Professor Craig S. Kaplan has been named a 2025 Fellow of the Fields Institute for Research in Mathematical Sciences, an honour that recognizes his exceptional contributions to mathematics.
Established in 2002 to mark the Institute’s 10th anniversary, the Fields Institute Fellow designation celebrates individuals who have made outstanding contributions to mathematics and in advancing the Institute’s mission. Each year, a select group of distinguished researchers is named to this prestigious fellowship.
“I’m very pleased to hear that Craig has been named a Fellow of the Fields Institute,” said Raouf Boutaba, University Professor and Director of the Cheriton School of Computer Science. “Over his career, he has made many contributions to mathematics, computational geometry and computer graphics, often exploring their intersections with art. Most recently, Craig’s research gained international attention through his co-discovery and computational proof of an aperiodic monotile, a breakthrough in tiling theory that captured the imagination of the mathematical community and public alike.”
Craig S. Kaplan, Professor at the Cheriton School of Computer Science, is interested in a broad range of interdisciplinary topics, with a particular focus on interactions between mathematics and art. He uses mathematical ideas to create tools and algorithms that generate ornamental patterns, and that empower artists and designers. His work frequently incorporates knowledge from computer graphics, classical and computational geometry, human-computer interaction, graph theory, symmetry and tiling theory, and perceptual psychology.
In addition to his research, Professor Kaplan serves as the associate editor of the Journal of Mathematics and the Arts, and serves on the board of the Bridges Organization, which oversees the annual Bridges conference.
Exploring mathematics, computer science and art
The hat and the spectre: Solving a six-decade puzzle in tiling theory
Professor Kaplan’s most celebrated accomplishment is the 2023 co-discovery and proof of an aperiodic monotile. Dubbed the hat, the 13-sided shape fills the infinite plane without gaps or overlaps, in a pattern that not only never repeats but — importantly — cannot be made to repeat. This breakthrough solved the long-standing ein stein or “one stone” problem in tiling theory, an open question for more than 60 years.
Professor Kaplan’s involvement began when David Smith, a self-described shape enthusiast, contacted him about a promising shape with apparent aperiodic properties. Together with collaborators Joseph Samuel Myers and Chaim Goodman-Strauss, the team solved the ein stein problem and published their proofs in a paper titled “An aperiodic monotile.” This paper was followed shortly by “A chiral aperiodic monotile,” which demonstrated that a related family of shapes, called spectres, tile the plane aperiodically using translations and rotations alone, meeting an even stricter definition of aperiodicity.
The ein stein discovery drew worldwide interest, with extensive coverage in media outlets including the New York Times, The Guardian, CNN, New Scientist, Phys.org, Smithsonian Magazine, Quanta Magazine and Scientific American. TIME magazine named the hat as one of the best inventions of 2023, a striking example of how mathematical discoveries can resonate far beyond academia.
The public celebrated the discovery through vibrant online discussion and events such as Hatfest, a conference exploring the aperiodic monotile convened at the University of Oxford’s Mathematical Institute. As co-author Chaim Goodman-Strauss observed, “What was so moving and significant to me was that people made this their own. They took on the hat as something they owned and that they will pass on and share with other people. It’s absolutely unique, in my experience, for a mathematical result to have a cultural life like that.”
The discovery continues to receive accolades. In a 2025 issue of BBC Science Focus magazine spotlighting the most significant breakthroughs of the century so far, Sir David Spiegelhalter described the solution to the ein stein problem as “the most important mathematical breakthrough of this century.”
Impossible solids: Insights that helped elucidate molecular structure
Perfect solids — the five Platonic and 92 Johnson polyhedra — have long fascinated mathematicians. While exploring them, Professor Kaplan became intrigued by what he called “near misses” — geometric oddballs tantalizingly close to being a mathematically perfect solid. He had documented these near-miss Johnson solids on his website, including one such shape built using 11-sided polygons, equilateral triangles and squares.
Some years later, a Japanese biologist stumbled upon his work while seeking to understand a molecular structure known as a TRAP-cage, a custom-engineered protein that can form an 11-sided ring structure. This molecular cage, formed from TRAP rings and bonded together with gold atoms, seemed to mirror the structure of the near-miss solid Professor Kaplan had documented on his site and built using paper shapes and tape. Through a computational analysis of the stability of the TRAP-cage, Professor Kaplan helped the research team confirm that the protein’s molecular structure was physio-chemically possible. The resulting interdisciplinary study was published in Nature in a paper titled “An ultra-stable gold-coordinated protein cage displaying reversible assembly.”
Islamic Star Patterns
Professor Kaplan has long held an interest in computer-generated Islamic geometric patterns, a field that he leads and where he has made significant contributions. His paper, “Islamic star patterns in absolute geometry,” remains the reference in the field. His software for generating these patterns is freely available and has been widely used in applications ranging from book covers and theatre sets to furniture and commissioned artworks.
Escherization
Professor Kaplan has played an important role in applying tiling theory in computer graphics. His paper titled “Escherization” helped draw the two fields closer together. The paper is frequently cited as a foundational contribution to the field of non-photorealistic rendering. Professor Kaplan remains a sought-after reviewer in this research area and continues to explore the artistic and mathematical potential of decorative tilings and Escher-style designs.
Bridges conference
Since 2005, Professor Kaplan has served on the board of the Bridges Organization, which oversees the annual Bridges conference on mathematics and art. In this role, he contributes to executive-level decision making as well as manages multiple operations that make the conferences a success. In 2017, marking the 50th anniversary of the Faculty of Mathematics, he hosted Bridges Waterloo 2017, a week-long event that brought together experts and enthusiasts in mathematics, art, music, architecture, education and culture.