Mathematicians at Freie Universität Berlin have demonstrated that the captivating beauty of tessellation patterns is deeply connected to powerful mathematical methods for solving complex problems. In a study published on June 13, 2025, in the journal Applicable Analysis, researchers Heinrich Begehr and Dajiang Wang reveal how repeating geometric reflections can bridge visual symmetry with analytical precision across different branches of mathematics. Their work shows that planar tiling, known as tessellation, serves as far more than a decorative arrangement of shapes—it is a practical tool with profound applications in mathematical physics and engineering.
Tessellations are patterns that cover a surface completely with repeating geometric forms, without any gaps or overlaps. The researchers focused on a key concept called the parqueting-reflection principle, which involves repeatedly reflecting geometric shapes across their edges to fill a plane with highly symmetric patterns. This technique may sound abstract, but many people have seen it in action through the famous artwork of M.C. Escher, whose intricate interlocking designs have fascinated the public for decades. However, Begehr and Wang demonstrate that this principle has far-reaching mathematical value beyond its aesthetic appeal, providing a systematic approach to classical boundary value problems that frequently arise in mathematical physics, including the Dirichlet and Neumann problems.
The study brings together ideas from complex analysis, partial differential equations, and geometric function theory to create something entirely new. One of the most important outcomes of this approach is the ability to derive explicit formulas for kernel functions, including Green, Neumann, and Schwarz kernels. These mathematical tools play a central role in solving boundary value problems encountered in both physics and engineering applications. According to Professor Heinrich Begehr, "Our research shows that beauty in mathematics is not only an aesthetic notion, but something with structural depth and efficiency." While previous research on tessellations has focused largely on how shapes can be used to tile a surface—such as the well-known work carried out by Nobel Prize winner Sir Roger Penrose—using the parqueting-reflection method to generate new tessellations opens up fresh possibilities for representing functions within these tiled regions.
Interest in the parqueting-reflection principle has grown steadily over the past decade, particularly among early-career researchers. Since the concept was first developed, it has been the subject of fifteen dissertations and final theses at Freie Universität Berlin, as well as seven additional doctoral dissertations completed by researchers at institutions abroad. Remarkably, the principle works not only in Euclidean space but also in hyperbolic geometries—the kinds used in theoretical physics and modern visualizations of spacetime. Last year, Begehr published an article titled "Hyperbolic Tessellation: Harmonic Green Function for a Schweikart Triangle in Hyperbolic Geometry" in the journal Complex Variables and Elliptic Equations, in which he demonstrated the use of the parqueting-reflection principle to construct the harmonic Green function for a Schweikart triangle in the hyperbolic plane. These special triangles, which feature one right angle and two zero angles, are named after amateur mathematician and law professor Ferdinand Kurt Schweikart and enable the complete, regular tiling of a circular disc.
The research highlights an often overlooked aspect of mathematics: it is not only an abstract discipline but also a visual science in which structure, symmetry, and aesthetics play a central role. For nearly two decades, Begehr's research group at Freie Universität Berlin's Institute of Mathematics has been studying what are known as the Berlin mirror tilings, a method based on the unified reflection principle developed by Berlin-based mathematician Hermann Amandus Schwarz, who lived from 1843 to 1921. In this approach, a circular polygon—a shape whose edges consist of pieces from straight lines and circular arcs—is reflected repeatedly until the entire plane is seamlessly and completely tiled, without any overlaps or gaps. These patterns are not only visually striking but also enable explicit integral representations of functions, which serve as a key tool for solving complex boundary value problems. As Begehr notes, "Mathematicians once had to use a three-part vanity mirror to produce an endless sequence of images. Nowadays, we can use iterative computer programs to generate the same effect—and we can complement this with exact mathematical formulas used in complex analysis."
The implications of this research extend far beyond pure mathematics. Dajiang Wang hopes that their results will resonate not only in mathematical physics but may also inspire ideas in fields like architecture or computer graphics. The aesthetic appeal of these patterns offers fresh inspiration for computer graphics artists and architects alike, while the underlying mathematical constructions provide powerful analytical methods for solving real-world problems. When paired with modern visualization techniques, graphics software, and digital tools, these insights become all the more relevant. The study successfully forges a clear link between geometric intuition and rigorous analytical methods, proving that in mathematics, beauty and utility are often two sides of the same coin.
