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This article is available for purchase or by subscription. See below.
Abstract
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László Fejes Tóth and Aladár Heppes proposed the following
generalization of the kissing number problem. Given a ball in
,
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.
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Keywords
kissing number, densest packings, packings of congruent
balls
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Mathematical Subject Classification
Primary: 05B40, 52C15, 52C26
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Milestones
Received: 29 May 2022
Revised: 29 August 2022
Accepted: 13 September 2022
Published: 15 October 2022
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