17-летний школьник из США доказал, что существует ровно 146 благородных многогранников

Most people encounter symmetrical polyhedra through familiar 3D shapes like cubes and pyramids. Perhaps you also know about the Platonic solids, which besides the cube, include the tetrahedron, octahedron, dodecahedron, and icosahedron. But geometry allows far stranger shapes, such as spiky structures whose faces pass through one another, yet remain so symmetrical that every face is equivalent and every corner or vertex has the same arrangement. Mathematicians call these perfectly symmetrical shapes noble polyhedra. They began cataloguing them seriously in the 1870s, but no one could determine how many existed. Connor Hill, a 17-year-old student from Pennsylvania, now says he has finished the list. The prodigious teenager wrote computer programs that transformed the seemingly endless hunt into a finite set of algebra problems. His 34-page mathematical preprint concludes that, beyond two infinite families, exactly 146 noble polyhedra exist under his definition. Mathematicians had catalogued only 61 by 2020, meaning his classification added 85 shapes to the published list. The work earned Hill the $250,000 top prize at the 2026 Regeneron Science Talent Search. If the proof withstands mathematical scrutiny, Hill has resolved a classification problem left open for more than a century — proving not merely that these 85 shapes exist, but that no others meeting the same rules remain to be found. This is the complete list of noble polyhedra. A Catalogue More Than a Century in the Making A polyhedron is a three-dimensional figure built from flat faces and straight edges. A polyhedron becomes “noble” when its symmetries treat every face alike and every vertex, or corner, alike. Take a cube: a rotation can move any face to the position of any other face, and any corner to the position of another corner But don’t let the cube’s simple shape fool you. Noble polyhedra can sometimes take wild shapes, which is why Hill’s classification allows nonconvex polyhedra and structures whose faces or edges intersect. The field itself grew from the ancient study of the five Platonic solids. In the early 1600s, Johannes Kepler showed that regular solids could also have star-shaped faces and intersect themselves. By 1813, Augustin-Louis Cauchy had proved that exactly four such regular star polyhedra exist. German mathematician Edmund Hess was one of the most prolific discoverers of noble polyhedra. Since he first embarked on a quest to find as many as possible in 1875, Hess found 16 nonregular examples and an infinite family called stephanoids, or crown polyhedra. Max Brückner added 10 more by 1907. Then the search largely stalled for a century until computers entered the picture. More recently, Robert Webb found another example in 2008 while experimenting with the polyhedron computer program Stella. Ulrich Mikloweit’s 2020 study using Stella4D reproduced earlier results, confirming them, and generated a few previously unpublished shapes. By that year, mathematicians knew of two infinite families and 61 isolated examples. But Mikloweit knew the hunt was unfinished. “I don’t think that this list is complete,” he later wrote. It was at this stage of progress that Hill began his own research. He encountered the unresolved classification question while browsing an online geometry community. He initially hoped to discover a handful of shapes. But as the teenager went deeper and deeper down the rabbit hole, the project expanded into an attempt to prove that none were missing. Turning Infinite Geometry Into Finite Algebra Hill’s key move was to stop searching through three-dimensional models directly. He sorted possible sets of vertices according to their symmetries, then translated a geometric condition — whether certain points lie in the same plane — into algebra. Four points lie in one plane when a particular mathematical determinant equals zero. Because of the way Hill parameterized the possible vertex arrangements, those determinants became polynomial equations of degree no higher than three. Their roots identified the special configurations worth examining. He then proved that the infinitely many possible arrangements could be divided into only finitely many relevant classes. “By translating the problem into a different mathematical framework, I could narrow it down to a finite set that could actually be solved,” Hill said. His software then performed the exhaustive part of the proof. Hill used Python libraries including SymPy and NumPy, along with the Wolfram Language, to handle the symbolic algebra, numerical root-finding and geometric tests. The search produced 146 isolated nonprismatic noble polyhedra. A separate mathematical argument showed that the only prismatic cases belong to the two already known infinite families. A $250,000 Proof Hill, a student at Delta High School in State College, prevailed over more than 2,600 entrants in the 85th Science Talent Search. Forty finalists shared more than $1.8 million in awards at the National Building Museum in Washington. He plans to study computational mathematics at the Massachusetts Institute of Technology this fall. His proof has now appeared on arXiv as a preprint, placing its definitions, algorithms and catalogue before the wider mathematical community for scrutiny. Hill also released his code and three-dimensional model files, allowing others to examine the computer-assisted portion of the argument. Edward Kang, 17, of Hackensack, New Jersey, took second place and $175,000 for RetinaMind, an artificial-intelligence system trained on retinal images to detect patterns associated with autism and ADHD. Iris Shen, 17, of The Woodlands, Texas, won third place and $150,000 for testing potential blood-cancer treatments in clams, which she investigated as a possible animal model for drug discovery. Other finalists tackled quantum physics, protein structure, wound healing in microgravity and Alzheimer’s disease.