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Moiré Pattern of Rotated Sierpiński Gaskets
math.MG
Sep 24, 2026 · v1
math.DS
TL;DR
Sections defining the rotated Sierpiński gasket intersection and its finite-type dimension results are formalised in Lean.
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 $\overline{\dim}_{\mathrm B}(\mathcal S_θ)\le 2\dim_{\mathrm H}(S)-2$. Finally, we conjecture that both dimensions equal this bound at every non-resonant angle. Sections 2 and 3 are formalised in Lean.
Problem
The moiré pattern formed by intersecting a planar Sierpiński gasket with a copy rotated about its centre is not well understood in terms of its dimensional properties.
Approach
The intersection is analyzed at resonant rotation angles, where it is shown to be of finite type. Hausdorff and Minkowski dimensions are computed from an explicit finite matrix. For Lebesgue-almost every angle, an upper bound on the upper Minkowski dimension is derived. Sections 2 and 3 of the work are formalised in Lean.
Results
At every resonant angle the intersection is of finite type with computable dimensions; for almost every angle the upper Minkowski dimension is bounded by 2·dim_H(S)−2, conjectured to be attained at every non-resonant angle.