A new study by mathematicians at Freie Universität Berlin shows that planar tiling, also known as tessellation, is far more than a decorative technique. Tessellations cover a surface with one or more geometric shapes without gaps or overlaps, and the researchers demonstrate that these structures can serve as precise tools for tackling difficult mathematical problems. The findings appear in the paper "Beauty in/of Mathematics: Tessellations and Their Formulas," written by Heinrich Begehr and Dajiang Wang and published in the journal Applicable Analysis. The work brings together ideas from complex analysis, partial differential equations, and geometric function theory.
At the heart of the research is the "parqueting-reflection principle." This method involves repeatedly reflecting geometric shapes across their edges to fill a plane, creating highly ordered and symmetrical patterns. Well-known visual examples of this kind of tiling can be found in the artwork of M.C. Escher. The researchers show that beyond their visual appeal, these reflections play a practical role in mathematical analysis. They can be used, for example, to help solve classic boundary value problems such as the Dirichlet problem or the Neumann problem.
Beauty With Structure and Purpose
"Our research shows that beauty in mathematics is not only an aesthetic notion, but something with structural depth and efficiency," says Professor Heinrich Begehr. "While previous research on tessellations has focused largely on how shapes can be used to tile or cover a surface -- for example, some well-known work carried out by Nobel Prize winner Sir Roger Penrose -- using the parqueting-reflection method to generate new tessellations opens up new possibilities. It is a practical tool for developing ways of representing functions within these tiled regions, which could be useful in areas such as mathematical physics and engineering."
One key outcome of this approach is the ability to derive exact formulas for kernel functions. These include the Green, Neumann, and Schwarz kernels, which are important tools for solving boundary value problems in physics and engineering. By linking geometric patterns with analytical formulas, the research bridges intuitive visual thinking and rigorous mathematical precision.
Growing Interest and Expanding Applications
The parqueting-reflection principle has attracted increasing attention for more than ten years and has become especially popular among early-career researchers. Since its introduction, fifteen dissertations and final theses at Freie Universität have focused on the topic, along with seven additional dissertations completed by researchers in other countries.
The method is not limited to familiar flat, or Euclidean, spaces. It also applies to hyperbolic geometries, which are commonly used in theoretical physics and modern models of spacetime. Interest in this area continues to grow. Last year, Begehr published a paper titled "Hyperbolic Tessellation: Harmonic Green Function for a Schweikart Triangle in Hyperbolic Geometry" in the journal Complex Variables and Elliptic Equations, where he demonstrated how the parqueting-reflection principle can be used to construct the harmonic Green function for a Schweikart triangle in the hyperbolic plane.
"We hope that our results will resonate not only in pure mathematics and mathematical physics," Dajiang Wang says, "but may even inspire ideas in fields like architecture or computer graphics."
The Tiling Tradition in Berlin
For almost twenty years, the research group led by Heinrich Begehr at Freie Universität Berlin's Institute of Mathematics has been investigating what are known as the "Berlin mirror tilings." This approach builds on the unified reflection principle developed by Berlin mathematician Hermann Amandus Schwarz (1843‒1921).
In this method, a circular polygon -- a shape whose edges consist of pieces from straight lines and circular arcs -- is reflected again and again until it fills the entire plane without overlaps or gaps. These designs stand out visually, but they also make it possible to write explicit integral representations of functions, which are essential for solving complex boundary value problems.
"Mathematicians once had to use a three-part vanity mirror to produce an endless sequence of images," says Begehr. "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."
Schweikart Triangles and Hyperbolic Geometry
Tessellations in hyperbolic spaces are especially striking but also especially challenging to analyze. These patterns often appear inside a circular disc and require sophisticated mathematical tools. A key concept in this area is the "Schweikart triangle," a special type of triangle with one right angle and two zero angles. It is named after amateur mathematician and law professor Ferdinand Kurt Schweikart (1780‒1857).
Schweikart triangles allow mathematicians to tile a circular disc completely and regularly. The resulting patterns are visually impressive and can inspire designers in fields such as computer graphics and architecture. At the same time, the mathematical foundations behind these constructions are highly advanced and demand careful analytical work.
Mathematics as a Visual Science
The team's findings highlight an aspect of mathematics that is often overlooked. Mathematics is not only an abstract discipline focused on symbols and equations. It is also a visual science, where structure, symmetry, and aesthetics play a crucial role. Combined with modern visualization tools, graphics software, and digital techniques, these ideas gain even greater relevance and practical impact.
