Ruslan Mizhaev, a freelance researcher and design engineer, has created a geometric model of a genus-three polyhedron. This object is essentially a surface with three holes, composed of eight flat, nine-sided polygonal faces. It features a total of 24 vertices and 36 edges, with three faces meeting at each vertex.

"Every face is an adjacent 'neighbor' to the seven other faces. All combined, they form a closed surface with three 'handles'," Mizhaev shared with New Scientist. He posted his work on the online repository arXiv in September.

In topology, the study of properties of shapes that remain unchanged when stretched, twisted, or compressed, a "handle" is fundamentally a hole or tunnel through a surface. For instance, a donut can be described as having one handle.

This characteristic makes the new discovery reminiscent of tetrahedra and the well-known Szilassi polyhedron, both of which also feature all faces adjacent to one another. However, in Mizhaev's design, some pairs of faces share two edges instead of just one.

Mizhaev explained that he stumbled upon this property while experimenting with polyhedral surfaces using computer-aided design (CAD) software, rather than intentionally seeking it out. He first described the fundamental structure in 2020.

His latest work assigns integer coordinates to all 24 vertices, accompanied by equations that allow mathematicians to verify its accuracy. This ensures the faces are perfectly flat, the surface is correctly closed, and there are no unintended self-intersections—a common issue with complex shapes of this kind.

Lars Schewe, a mathematician at the University of Edinburgh, noted that this discovery is an interesting piece of a larger mathematical puzzle. Shapes like this do not overturn existing theories but are difficult to create due to the lack of reliable general methods. One approach involves establishing complex equations to define the position of each vertex and how the faces fit together, but Schewe believes this is practically impossible.

Therefore, researchers often rely on computer calculations or mathematical intuition. Mizhaev also mentioned using ChatGPT to help check some calculations and write Python code for the new version of the shape. However, he affirmed that the original design was developed in 2020, before he began using AI.

Typical images showing the results of integer calculations in Mizhaev's work. Photo: Ruslan Mizhaev

By Khanh Linh