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Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem

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New Trends in Intuitive Geometry

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 27))

Abstract

The problem of twelve spheres is to understand, as a function of \(r \in (0,r_{max}(12)]\), 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 \(3 \le N \le 14\). The problem of determining the maximal radius \(r_{max}(N)\) is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.

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Notes

  1. 1.

    “Iam si ad structuram solidorum quam potest fieri arctissimam progredaris, ordinesque ordinibus superponas, in plano prius coaptatos aut ii erunt quadrati A aut trigonici B: si quadrati aut singuli globi ordinis superioris singulis superstabunt ordinis inferioris aut contra singuli ordinis superioris sedebunt inter quaternos ordinis inferioris. Priori modo tangitur quilibit globis a quattuour cirucmstantibus in eodem plano, ab uno supra se, et ab uno infra se: et sic in universum a six aliis, eritque ordo cubicus, et compressione facta fient cubi: sed no erit arctissima coaptatio. Posteriori modo praeterquam quod quileibet globus a quattuor circumstantibus in eodem plano tangitur etiam a quattuor infra se, et a quattuor supra se, et sic in universum a duodecim tangetur; fientque compressione ex globosis rhombica. Ordo hic magis assimilabitur octahedro et pyramidi. Coaptatio fiet arctissima, ut nullo praetera ordine plures globuli in idem vas compingi queant.” [Translation: Colin Hardie [61, p. 15]]

  2. 2.

    “Esto enim B copula trium globorum. Ei superpone A unum pro apice; esto et alia copula senum globorum C, et alia denum D et alia quindenum E. Impone semper angustiorem latiori, ut fiat figura pyramidis. Etsi igitur per hanc impositionem singuli superiores sederunt into trinos inferiores: tamen iam versa figura, ut non apex sed integrum latus pyramidis sit loc superiori, quoties unum globulum deglberis e summis, infra stabunt quattuor ordine quadrato. Et rursum tangetur unus globus ut prius, et duodecim aliis, a sex nempe circumstantibus in eodem plano tribus supra et tribus infra. Ita in solida coaptatione arctissima non potest ess ordo triangularis sine quadrangulari, nec vicissim. Patet igitur, acinos punici mali, materiali necessitate concurrente cum rationaibus incrementi acinorum, exprimi in figuri rhombici corporis...” [Translation by Colin Hardie [61, p. 17]].

  3. 3.

    “Ut noscatur quot sunt stellae magnitudinis 1 ae, 2 dae, 3 ae & c. considerando quot spherae proximae, seundae ab his 3 ae & c. spheram in spatio trium dimensionis circumstent: erunt 13 primae, \(4 \times 13\) 2-dae, \(9 \times 4 \times 13\) 3 ae.”

  4. 4.

    This notebook is at Christ Church, Oxford, according to J. Leech  [66].

  5. 5.

    John Keill (1671–1721) succeeded Gregory as Savilian Professor.

  6. 6.

    This is Kepler [62].

  7. 7.

    Aprés Charles Radin

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Acknowledgements

The authors were each supported by ICERM in the Spring 2015 program on “Phase Transitions and Emergent Properties.” R. K. was also supported by the University of Pennsylvania Mathematics Department sabbatical visitor fund and by MSRI via NSF grant DMS-1440140. W. K. was also supported by Austrian Science Fund (FWF) Project 5503. J. L. was supported by NSF grant DMS-1401224 and by a Clay Senior Fellowship at ICERM. Part of the work of S. S. has been carried out in the framework of the Labex Archimede (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). Part of the work of S. S. has been carried out at IITP RAS. The support of Russian Foundation for Sciences (Project No. 14-50-00150) is gratefully acknowledged. The authors thank Bob Connelly, Sharon Glotzer, Mark Goresky, Tom Hales and Oleg Musin for helpful comments. Parts of Sect. 4.1 are adapted from unpublished notes by R. K. and John Sullivan (MSRI, 1994) about critical configurations of “electrons” on the sphere.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Rob Kusner

  2. Institute of Analysis and Number Theory, Graz University of Technology, 8010, Graz, Austria

    Wöden Kusner

  3. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Wöden Kusner

  4. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA

    Jeffrey C. Lagarias

  5. Skolkovo Institute of Science and Technology, Moscow, Russia

    Senya Shlosman

  6. Aix Marseille Université, Université de Toulon, CNRS, CPT, UMR 7332, 13288, Marseille, France

    Senya Shlosman

  7. Institute for Information Transmission Problems, RAS, Moscow, Russia

    Senya Shlosman

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Editors and Affiliations

  1. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gergely Ambrus

  2. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Imre Bárány

  3. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Károly J. Böröczky

  4. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gábor Fejes Tóth

  5. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    János Pach

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Kusner, R., Kusner, W., Lagarias, J.C., Shlosman, S. (2018). Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem. In: Ambrus, G., Bárány, I., Böröczky, K., Fejes Tóth, G., Pach, J. (eds) New Trends in Intuitive Geometry. Bolyai Society Mathematical Studies, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57413-3_10

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