Robert C. Sargent | Searcharxiv

arXiv2025

A gasket construction of the Koch snowflake and variations

We introduce a construction of the Koch snowflake that is not inherently six-way symmetrical, based on iteratively placing similar rhombi. This construction naturally splits the snowflake into four identical self-similar curves, in contrast to the typical decomposition into three Koch curves. Varying the shape of the rhombi creates a continuous family of new fractal curves with rectangular symmetry. We compute the Hausdorff dimension of the generalized curve and show that it attains a maximum at the original Koch snowflake.

math.MG↗

arXiv2024

Minimum-distortion continuous cartograms by numerically optimized meshes

We present an algorithm for creating contiguous cartograms using meshes. We use numerical optimization to minimize cartographic error and distortion by transforming the mesh vertices. The vertices can either be optimized in the plane or optimized on the unit sphere and subsequently projected to the plane. We also present a hybrid "best of both worlds" method, where the vertices are optimized on the sphere while anticipating the distortion caused by the final projection to the plane. We show a significant improvement in the preservation of region shapes compared to existing automated methods. Outside the realm of cartograms, we apply this hybrid technique to optimized map projections, creating the Liquid Earth projection.

cs.CG↗