To highlight the science behind the world s most popular sport, the National Museum of Mathematics (MoMath) in New York is hosting a special educational program titled "The Hidden Geometry of Soccer." The session explores how mathematical principles and geometric symmetry influence the design of footballs used at the highest level of the game.
Mathematician Dr. Chaim Goodman-Strauss, who leads the presentation, says the event demonstrates how mathematical patterns appear not only in footballs but also in architecture, gates, shadows, and everyday objects.
According to Goodman-Strauss, the traditional black-and-white football made from 32 panels represents one of the finest examples of geometric symmetry. Its arrangement of polygons creates one of the most balanced spherical designs possible.
Hannes Schefk, Head of Football Innovation at Adidas, agrees that mathematics is at the heart of football design.
Speaking in an interview, Schefk said a football represents a perfectly balanced engineering system, maintaining its symmetry regardless of how it is rotated.
He explained that the design process begins with a fundamental mathematical challenge: how to distribute points evenly across the surface of a sphere while maintaining consistent shape and performance.
Mathematicians note that while symmetry is often understood as visual balance, it has a deeper scientific meaning. According to Professor Emeritus Doris Schattschneider of Moravian University, an object is considered symmetrical if it remains unchanged after certain transformations, such as rotation or reflection.
The iconic Adidas Telstar ball, used during the 1970 and 1974 FIFA World Cups, featured 12 black pentagons and 20 white hexagons. Experts say rotating and reflecting a single geometric pattern creates the complete football design.
For the 2026 FIFA World Cup, Adidas has introduced a new official match ball called Trionda, a name inspired by the tournament s three host nations—the United States, Canada, and Mexico.
Unlike traditional footballs, Trionda features curved, wave-like S-shaped panels instead of straight-edged polygons. According to Dr. Goodman-Strauss, rotating the ball around specific points reveals that its geometric symmetry remains perfectly preserved despite the modern design.
Beyond official match balls, designers have also experimented with optical illusions.
Texas craftsman John Paul Wheatley created the Impossible Ball, inspired by a British road sign depicting a football made entirely of hexagons—a shape considered mathematically impossible for a closed sphere.
Wheatley explained that although the ball appears to consist only of hexagonal panels from one angle, carefully curved panel shapes create the visual illusion without violating geometric principles.
He also designed the Glitch Ball, an optical illusion football that appears blurred and distorted, confusing the viewer s perception.
Football fans on social media have joked that the Glitch Ball could be even more difficult for goalkeepers than the infamous Jabulani ball used at the 2010 FIFA World Cup, which became notorious for its unpredictable flight due to its exceptionally smooth surface.
The influence of football geometry has even extended beyond sports. A New York bakery has created a football-inspired doughnut based on geometric football patterns, demonstrating how the same mathematical principles apply to everyday objects as well as elite sports equipment.
