Delimiting Maximal Kissing Configurations in Four Dimensions

TL;DR

This paper explores the maximum arrangements of four-dimensional spheres touching a central sphere, using a novel physical approach to delimit possible configurations and investigate the uniqueness of the 24-cell structure.

Contribution

Introduces a physical method utilizing the Hopf map to delimit potential kissing configurations in four dimensions, advancing understanding of sphere packings beyond known solutions.

Findings

Delimited possible kissing configurations in four dimensions.

Provided evidence regarding the uniqueness of the 24-cell configuration.

First physical approach to explore four-dimensional kissing arrangements.

Abstract

How many unit $n-$dimensional spheres can simultaneously touch or kiss a central $n-$dimensional unit sphere? Beyond mathematics this question has implications for fields such as cryptography and the structure of biologic and chemical macromolecules. The kissing number is only known for dimensions 1-4, 8 and 24 (2, 6, 12, 24, 240, 19650, respectively) and only particularly obvious for dimensions one and two. Indeed, in four dimensions it is not even known if Platonic polytope unique to that dimension known as the 24-cell is the unique kissing configuration. We have not been able to prove that the 24-cell is unique, but, using a physical approach utilizing the hopf map from four to three dimensions, we for the first time delimit the possible other configurations which could be kissing in four dimensions.