+- MKLab (https://mklab.gr)
+-- Forum: [INDEX] (https://mklab.gr/forumdisplay.php?fid=1)
+--- Forum: MATHEMATICS (https://mklab.gr/forumdisplay.php?fid=3)
+---- Forum: ARTICLES (https://mklab.gr/forumdisplay.php?fid=13)
+---- Thread: A fractal arbelos [Eppstein] (/showthread.php?tid=425)
A fractal arbelos [Eppstein] - mklabgr - 06-16-2026
A fractal arbelos
BY David Eppstein
Summary
In this article, mathematician David Eppstein introduces a “fractal arbelos,” a recursive geometric construction inspired by the classical arbelos—the region bounded by three tangent semicircles. Starting from a hierarchy of nested polygons known since Archimedes, he replaces each triangle with an arbelos, creating a self-similar tessellation that can be viewed either as a subdivision of a semicircle or as a tiling of the hyperbolic plane by ideal triangles.
The article explores how this structure resembles, but differs from, a uniform hyperbolic tiling: although it shares the same combinatorial pattern, its geometric symmetries are much more limited. Eppstein also shows how the vertices and tiles can be labeled using binary numbers and dyadic rationals, revealing elegant arithmetic relationships between neighboring tiles and highlighting deep connections among fractal geometry, hyperbolic tessellations, binary representations, and infinite graph structures.
ARTICLE