OFFSET
1,1
COMMENTS
For a circle, curvature = 1/radius. The curvatures of a quarter, nickel, and dime are approximately proportional to 21, 24, and 28, respectively. Three mutually tangent circles with curvatures 21, 24, 28 can be inscribed in a circle of curvature 11.
Apollonius's and Descartes's Theorems say that, given three mutually tangent circles of curvatures a, b, c, there are exactly two circles tangent to all three, and their curvatures are a + b + c +- 2*sqrt(ab + ac + bc). (Here negative curvature of one of the two circles means that the three circles are inscribed in it.)
Fuchs (2009) says "An Apollonian circle packing ... is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well." That is because if a + b + c - 2s*qrt(ab + ac + bc) is an integer, then so is a + b + c + 2*sqrt(ab + ac + bc).
For n > 1, the n-th generation of the packing has 4*3^(n-2) circles.
Infinitely many of the curvatures are prime numbers A189227. In fact, in any integral Apollonian circle packing that is primitive (i.e., the curvatures have no common factor), the prime curvatures constitute a positive fraction of all primes (Bourgain 2012) and there are infinitely many pairs of tangent circles both of whose curvatures are prime (Sarnak 2007, 2011).
Fuchs and Sanden (2012) report on experiments with the nickel-dime-quarter Apollonian circle packing, which they call the coins packing P_C.
LINKS
D. Austin, When Kissing Involves Trigonometry, AMS feature column March 2006.
J. Bourgain, Integral Apollonian circle packings and prime curvatures, arXiv:1105.5127 [math.NT], 2011-2012.
J. Bourgain and A. Kontorovich, On the Strong Density Conjecture for Integral Apollonian Circle Packings, arXiv:1205.4416 [math.NT], 2012-2013. See figure 1.
S. Butler, R. Graham, G. Guettler and C. Mallows, Irreducible Apollonian configurations and packings, Discrete & Computational Geometry, 44 (2010), 487-507.
E. Fuchs, Arithmetic Properties of Apollonian Circle Packings, Ph.D. thesis 2009.
E. Fuchs and K. Sanden, Some experiments with integral Apollonian circle packings, Experiment. Math. 20 (2011), 380-399.
R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks, and C. H. Yan, Apollonian Circle Packings: Number Theory, J. Number Theory, 100 (2003), 1-45.
R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks, and C. H. Yan, Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group., Discrete & Computational Geometry, 34 (2005), no. 4, 547-585.
K. E. Hirst, The Apollonian Packing of Circles, J. London Math. Soc. s1-42(1) (1967), 281-291.
E. Kasner and F. Supnick, The Apollonian packing of circles, Proc. Nat. Acad. Sci. U.S.A. 29 (1943), 378-384.
A. Kontorovich, From Apollonius to Zaremba: Local-global phenomena in thin orbits, Bull. Amer. Math. Soc., 50 (2013), 187-228.
J. C. Lagarias, C. L. Mallows, and Allan Wilks, Beyond the Descartes Circle Theorem, Amer. Math Monthly, 109 (2002), 338-361.
L. Levine, W. Pegden, C. K. Smart, Apollonian Structure in the Abelian Sandpile, arXiv:1208.4839 [math.AP], 2012-2014.
D. Mackenzie, A Tisket, a Tasket, an Apollonian Gasket, American Scientist, 98 (2010).
C. L. Mallows, Growing Apollonian Packings, J. Integer Sequences, 12 (2009), article 09.2.1.
I. Peterson, Circle game, Science News, 4/21/01.
I. Peterson, Temple circles, Math Trek, 4/23/01.
P. Sarnak, Letter to Lagarias on integral Apollonian packings, June, 2007.
P. Sarnak, Integral Apollonian packings, MAA Lecture, Jan 2009.
P. Sarnak, Integral Apollonian packings, Amer. Math. Monthly, 118 (2011), 291-306.
K. E. Stange, The sensual Apollonian circle packing, arXiv:1208.4836 [math.NT], 2012-2014.
Wikipedia, Integral Apollonian circle packings
Wikipedia, Descartes' theorem
FORMULA
a(n) == 0, 4, 12, 13, 16, or 21 (mod 24).
EXAMPLE
The 1st-generation curvatures are -11, 21, 24, 28, the 2nd are 40, 52, 61, 157, and the 3rd are 76, 85, 96, 117, 120, 132, 181, 213, 237, 376, 388, 397. The 4th generation begins 132, 156, 160, 189, 204, 205, 216, ....
As 21 + 24 + 28 +- 2*sqrt(21*24 + 21*28 + 24*28) = 157 or -11, the sequence begins -11, 21, 24, 28, ... and 157 is in it.
The primes 157 and 397 are the curvatures of two circles that are tangent.
MATHEMATICA
root = {-11, 21, 24, 28};
triples = Subsets[root, {3}];
a = {root};
Do[
ng = Table[Total@t + 2 Sqrt@Total[Times @@@ Subsets[t, {2}]], {t, triples}];
AppendTo[a, Sort@ng];
triples = Join @@ Table[{t, r} = tr; Table[Append[p, r], {p, Subsets[t, {2}]}], {tr, Transpose@{triples, ng}}]
, {k, 3}];
Flatten@a (* Andrey Zabolotskiy, May 29 2022 *)
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Jonathan Sondow, Apr 18 2011
EXTENSIONS
Terms a(28) and beyond from Andrey Zabolotskiy, May 29 2022
STATUS
approved