Mathematician constructs rare eight-faced all-neighbor shape — SkimNews
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- Ruslan Mizhaev, an independent researcher and design engineer, constructed a genus-3 polyhedron with eight flat, nine-sided faces, 24 vertices and 36 edges, where each face shares an edge with all seven others and three faces meet at every vertex.
- Mizhaev's shape uses integer coordinates for all 24 vertices, alongside equations that let other mathematicians verify the faces are flat, the surface closes properly and no parts accidentally intersect — a common pitfall for such shapes.
- The design echoes the Szilassi polyhedron and tetrahedron, which also have all-neighbor faces, but Mizhaev's version has some face pairs sharing two edges rather than exactly one, with the surface pierced by three handles (holes).
- Mizhaev first described the underlying structure in 2020 but says he stumbled across the all-neighbor property while experimenting with polyhedral surfaces in CAD software, rather than deliberately seeking it out.
- Lars Schewe at the University of Edinburgh called the result "a small piece that we didn't have before," noting it does not overturn existing conjectures but that constructing such shapes remains extremely difficult because no reliable general method exists.
- Mizhaev used ChatGPT to help with verification calculations and to write Python scripts for the new version, though the original shape predates his use of the tool.
Why it matters: The discovery gives topology researchers a verifiable, integer-coordinate example of an all-neighbor polyhedron — a class of shapes so hard to build that brute-force equation-solving rapidly becomes impractical, as Schewe explains. With no general method available, each new specimen is a bespoke achievement rather than a routine output.
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