A scientist has discovered a strange new eight-faced shape in which every face touches every other face, using a set of simple integer coordinates.

Mathematicians can define abstract patterns, describing which vertices, edges and faces should connect. But working out whether those rules can actually be turned into an object in three-dimensional space, without parts intersecting where they shouldn’t, can be extremely difficult.

Ruslan Mizhaev, an independent researcher and design engineer, has produced a geometric realisation of a genus-3 polyhedron – essentially a surface with three holes – made from eight flat, nine-sided faces. It has 24 vertices and 36 edges, with three faces meeting at every vertex.

“There are eight flat, polygonal faces, and each is a neighbour of all seven others along an edge,” says Mizhaev. “Together, they form a closed surface with three handles.”

In topology – the study of a shape’s properties that can’t be changed by stretching, twisting or squeezing the shape – a handle is essentially a hole or tunnel through a surface. A doughnut, for example, has one handle.

That makes the object reminiscent of a tetrahedron and the better-known Szilassi polyhedron, which also have faces that are all neighbours. However, in Mizhaev’s design, some pairs of faces share two edges, rather than exactly one.

Mizhaev says he stumbled across the property, rather than setting out to find it, while experimenting with polyhedral surfaces in computer-aided design (CAD) software. The underlying structure isn’t entirely new – it is something Mizhaev first described it in 2020.

But the new work gives all 24 vertices integer coordinates, alongside equations that allow other mathematicians to check the shape, ensuring that the faces are flat, the surface closes properly and no parts accidentally intersect (which can be a common issue in shapes like this).

The discovery is an interesting one, but should be seen as a piece of a larger puzzle, rather than an entirely new innovation, says Lars Schewe at the University of Edinburgh, UK. “It’s more a small piece that we didn’t have before,” says Schewe. The result doesn’t overturn existing conjectures, he says, but “constructing these has always been a difficult task”.

The reason it is so difficult to construct such shapes is that there is no reliable way to do so. One brute-force approach involves writing down a huge set of equations specifying where every vertex can sit and how all the faces must fit together, says Schewe – but that approach can rapidly become practically impossible to solve.

Researchers instead often depend on computer searches, geometric intuition or even physical models. Mizhaev used ChatGPT to help with some verification calculations and to write Python scripts for the new version of his shape. But he says the original shape was developed in 2020, before he began using ChatGPT.

For Schewe, the gap between describing which parts of a shape can connect and actually building it is what makes these problems interesting. “I can tell you these are the rules,” he says. “But when I try to build it and I try to draw it, I can’t do it.”