Twenty-four integer points build a three-holed polyhedron whose eight faces all touch

The published counts pass the first test any claimed polyhedron should face. Vertices minus edges plus faces gives 24 - 36 + 8 = -4 ^[1]. A closed surface with that Euler characteristic has genus (2 - (-4))/2 = 3 ^[2]. That means three handles, in the topologist's sense of holes or tunnels through a surface, of which a doughnut has one ^[4]. The faces and corners agree too. Eight nine-sided faces have 72 sides between them, and each edge borders two faces, so there are 36 edges; 24 vertices with three edges apiece give the same 72 edge-ends ^[3].

Mizhaev states the defining property plainly. "There are eight flat, polygonal faces, and each is a neighbour of all seven others along an edge," he said ^[3]. The tetrahedron and the Szilassi polyhedron also have faces that are all neighbours ^[5]. According to New Scientist, though, some pairs of faces in Mizhaev's design share two edges instead of one ^[5]. Counting gives the size of that difference. Eight faces form 28 pairs, and 36 edges leave eight more than one per pair ^[4].

Integer coordinates make the checks exact. Whether the nine corners of each face lie in one plane becomes a question of whole-number arithmetic with a yes-or-no answer and no rounding tolerance to argue about ^[8]. Accidental intersections are a common problem in shapes like this one, according to New Scientist, and the supplied equations test for them directly ^[8].

Lars Schewe at the University of Edinburgh was measured about the result. "It's more a small piece that we didn't have before," he said ^[9]. He said it does not overturn existing conjectures, but "constructing these has always been a difficult task" ^[10]. There is no reliable way to do it ^[11]. Writing down equations for where every vertex can sit and how every face must fit can rapidly become practically impossible to solve, Schewe said, so researchers lean on computer searches, geometric intuition and physical models ^[11][12].

"I can tell you these are the rules," he said. "But when I try to build it and I try to draw it, I can't do it." ^[14]

Mizhaev used ChatGPT to help with some verification calculations and to write Python scripts for the new version ^[13]. He says the original shape dates from 2020, before he began using the chatbot ^[13]. The integer vertices and checking equations can be run again without those scripts ^[8]. The report does not say whether anyone outside the project has done that yet.