Fixed a math problem that had been looking for a solution for 70 years

Fixed a math problem that had been looking for a solution for 70 years

What is the maximum number of equiangular lines in a space with many dimensions? Today, a group from MIT answers the mathematical question that has been debating for almost 70 years

(photo: Gerd Altmann via Pixabay) Mathematics - or rather, mathematicians - are striking again. A team from MIT (Massachusetts Institute of Technology) has solved a geometry problem that has been looking for a solution for almost 70 years. The problem concerns equiangular lines and today researchers have determined for the first time how many lines of this kind can exist in a multi-dimensional space and have shown that the number is not infinite. The result fits into the so-called "graph theory", mathematical objects that are used in real life to schematize different processes, circuits, computer networks and maps. The study will be published in January 2022 in Annals of Mathematics and is now available in pre-print on arXiv. The mathematics of equiangular lines has many practical applications for algorithms, in the field of information technology and communications.

What are equiangular lines

A set of lines, in 2, 3 ( as in the image below) or more dimensions, they are equiangular when they all intersect at the same point and at the same time form, when considered in pairs, equal angles. Establishing the maximum number of equiangular lines in a multi-dimensional space has been discussed for decades and never resolved. If we start from a 2-dimensional space, we know that these lines are at most 3.

For example, taking a regular hexagon, the straight lines which, if considered in pairs, intersect at the same point and form equal angles are at most 3 and are the diagonals each at an angle of 120 ° from the others two. While in 3D, in this case in an icosahedron (the 3D solid represented in the image, polyhedron with 20 faces which are equilateral triangles) these lines are at most 6.

Icosahedron in purple with the 6 lines at the same time equiangular represented in red (photo: Zilin Jiang)

The problem in a multi-dimensional space

Problems arise when we increase the dimensions. The calculation is not simple, so much so that mathematicians have been wondering for years what the solution is. “In [a space a - ed] many dimensions things become really interesting - says Yufei Zhao, math teacher at MIT, co-author of the work - and the possibilities can seem endless”, even if they are not. The first formulation of this argument is due to the mathematicians Lemmens and Seidel in a study of 1973, even if the question had been talked about for at least a decade. In the following decades, several good ideas and hypotheses have accumulated, although everything has stood still until the progress made by the mathematician of the Federal Institute of Technology in Zurich Benny Sudakov, illustrated in 2018.

The theory of graphs is to help

Today the MIT group goes even further and finds a possible valid solution to the problem of equiangular lines, faced with graph theory. In the field of mathematics, computer science and combinatorial geometry, this theory studies graphs, discrete (mathematical) objects that allow the representation of various processes with models and are useful for algorithms and for representing circuits, geographic networks and maps. For example, the problem of the seven Königsberg bridges was formulated by Euler with graphs.

In this case we use the spectral theory of graphs, which provides the mathematical tools to study networks and which has allowed the development of important algorithms such as Google PageRank at the base of the search engine. The study and understanding of the problem of equiangular lines, carried out today by the mathematicians of MIT, has significant implications in the field of communications and programming in the computer sector.

The solution of the problem

Following a complex theoretical elaboration, the researchers provide a formula for the maximum number of lines taken in pairs forming the same angle. If d is the dimension of the space and it is large enough, the formula (to be deepened directly on the paper) is the following:

Nα (d) = ⌊k (d − 1) / (k − 1) ⌋

where α is a parameter fixed between 0 and 1 and k the minimum number of vertices (not infinite) of a particular graph, whose characteristics the authors explain in the text. There are also formulas that apply in other situations.

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