A scientist has discovered a strange new octahedral shape where each face touches all the others, writes the magazine "New Scientist". To do this, he used a set of simple integer coordinates, BTA explains.
Mathematicians can define abstract models that describe which vertices, edges, and faces should be connected. Determining whether these rules can actually be translated 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 created a geometric realization of a polyhedron of genus 3 - essentially a three-holed surface made up of eight flat nine-sided faces. It has 24 vertices and 36 edges, with three faces meeting at each vertex.
Mathematician discovers rare eight-faced shape
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It isn’t easy to demonstrate that a new geometric shape can actually exist in the real world, but one researcher argues that the genus-3 polyhedron canhttps://t.co/DhbYmtziOchttps://t.co/XzeXswxAfg pic.twitter.com/cyyLMd9jRH
"We have eight flat, polygonal walls, each of which borders the other seven along the length of the edge. Together they form a closed surface with three handles", says Mizhaev.
In topology - the science that studies the properties of shapes that cannot be changed by stretching, twisting or squeezing, a handle is essentially an opening or tunnel through a surface. For example, the donut has one handle.
This makes the object resemble a tetrahedron and the more famous Silasy polyhedron, which also have adjacent faces. However, in Mizhaev's design, some pairs of faces share two edges, instead of exactly one.
Ruslan Mizhaev says he came across this property by accident, rather than by looking for it purposefully, while experimenting with polyhedra in computer-aided design (CAD) software. The structure itself is not entirely new - Mizhaev first described it in 2020.
However, the new development assignsinteger coordinates to all 24 vertices, as well as equations that allow other mathematicians to verify the shape.
The discovery is interesting, but should be seen as part of a larger puzzle, not something completely new, says Lars Scheve of the University of Edinburgh, UK. “It’s more of a small piece that we’ve been missing,” he says.
The reason it’s so hard to create such figures is that there’s no reliable way to do it.
One approach, for example, involves writing out a huge set of equations, but that can quickly become virtually impossible to solve, Scheve notes.
Instead, researchers often rely 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 figure. However, he notes that the original figure was developed in 2020, before he started using ChatGPT, according to New Scientist.