Geodesic laminations as geometric realizations of Pisot substitutions

We construct a geodesic lamination on the hyperbolic disk and a dynamical system defined on this lamination. We prove that this dynamical system is a geometrical realization of the symbolic dynamical system that arises from the following Pisot substitution: $1\rightarrow 12, \dotsc, (n-1) \rightarrow 1n, n\rightarrow 1$.

Exact geometrical realization of the Cabibbo–Kobayashi–Maskawa matrix: The catamorphy of quark–lepton families

It is suggested that the number of quark–lepton families is three as the result of a catamorphy; this is illustrated by a geometrical representation that limits the number of families to three, that most naturally generates 15 quark–lepton fields per family, and that exactly realizes the Cabibbo–Kobayashi–Maskawa matrix. The geometrical representation uniquely requires a 1:λ:λ2:λ3 hierarchy for the weak quark–quark linkages and, together with a supplementary Ansatz that derives from the geometrical representation, entrains CP violation in the [Formula: see text]-system and yields mt = 123 ± 65 GeV/c2 from an analysis of Cabibbo–Kobayashi–Maskawa phenomenology.