Moiré Pattern of Rotated Sierpiński Gaskets

Abstract

We study the moiré pattern formed by the intersection of a planar Sierpiński gasket and a copy rotated about its centre. At every resonant angle, we prove that the intersection is of finite type and compute its Hausdorff and Minkowski dimensions from an explicit finite matrix; for Lebesgue-almost every angle, we bound its upper Minkowski dimension by ${\overline{\dim }}_{\mathrm{B}}({\mathcal{S}}_{\theta })\le 2{\dim }_{\mathrm{H}}(S)-2$. Finally, we conjecture that both dimensions equal this bound at every non-resonant angle. Sections 2 and 3 are formalised in Lean.

arXiv comment16 pages, 10 figures