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Apollonian circle packings:number theory
, 2000
"... Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper st ..."
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Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper
A generalization of Apollonian packing of circles
 Journal of Combinatorics
"... Three circles touching one another at distinct points form two curvilinear triangles. Into one of these we can pack three new circles, touching each other, with each new circle touching two of the original circles. In such a sextuple of circles there are three pairs of circles, with each of the cir ..."
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of the circles in a pair touching all four circles in the other two pairs. Repeating the construction in each curvilinear triangle that is formed results in a generalized Apollonian packing. We can invert the whole packing in every circle in it, getting a \generalized Apollonian superpacking". Many
Integral Apollonian Packings
"... Abstract. We review the construction of integral Apollonian circle packings. There are a number of Diophantine problems that arise in the context of such packings. We discuss some of them and describe some recent advances. 1. AN INTEGRAL PACKING. The quarter, nickel, and dime in Figure 1 are placed ..."
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Abstract. We review the construction of integral Apollonian circle packings. There are a number of Diophantine problems that arise in the context of such packings. We discuss some of them and describe some recent advances. 1. AN INTEGRAL PACKING. The quarter, nickel, and dime in Figure 1 are placed
Apollonian Circle Packings: Geometry and Group Theory II. SuperApollonian Group and Integral Packings
, 2006
"... Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain, where a Descartes configuration is a set of four mutually tangent circles ..."
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Cited by 31 (4 self)
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circles in the Riemann sphere, having disjoint interiors. Part I showed there exists a discrete group, the Apollonian group, acting on a parameter space of (ordered, oriented) Descartes configurations, such that the Descartes configurations in a packing formed an orbit under the action of this group
FROM APOLLONIAN PACKINGS TO HOMOGENEOUS SETS
"... Abstract. We extend fundamental results concerning Apollonian packings, which constitute a major object of study in number theory, to certain homogeneous sets that arise naturally in complex dynamics and geometric group theory. In particular, we give an analogue of D. W. Boyd’s theorem (relating th ..."
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Abstract. We extend fundamental results concerning Apollonian packings, which constitute a major object of study in number theory, to certain homogeneous sets that arise naturally in complex dynamics and geometric group theory. In particular, we give an analogue of D. W. Boyd’s theorem (relating
Geometric Sequences Of Discs In The Apollonian Packing
 ENGLISH VERSION: ST. PETERSBURG MATH. J
, 1997
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APOLLONIAN STRUCTURE IN THE ABELIAN SANDPILE
, 2012
"... We state a conjecture relating integervalued superharmonic functions on Z² to an Apollonian circle packing of R². The conjecture is motivated by the Abelian sandpile process, which evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing on ..."
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We state a conjecture relating integervalued superharmonic functions on Z² to an Apollonian circle packing of R². The conjecture is motivated by the Abelian sandpile process, which evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing
Apollonian Equilateral Triangles
, 2012
"... Given an equilateral triangle with a the square of its side length and a point in its plane with b, c, d the squares of the distances from the point to the vertices of the triangle, it can be computed that a, b, c, d satisfy 3(a2+b2+ c2+d2) = (a+b+ c+d)2. This paper derives properties of quadruples ..."
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. We examine in detail the triangle group, the group with these operations as generators, and completely classify the orbits of quadruples with respect to the triangle group action. We also compute the number of triangle quadruples generated after a certain number of operations and approximate
Results 1  10
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