Abstract
In this article, using the computer, we enumerate all locally-rigid packings by N congruent circles (spherical caps) on the unit sphere \( \mathbb{S} \) 2 with N < 12. This is equivalent to the enumeration of irreducible spherical contact graphs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Aigner and G. M. Ziegler, Proofs from THE BOOK, Springer, Berlin (1998, 2002).
K. Anstreicher, “The thirteen spheres: A new proof,” Discrete Comput. Geom., 31, 613–625 (2004).
A. Barg and O. R. Musin, “Codes in spherical caps,” Adv. Math. Commun., 1, 131–149 (2007).
K. Böröczky, “The problem of Tammes for n = 11,” Stud. Sci. Math. Hung., 18 165–171 (1983).
K. Böröczky, “The Newton–Gregory problem revisited,” in: A. Bezdek, ed., Discrete Geometry, Marcel Dekker, New York (2003), pp. 103–110.
K. Böröczky and L. Szabó, “Arrangements of 13 points on a sphere,” in: A. Bezdek, ed., Discrete Geometry, Marcel Dekker, New York (2003), pp. 111–184.
K. Böröczky and L. Szabó, “Arrangements of 14, 15, 16 and 17 points on a sphere,” Stud. Sci. Math. Hung., 40, 407–421 (2003).
P. Boyvalenkov, S. Dodunekov, and O. R. Musin, “A survey on the kissing numbers,” Serdica Math. J., 38, 507–522 (2012).
P. Brass, W. O. J. Moser, and J. Pach, Research Problems in Discrete Geometry, Springer, Berlin (2005).
G. Brinkmann and B. D. McKay, Fast Generation of Planar Graphs (expanded edition), http://cs.anu.edu.au/~bdm/papers/plantri-full.pdf.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, Springer, New York (1999).
A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, “Jamming in hard sphere and disk packings,” J. Appl. Phys., 95, 989–999 (2004).
L. Danzer, “Finite point–sets on S 2 with minimum distance as large as possible,” Discrete Math., 60, 3–66 (1986).
L. Fejes Tóth, “Über die Abschätzung des kürzesten Abstandes zweier Punkte eines auf einer Kugelfläche liegenden Punktsystems,” Jber. Deutsch. Math. Verein., 53, 66–68 (1943).
L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und in Raum, Springer, Berlin (1953).
W. Habicht und B. L. van der Waerden, “Lagerungen von Punkten auf der Kugel,” Math. Ann., 123, 223–234 (1951).
A. B. Hopkins, F. H. Stillinger, and S. Torquato, “Densest local sphere—packing diversity: General concepts and application to two dimensions,” Phys. Rev. E, 81, 041305 (2010).
W.-Y. Hsiang, Least Action Principle of Crystal Formation of Dense Packing Type and Kepler’s Conjecture, World Scientific, New York (2001).
J. Leech, “The problem of the thirteen spheres,” Math. Gazette, 41, 22–23 (1956).
H. Maehara, “Isoperimetric theorem for spherical polygons and the problem of 13 spheres,” Ryukyu Math. J., 14, 41–57 (2001).
H. Maehara, “The problem of thirteen spheres—a proof for undergraduates,” Eur. J. Combin., 28, 1770–1778 (2007).
O. R. Musin, “The problem of the twenty–five spheres,” Russ. Math. Surv., 58, 794–795 (2003).
O. R. Musin, “The kissing problem in three dimensions,” Discrete Comput. Geom., 35, 375–384 (2006).
O. R. Musin, “The one–sided kissing number in four dimensions,” Period. Math. Hung., 53, 209–225 (2006).
O. R. Musin, “The kissing number in four dimensions,” Ann. Math., 168, 1–32 (2008).
O. R. Musin, “Bounds for codes by semidefinite programming,” Proc. Steklov Inst. Math., 263, 134–149 (2008).
O. R. Musin, “Positive definite functions in distance geometry,” in: European Congress of Mathematics, Amsterdam, 14–18 July, 2008, EMS Publ. (2010), pp. 115–134.
O. R. Musin and A. S. Tarasov, “The strong thirteen spheres problem,” Discrete Comput. Geom., 48, 128–141 (2012).
plantri and fullgen, http://cs.anu.edu.au/~bdm/plantri/.
R. M. Robinson, “Arrangement of 24 circles on a sphere,” Math. Ann., 144, 17–48 (1961).
K. Schütte and B. L. van der Waerden, “Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand 1 Platz?” Math. Ann., 123, 96–124 (1951).
K. Schütte and B. L. van der Waerden, “Das Problem der dreizehn Kugeln,” Math. Ann., 125, 325–334 (1953).
R. M. L. Tammes, “On the origin number and arrangement of the places of exits on the surface of pollengrains,” Rec. Trv. Bot. Neerl., 27, 1–84 (1930).
B. L. van der Waerden, “Punkte auf der Kugel. Drei Zusätze,” Math. Ann., 125, 213–222 (1952).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 2, pp. 125–145, 2013.
Rights and permissions
About this article
Cite this article
Musin, O.R., Tarasov, A.S. Enumeration of Irreducible Contact Graphs on the Sphere. J Math Sci 203, 837–850 (2014). https://doi.org/10.1007/s10958-014-2174-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-2174-7