Apollonian circle packings: number theory
نویسندگان
چکیده
منابع مشابه
Apollonian Circle Packings: Number Theory
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 studies number-theoretic properties of the set of integer curvatures appearing in such packings. Ea...
متن کاملApollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In Euclidean space 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. There are infinitely many different integral packings; these were studied in the paper [8]....
متن کاملBeyond Apollonian Circle Packings: Expander Graphs, Number Theory and Geometry
In the last decade tremendous effort has been put in the study of the Apollonian circle packings. Given the great variety of mathematics it exhibits, this topic has attracted experts from different fields: number theory, homogeneous dynamics, expander graphs, group theory, to name a few. The principle investigator (PI) contributed to this program in his PhD studies. The scenery along the way fo...
متن کاملApollonian Circle Packings : Geometry and Group Theory
Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)×(center) is an integer vector. This series of papers explain such...
متن کاملApollonian Circle Packings
Figure 1: An Apollonian Circle Packing Apollonius’s Theorem states that given three mutually tangent circles, there are exactly two circles which are tangent to all three. Apollonian circle packings are produced by repeating the construction of mutually tangent circles to fill all remaining spaces. A remarkable consequence of Descartes’ Theorem is if the initial four tangent circles have integr...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2003
ISSN: 0022-314X
DOI: 10.1016/s0022-314x(03)00015-5