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 $$\overline{\dim}<em>{\mathrm B}(\mathcal S</em>θ)\le 2\dim_{\mathrm H}(S)-2$$. Finally, we conjecture that both dimensions equal this bound at every non-resonant angle. Sections 2 and 3 are formalised in Lean.