field $\mathbb{K}=\mathbb{Q}(\sqrt{5})$, is fixed. The
translations are specified by affine functions of the parameters, with
coefficients in $\mathbb{K}$. In each case the parameters range over a convex
domain. In one scenario the PWI is self-similar if and only if one parameter
belongs to $\mathbb{K}$, while the other is free. Such a continuous
self-similarity is due to the possibility of merging adjacent atoms of an
induced PWI, a common phenomenon in the Rauzy-Veech induction for interval
exchange transformations. In the second scenario, the phase space splits into
several disjoint (non-convex) invariant components. We show that each component
has continuous self-similarity, but due to the transversality of the
corresponding foliations, full self-similarity in phase space is achieved if
and only if both parameters belong to $\mathbb{K}$. All our computations are
exact, using algebraic numbers.
