In the realm of mathematical curiosities, few questions have intrigued thinkers as much as the one posed by Prince Rupert of the Rhine in the 17th century: Can you drill a hole through a cube large enough that an identical cube could pass through it? Remarkably, the answer is yes. Imagine positioning a cube on its corner and boring a square hole vertically through it; if done correctly, an identical cube could slip entirely through this opening. This seemingly simple puzzle sparked centuries of exploration into the properties of three-dimensional shapes, particularly focusing on a property now known as “Rupert’s property.”
Over time, mathematicians discovered a variety of three-dimensional solids sharing this intriguing feature. Shapes possessing Rupert’s property can accommodate a hole through which a congruent, identical shape can pass without breaking or deforming. These shapes, which are convex polyhedrons—three-dimensional figures with flat faces and no indentations—have fascinated researchers for decades. In 2017, a formal conjecture emerged suggesting that every convex polyhedron exhibits Rupert’s property. Despite intensive efforts, no one had managed to disprove this hypothesis—until very recently.
Enter the “noperthedron,” a groundbreaking new shape that challenges this long-standing conjecture. Coined by computer science researcher Tom Murphy VII, the term “nopert” means “not Rupert,” describing shapes that do not allow an identical copy to pass through a hole bored into them. The noperthedron is a convex polyhedron characterized by an intricate structure: it has 90 vertices, 240 edges, and 152 faces. This complex geometry is significant not only for its size but for its unique property of being the first proven convex polyhedron without Rupert’s property.
The researchers behind this discovery are Sergey Yurkevich, affiliated with the Austrian technology company A&R Tech, and Jakob Steininger of Statistics Austria, the country’s national statistical institute. Their work, recently published on the preprint server arXiv.org, introduces the noperthedron to the mathematical community. While previous shapes had been suspected to lack Rupert’s property, none had been definitively proven until Yurkevich and Steininger’s study. Their approach involved designing the noperthedron with certain properties that made rigorous proof more manageable.
To verify the noperthedron’s unique property, the researchers utilized a custom-built computer program. This software tested every possible configuration, rotation, and translation of two identical noperthedrons to determine if one could pass through a hole in the other. The result was conclusive: no matter how the shapes were positioned, it was impossible for one noperthedron to fall through a hole in its twin. This computational verification marks a significant milestone in the study of convex polyhedrons and Rupert’s property.
The collaboration between Yurkevich and Steininger is itself a story of enduring partnership and shared passion for mathematics. The two met as teenagers preparing for math olympiads and have since combined their strengths to tackle complex problems. Their mutual respect and candid communication have been key to their success; as Steininger remarks, they are unafraid to admit confusion and clarify ideas with each other, fostering a productive and honest working relationship.
Their fascination with Rupert’s property was sparked during their university years when they came across videos about Prince Rupert’s cube on YouTube. Recognizing the depth and open-ended nature of this mathematical question, they embarked on a journey to explore the prevalence of Rupert’s property among convex polyhedrons. In 2020, they publicly conjectured that not all convex polyhedrons possess Rupert’s property—a bold hypothesis that challenged established thinking. Five years later, with the introduction and proof of the noperthedron, they have turned this conjecture into a verified fact.
An essential part of their methodology involved conceptualizing the problem in higher-dimensional terms. Specifically, they represented the set of all possible holes through the noperthedron as a five-dimensional cube. Each axis in this abstract cube corresponds to a different rotation or orientation of the polyhedron. By combining mathematical reasoning with exhaustive computer-aided analysis, the researchers systematically eliminated every potential orientation that could allow one noperthedron to pass through another. This innovative blend of theory and computation has impressed other mathematicians in the field.
Pongbunthit Tonpho, a mathematician at Chulalongkorn University in Thailand who also studies Rupert’s property, praised the work as both “creative and rigorous.” Tonpho expressed surprise that someone could disprove the