Topic
Ford circle
About: Ford circle is a research topic. Over the lifetime, 45 publications have been published within this topic receiving 505 citations.
Papers
TL;DR: In this paper, it is shown that the set of integer radii of curvatures that appear in an integral Apollonian circle packing has positive density, and in fact represents all sufficiently large integers not excluded by congruence conditions.
169 citations
TL;DR: The construction provides a generalization of the Farey series and the associated Ford circles and the quantities bx/ √ 2 and by are integers.
Abstract: 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 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 super-packing”. Many of the properties of the Descartes configuration and the standard Apollonian packing carry over to this case. In particular, there is an equation of degree 2 connecting the bends (curvatures) of a sextuple; all the bends can be integers; and if they are, the packing can be placed in the plane so that for each circle with bend b and center (x, y), the quantities bx/ √ 2 and by are integers. The construction provides a generalization of the Farey series and the associated Ford circles.
32 citations
1 Jan 1995
TL;DR: In this article, the authors give a short sketch of a relation between continued fractions and the hyperbolic geometry on the upper half plane, (the simple continued fractions case and a generalized case).
Abstract: A purpose of this paper is to give a short sketch of a relation between continued fractions and the hyperbolic geometry on the upper half plane, (the simple continued fractions case and a generalized case). Relations between continued fractions and the geodesic flows on the modular surface are well-known. For example, Adler and Flatto [1] showed that the continued fraction transformation is obtained as a cross-section map of the geodesic flow. Another interesting one is due to Moeckel [8], who proved a metrical property of continued fractions concerning to a distribution of digits by using the Farey tessellation and the ergodicity of geodesic flows.
30 citations
Posted Content•
TL;DR: In this paper, a formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found, and the general theorem for n-spheres is also considered.
Abstract: A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an analytic solution to the Apollonian problem. The general theorem for n-spheres is also considered.
26 citations
TL;DR: In this paper, the authors prove a conjecture of Mallows by giving a closed formula for the number of primitive root quadruples with minimum curvature -n, which is called strongly integral Apollonian circle packing.
15 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2020 | 3 |
| 2019 | 2 |
| 2018 | 4 |
| 2017 | 1 |
| 2016 | 2 |