Researchers uncover the mathematical structure behind mesmerizing tiling patterns, linking their visual appeal to the complexity of mathematical research.
In a new study, mathematicians at Freie Universität Berlin show that planar tiling, also known as tessellation, is far more than a decorative arrangement of shapes.
Tessellations cover a surface completely with repeating geometric forms, without gaps or overlaps, and the researchers demonstrate that these structures can also serve as powerful tools for tackling difficult mathematical problems. The study, conducted by Heinrich Begehr and Dajiang Wang, was published in the journal Applicable Analysis. The work brings together ideas from complex analysis, partial differential equations, and geometric function theory.
A key concept explored in the study is the “parqueting-reflection principle.” This method involves repeatedly reflecting geometric shapes across their edges to fill a plane, creating highly symmetric patterns.
Such tessellations are visually familiar from the artwork of M.C. Escher, but the researchers show that the principle has far-reaching mathematical value. In particular, it provides a systematic way to approach classical boundary value problems, including the Dirichlet and Neumann problems, which arise frequently in mathematical physics.
“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 important outcome 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 physics and engineering. In this way, the study forges a clear link between geometric intuition and rigorous analytical methods.
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. 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.”
The Tiling Tradition in Berlin
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.”
Schweikart Triangles and Hyperbolic Beauty
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.
Mathematics as a Visual Science
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.
References:
“Beauty in/of mathematics: tessellations and their formulas” by H. Begehr and D. Wang, 13 June 2025, Applicable Analysis.
DOI: 10.1080/00036811.2025.2510472
“Hyperbolic tessellation: harmonic Green function for a Schweikart triangle in hyperbolic geometry” by Heinrich Begehr, 15 October 2024, Complex Variables and Elliptic Equations.
DOI: 10.1080/17476933.2024.2408729
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