When mathematics meets aesthetics: Tessellations as a precise tool for solving complex problems

mathematicstessellationscomplex-analysisboundary-value-problems

Abstraction: Tessellation parqueting-reflection principle solves boundary value problems analytically

Key points:

Mathematicians at Freie Universität Berlin show planar tessellation (parqueting-reflection principle) is a rigorous tool, not just aesthetics — published in Applicable Analysis (DOI: 10.1080/00036811.2025.2510472)
Parqueting-reflection: repeated reflections of geometric shapes across edges tile a plane, enabling explicit integral representations of functions
Enables derivation of Green, Neumann, and Schwarz kernel functions used in boundary value problems (Dirichlet, Neumann problems) in physics and engineering
Works in both Euclidean and hyperbolic geometries; Begehr extended it to Schweikart triangles in hyperbolic planes (harmonic Green functions)
15 dissertations at Freie Universität plus 7 abroad have explored the principle since its development; active research area especially for early-career scholars
Potential applications in architecture, computer graphics, and mathematical physics

Connections: Freie Universitaet Berlin · Heinrich Begehr · Computational Geometry · Mathematical Physics