Researchers at Freie Universität Berlin reveal the mathematics behind mesmerizing patterns / New study links the beauty of tiling patterns to the structure and complexity of mathematical research

№ 158/2025 from Oct 10, 2025

In a recent study, mathematicians from Freie Universität Berlin have demonstrated that planar tiling, or tessellation, is much more than a way to create a pretty pattern. Consisting of a surface covered by one or more geometric shapes with no gaps and no overlaps, tessellations can also be used as a precise tool for solving complex mathematical problems. This is one of the key findings of the study, “Beauty in/of Mathematics: Tessellations and Their Formulas,” authored by Heinrich Begehr and Dajiang Wang and recently published in the scientific journal Applicable Analysis. The study combines results from the field of complex analysis, the theory of partial differential equations, and geometric function theory.

A central focus of the study is the “parqueting-reflection principle.” This refers to the use of repeated reflections of geometric shapes across their edges to tile a plane, resulting in highly symmetrical patterns. Aesthetic examples of planar tessellations can be seen in the work of M.C. Escher. Beyond its visual appeal, the principle has applications in mathematical analysis – for example, as a basis for solving classic boundary value problems such as the Dirichlet problem or the Neumann problem.

“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.”

In particular, it can be used to derive specific formulas for kernel functions – including the Green, Neumann, and Schwarz kernels, some of the tools that can be used to solve boundary value problems in physics and engineering. As such, this work creates an elegant connection between geometric intuition and analytical precision.

The parqueting-reflection principle has been gaining renown for more than a decade now and is particularly popular as a topic of research among early-career scholars. Since its initial development, a total of fifteen dissertations and final theses at Freie Universität have explored the subject, along with an additional seven dissertations by researchers 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. Interest in the principle remains high. Last year, Begehr published an article, “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.

“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.”

For nearly two decades, the research group led by Heinrich Begehr 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 (1843‒1921).

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 – a key tool 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.”

While they are considered very aesthetically impressive, tessellations in hyperbolic spaces – for example, within a circular disc – represent a particular challenge for mathematicians. This is where “Schweikart triangles” come into play: special triangles featuring one right angle and two zero angles, named after amateur mathematician and law professor Ferdinand Kurt Schweikart (1780‒1857).

These triangles enable the complete, regular tiling of a circular disc – producing patterns with an aesthetic appeal that offers fresh inspiration for computer graphics artists and architects alike. At the same time, the underlying mathematical constructions are highly complex and require advanced analytical methods.

The findings of the team highlight an often overlooked aspect of mathematics: it is not only an abstract discipline, but also a visual science – one in which structure, symmetry, and aesthetics play a central role. When paired with modern visualization techniques, graphics software, and digital tools, these insights become all the more relevant.

• Begehr, H., and D. Wang. 2025. “Beauty in/of Mathematics: Tessellations and Their Formulas.” Applicable Analysis, June, 1–47. https://doi.org/10.1080/00036811.2025.2510472.

• Begehr, Heinrich. 2024. “Hyperbolic Tessellation: Harmonic Green Function for a Schweikart Triangle in Hyperbolic Geometry*.” Complex Variables and Elliptic Equations, October, 1–46. https://doi.org/10.1080/17476933.2024.2408729.