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On the maximum size packings of disks with kissing radius 3

### Alexander Golovanov

Vol. 11 (2022), No. 3, 263–286
##### Abstract

László Fejes Tóth and Aladár Heppes proposed the following generalization of the kissing number problem. Given a ball in ${ℝ}^{d}$, consider a family of balls touching it, and another family of balls touching the first family. Find the maximal possible number of balls in this arrangement, provided that no two balls intersect by interiors, and all balls are congruent. They showed that the answer for disks on the plane is 19. They also conjectured that if there are three families of disks instead of two, the answer is 37. We confirm this conjecture.

##### Keywords
kissing number, densest packings, packings of congruent balls
##### Mathematical Subject Classification
Primary: 05B40, 52C15, 52C26