Researcher constructs polyhedron with eight mutually adjacent faces
An independent design engineer has created a genus-3 polyhedron with eight nine-sided faces, each sharing an edge with all seven others. The shape has 24 vertices and 36 edges, and all vertex coordinates are integers, enabling independent verification. It resembles the Szilassi polyhedron but differs in that some face pairs share two edges.
The shape's construction builds on Mizhaev's 2020 work, with the new contribution being integer coordinates for all 24 vertices. This allows independent verification that faces remain flat, the surface closes properly, and no unintended intersections occur—a common failure point in such constructions. The polyhedron's genus-3 topology means it contains three holes, distinguishing it from simpler neighbor-sharing shapes like the tetrahedron.
Schewe notes that constructing such shapes has always been difficult because no reliable method exists. Brute-force approaches requiring vast equation systems quickly become impractical, so researchers rely on computer searches, geometric intuition, or physical models. Mizhaev employed ChatGPT for verification calculations and Python scripting, though the original design predates his use of such tools.
The discovery may interest mathematicians studying polyhedral structures and topology, potentially offering a new example for testing conjectures about face adjacency. It could also demonstrate how AI tools like ChatGPT assist in verification work, though the core insight came from human intuition. For society broadly, the impact is likely limited to academic circles, though accessible integer coordinates may make the shape useful in education or 3D printing demonstrations. The work underscores how independent researchers can contribute to mathematics outside traditional academic settings.