An octahedral polyhedron with three holes, 24 vertices and 36 edges has been presented by independent researcher and design engineer Ruslan Mizhaev. The unusual thing about the figure is that each of the eight flat nine-sided faces borders all seven others along the length of an edge.
„We have eight flat polygonal faces. Each of them borders the other seven along the length of an edge. Together they form a closed surface with three handles“, says Mizhaev.
Figure with 24 vertices and 36 edges
The model represents a polyhedron of genus 3. In topology, this means a closed surface with three holes or tunnels. For comparison, the surface of a doughnut has one handle.
Three walls meet at each vertex of the new structure. All 24 vertices are described with exact integer coordinates, and equations have been added to them that allow other mathematicians to check whether the walls are flat, the surface is closed, and the individual parts do not intersect incorrectly.
The mathematical model is known as a realization of a polyhedron of genus 3. The scientific description states that each pair of walls has a common edge, with 20 pairs sharing one edge, and eight pairs having two common edges.
The structure was described back in 2020.
Mizhaev says he came across the unusual property by accident while experimenting with computer-aided design software. The structure itself is not entirely new - he first described it in 2020.
The new development differs in that it assigns specific coordinates to all vertices and provides mathematical means for checking the figure. Some pairs of faces share two edges, which distinguishes the construction from the tetrahedron and the famous Silasy polyhedron, in which the faces are also adjacent to each other.
“This is more of a small part that we have been missing so far“, commented Lars Scheve from the University of Edinburgh. According to him, creating such figures is difficult, since there is no reliable universal method for finding them, and the systems of equations are often too complex to solve.
Artificial intelligence was used for verification
When developing the new version, Mizhaev used ChatGPT to check some of the calculations and write scripts. He specifies that the original model was created in 2020 - before it started using artificial intelligence.
Thus, the unusual polyhedron combines an abstract topological structure with a specific geometric description that can be verified through coordinates and equations.
Source: BTA