Eight faces, twenty single edges, eight double edges
On 15 September Ruslan Mizhaev submitted a preprint to arXiv that gives integer coordinates for a genus-3 polyhedron. The surface has 8 flat faces, each with 9 sides, 24 vertices and 36 edges in all, and 3 faces meet at every vertex. New Scientist reported it on 25 September and did not present the text as a peer-reviewed journal paper.[1]
Each face meets the other 7 along an edge. 20 face pairs share one edge. 8 pairs share two edges and pass across each other along the line where their planes meet. Mizhaev said the eight flat faces together form a closed surface with three handles.[1]
The list you can check
The preprint gives the vertex coordinates, the face walks and the plane equations. Verification splits into four exact integer checks: planarity of the faces, a closed orientable surface, absence of unintended intersections, and four-fold rotational symmetry (C4). The construction keeps the incidence of the skeleton Mizhaev described in 2020. The new version adds a coordinate certificate that can be reproduced for that same incidence. The same four checks can be rerun in a computer-algebra session from the published coordinates.[1]
Lars Schewe at the University of Edinburgh said he sees the shape as a small piece of a larger puzzle and that constructing these objects has always been difficult. He added that existing conjectures still stand. Mizhaev said he found the shape while trying polyhedral surfaces in computer-aided design, built the skeleton in 2020, and used ChatGPT for some verification calculations and Python scripts on the new version. The original shape predates that use. The part a reader can rebuild by hand is the integer list and the plane equations.[1]