Mathematicians from Kazakhstan have improved a long-standing world result in one of the fundamental problems of discrete geometry, surpassing a benchmark that had remained unbeaten for more than half a century. Rustem Takhanov, Assistant Professor at Nazarbayev University, together with Stanislav Yun, NU Master’s student, and Zhenisbek Assylbekov, former Professor at the NU School of Sciences and Humanities, constructed a configuration of 841 equal spheres in 12-dimensional space, improving on a record established in 1971 by British mathematicians John Leech and Neil Sloane.
The result concerns the famous kissing number problem, which asks how many non-overlapping unit spheres can simultaneously touch a central unit sphere. In 12 dimensions, the best known lower bound had remained at 840 spheres for 55 years. The Kazakhstani researchers succeeded in increasing that number to 841.
The work began in January 2026. By analyzing the existing 840-sphere configuration, the team discovered that it possessed greater structural flexibility than previously recognized. This insight enabled them to reorganize the arrangement and construct a new configuration that satisfies all geometric constraints while accommodating one additional sphere.
“In short, this breakthrough reflects a new era in mathematical research, where human intuition is amplified by rapid computational experimentation using vibe coding and by the growing computational power of graphics processing units (GPUs),” Takhanov said.
The study has been published in the open-access research repository arXiv. Following the publication, Henry Cohn, a leading expert on sphere packings and spherical codes at the Massachusetts Institute of Technology, updated the international tables of kissing numbers, establishing 841 as the new lower bound for 12-dimensional space.
The kissing number problem is one of the classical challenges of discrete geometry. Beyond its theoretical importance, research on sphere packings and related structures has applications in coding theory and information transmission, where high-dimensional arrangements help improve error detection and correction methods.
The achievement has attracted additional attention because similar configurations have recently been explored by advanced artificial intelligence systems, which had not succeeded in finding the new result. The work represents one of the most significant mathematical advances by researchers from Kazakhstan in recent years and improves a global benchmark that had stood unchanged since 1971.
“I had always wanted to make a contribution to pure geometry. When I learned that Google’s AlphaEvolve agent had improved the kissing number record in 11 dimensions, I became seriously interested in the problem. Around the same time, my research grant enabled the purchase of six GPUs, and I was eager to see what they could do,” Takhanov said.
Rustem Takhanov was born in Zhezkazgan, Kazakhstan. He graduated from the Moscow Institute of Physics and Technology (MIPT) and earned his PhD at the Computing Center of the Russian Academy of Sciences under the supervision of Academician Konstantin Rudakov. His academic training at the intersection of mathematics, physics, and computer science shaped the interdisciplinary approach that characterizes his research. After spending four and a half years as a postdoctoral researcher at European universities, he joined the Department of Mathematics, School of Sciences and Humanities (SSH), Nazarbayev University, as an assistant professor in 2015, where he continues his research today.
