Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. Dense Sphere Packings
  • Cited by 51
Publisher:
Cambridge University Press
Online publication date:
October 2012
Print publication year:
2012
Online ISBN:
9781139193894
DOI:
https://doi.org/10.1017/CBO9781139193894
Subjects:
Mathematics (general), Mathematics, Algorithmics, Complexity, Computer Algebra, Computational Geometry, Computer Science, Geometry and Topology
Series:
London Mathematical Society Lecture Note Series (400)
Information

Book description

The 400-year-old Kepler conjecture asserts that no packing of congruent balls in three dimensions can have a density exceeding the familiar pyramid-shaped cannonball arrangement. In this book, a new proof of the conjecture is presented that makes it accessible for the first time to a broad mathematical audience. The book also presents solutions to other previously unresolved conjectures in discrete geometry, including the strong dodecahedral conjecture on the smallest surface area of a Voronoi cell in a sphere packing. This book is also currently being used as a blueprint for a large-scale formal proof project, which aims to check every logical inference of the proof of the Kepler conjecture by computer. This is an indispensable resource for those who want to be brought up to date with research on the Kepler conjecture.

Reviews

'… interesting and unusual book … beautifully written and is full of interesting historical notes. Moreover, each chapter is equipped with a very helpful summary, and many technical arguments are accompanied by a conceptual informal discussion. The book also features a detailed index and a nice bibliography. It is bound to become an indispensable resource for anyone wishing to study Kepler's conjecture.'

