TY - CHAP
T1 - Configuration spaces of equal spheres touching a given sphere
T2 - The twelve spheres problem
AU - Kusner, Rob
AU - Kusner, Wöden
AU - Lagarias, Jeffrey C.
AU - Shlosman, Senya
PY - 2018/1/1
Y1 - 2018/1/1
N2 - The problem of twelve spheres is to understand, as a function of r ϵ (0,rmax(12)], the configuration space of 12 non-overlapping equal spheres of radius r touching a central unit sphere. It considers to what extent, and in what fashion, touching spheres can be varied, subject to the constraint of always touching the central sphere. Such constrained motion problems are of interest in physics and materials science, and the problem involves topology and geometry. This paper reviews the history of work on this problem, presents some new results, and formulates some conjectures. It also presents general results on configuration spaces of N spheres of radius r touching a central unit sphere, with emphasis on 3 ≤ N ≤ 14. The problem of determining the maximal radius rmax(N) is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.
AB - The problem of twelve spheres is to understand, as a function of r ϵ (0,rmax(12)], the configuration space of 12 non-overlapping equal spheres of radius r touching a central unit sphere. It considers to what extent, and in what fashion, touching spheres can be varied, subject to the constraint of always touching the central sphere. Such constrained motion problems are of interest in physics and materials science, and the problem involves topology and geometry. This paper reviews the history of work on this problem, presents some new results, and formulates some conjectures. It also presents general results on configuration spaces of N spheres of radius r touching a central unit sphere, with emphasis on 3 ≤ N ≤ 14. The problem of determining the maximal radius rmax(N) is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.
KW - Configuration spaces
KW - Constrained optimization
KW - Criticality
KW - Discrete geometry
KW - Materials science
KW - Morse theory
UR - http://www.scopus.com/inward/record.url?scp=85056318530&partnerID=8YFLogxK
U2 - 10.1007/978-3-662-57413-3_10
DO - 10.1007/978-3-662-57413-3_10
M3 - Chapter
AN - SCOPUS:85056318530
T3 - Bolyai Society Mathematical Studies
SP - 219
EP - 277
BT - Bolyai Society Mathematical Studies
PB - Springer Berlin - Heidelberg
ER -