  We present a geometric realization for all mutation classes of quivers of
rank $3$ with real weights. This realization is via linear reflection groups
for acyclic mutation classes and via groups generated by $\pi$-rotations for
the cyclic ones. The geometric behavior of the model turns out to be controlled
by the Markov constant $p^2+q^2+r^2-pqr$, where $p,q,r$ are the elements of
exchange matrix. We also classify skew-symmetric mutation-finite real $3\times
3$ matrices and explore the structure of acyclic representatives in finite and
infinite mutation classes.

Combinatorics;Metric Geometry;Rings and Algebras