q-rationals are realized as circles in the plane with Springborn operations defined geometrically as homothety centers, producing a q-deformed midpoint formula and a new q-version of Markov numbers.
Farey boat I. Continued fractions and triangulations, modular group and polygon dissections
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abstract
We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections appear when extending these theorems for elements of the modular group $PSL(2,\mathbb{Z})$. These polygon dissections are interpreted as walks in the Farey tessellation. The combinatorial model of continued fractions can be further developed to obtain a canonical presentation of elements of $PSL(2,\mathbb{Z})$.
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Plane geometry of $q$-rationals and Springborn Operations
q-rationals are realized as circles in the plane with Springborn operations defined geometrically as homothety centers, producing a q-deformed midpoint formula and a new q-version of Markov numbers.