A sufficient condition for aperiodicity of striped Wang tiles is established via parabola geometry, enabling explicit constructions of aperiodic sets for every pair of quadratic irrationals as vertical and horizontal stripe densities.
Pytheas Fogg.Substitutions in dynamics, arithmetics and combinatorics, volume 1794 of Lecture Notes in Mathematics
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An aperiodic set of Wang tiles for every quadratic irrational
A sufficient condition for aperiodicity of striped Wang tiles is established via parabola geometry, enabling explicit constructions of aperiodic sets for every pair of quadratic irrationals as vertical and horizontal stripe densities.
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Transcendence of simple geodesics on finite modular covers
Conjectures that simple geodesics on finite covers of the modular orbifold have rational, quadratic, or transcendental endpoints, with proofs for minimal geodesic laminations and the modular torus cover.