Source: Zentralblatt MATH

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
References
References
[1] Aristotle., On the heaven. translated by J.L., Stocks, http://classics.mit.edu/Aristotle/heavens.html, 350BC.
[2] A., Barvinok. A Course in Convexity, volume 54 of Graduate Studies in Mathematics. American Mathematical Society, 2002.
[3] J., Beery and J., Stedall, editors. Thomas Harriot's Doctrine of Triangular Numbers: the ‘Magisteria Magna’. European Math. Soc., 2008.
[4] K., Bezdek. On a stronger form of Rogers' lemma and the minimum surface area of Voronoi cells in unit ball packings. J. Reine Angew. Math., 518:131–143, 2000.
[5] K., Bezdek and E., Daróczy-Kiss. Finding the best face on a Voronoi polyhedron – the strong dodecahedral conjecture revisited. Monatshefte für Mathematik, 145:145–2006, 2005.
[6] M., Bhubaneswar. Computational Real Algebraic Geometry. CRC Press, 1997.
[7] N., Bourbaki. Elements of Sets. Addison-Wesley, Boston MA, 1968.
[8] W., Casselman. Packing pennies in the plane, an illustrated proof of Kepler's conjecture in 2d. AMS Feature Column Archive, http://www.ams.org/featurecolumn/archive/cass1.html, December 2000.
[9] J. H., Conway and N. J. A., Sloane. What are all the best sphere packings in low dimensions?DCG, 13:13–403, 1995.
[10] A., Doxiadis and C. H., Papadimitriou. Logicomix An Epic Search for Truth. Worzalla Publishing, 2009.
[11] L., Euler. Variae speculationes super area triangulorum sphaericorum. N. Acta Ac. Petrop., pages 47–62, 1797. E698 (Erneström index).
[12] L. Fejes, Tóth. Lagerungen in der Ebene auf der Kugel und im Raum. Springer-Verlag, Berlin-New York, first edition, 1953.
[13] W., Fulton. Introduction to Toric Varieties. Princeton University Press, Princeton NJ, 1993.
[14] C. F., Gauss. Untersuchungen über die Eigenscahften der positiven ternären quadratischen Formen von Ludwig August Seber. Göttingische gelehrte Anzeigen, July 1831. also published in J. Reine Angew. Math. 20 (1840), 312–320, and Werke, vol. 2, Königliche Gesellschaft der Wissenschaften, Göttingen, 1876, 188–196.
[15] G., Gonthier. A computer-checked proof of the four colour theorem. Unpublished manuscript, 2005.
[16] G., Gonthier. Formal proof – the four colour theorem. Notices of the AMS, 55(11):1382–1393, December 2008.
[17] B., Gracián. The art of worldly wisdom. Frederick Ungar Publishing, New York NY, 1967.
[18] T. C., Hales. The sphere packing problem. In Journal of Computational and Applied Math, volume 44, pages 41–76, 1992.
[19] T. C., Hales. Cannonballs and honeycombs. Notices of the AMS, 47(4):440–449, 2000.
[20] T. C., Hales. An overview of the Kepler conjecture. Discrete and Computational Geometry, 36(1):5–20, 2006.
[21] T. C., Hales. The Flyspeck Project, 2012. http://code.google.com/p/flyspeck.
[22] T. C., Hales and S. P., Ferguson. The Kepler conjecture. Discrete and Computational Geometry, 36(1):1–269, 2006.
[23] T. C., Hales and S., McLaughlin. A proof of the dodecahedral conjecture. Journal of the AMS, 23:23–344, 2010. http://arxiv.org/abs/math/9811079.
[24] T. C., Hales, J., Harrison, S., McLaughlin, T., Nipkow, S., Obua, and R., Zumkeller. A revision of the proof of the Kepler Conjecture. DCG, 2009.
[25] J., Harrison. Formalizing an analytic proof of the prime number theorem. Journal of Automated Reasoning, 43:43–261, 2009.
[26] J., Harrison. The HOL Light theorem prover, 2010. http://www.cl.cam.ac.uk/~jrh13/hol-light/index.html.
[27] R., Kargon. Atomism in England from Hariot to Newton. Clarendon Press, Oxford, 1966.
[28] J., Kepler. The Six-cornered snowflake. Oxford Clarendon Press, 1966. forward by L. L., Whyte.
[29] J., Lanier. You are not a gadget. Alfred A. Knopf, New York, 2010.
[30] J., Leech. The problem of the thirteen spheres. In Mathematical Gazette, pages 22–23, February 1956.
[31] C., Marchal. Study of the Kepler's conjecture: the problem of the closest packing. Mathematische Zeitschrift, December 2009.
[32] O. R., Musin and A. S., Tarasov. The strong thirteen spheres problem. preprint http://arxiv.org/abs/1002.1439, February 2010.
[33] T., Nipkow, G., Bauer, and P., Schultz. Flyspeck I: Tame Graphs. In Ulrich, Furbach and Natarajan, Shankar, editors, International Joint Conference on Automated Reasoning, volume 4130 of Lect. Notes in Comp. Sci., pages 21–35. Springer-Verlag, 2006.
[34] S., Obua. Proving bounds for real linear programs in Isabelle/HOL. In J., Hurd and T. F., Melham, editors, Theorem Proving in Higher Order Logics, volume 3603 of Lect. Notes in Comp. Sci., pages 227–244. Springer-Verlag, 2005.
[35] K., Plofker, January 2000. private communication.
[36] C. A., Rogers. The packing of equal spheres. Journal of the London Mathematical Society, 3/8:8–620, 1958.
[37] K., Schütte and B. L., van der Waerden. Auf welcher Kugel haben 5, 6, 7, oder 9 Punkte mit Mindestabstand Eins Platz. Math. Annalen, 123:123–124, 1951.
[38] K., Schütte and B. L., van der Waerden. Das Problem der dreizehn Kugeln. Math. Annalen, 125:125–334, 1953.
[39] W., Shirley. Thomas Harriot: a biography. Oxford, 1983.
[40] K. S., Shukla. The Āryabhaṭīya of Āryabhaṭa with the Commentary of Bhāskara I and Someśvara. New Delhi: Indian National Science Academy, 1976.
[41] A., Solovyev and T. C., Hales. Efficient formal verification of bounds of linear programs, volume 6824 of LNCS, pages 123–132. Springer-Verlag, 2011.
[42] J., Spolsky. Joel on Software. Apress, 2004.
[43] G. G., Szpiro. Kepler's Conjecture: How Some of the Greatest Minds in History Helped to Solve one of the Oldest Math Problems in the Worlds. John Wiley and Sons, New York NY, 2003.
[44] A., Tarski. A decision method for elementary algebra and geometry. University of California Press, Berkeley and Los Angeles, Calif., 1951. 2nd ed.
[45] A., Thue. Om nogle geometrisk taltheoretiske theoremer. Forandlingerneved de Skandinaviske Naturforskeres, 14:14–353, 1892.
[46] A., Thue. über die dichteste Zusammenstellung von kongruenten Kreisen in der Ebene. Christinia Vid. Selsk. Skr., 1:1–9, 1910.
[47] W. T., Tutte. Graph Theory. Encyclopedia of Mathematics and Its Applications. Addison-Wesley Publishing, 1984.
[48] R. J., Webster. Convexity. Oxford University Press, 1994.
[49] F., Wiedijk. Jordan curve theorem. http://www.cs.ru.nl/~freek/jordan/index.html, referenced 2010.
[50] F., Wiedijk. Formalizing 100 theorems. http://www.cs.ru.nl/~freek/100/, referenced 2012.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.