Apollonian tiling, the Lorentz group, and regular trees.
نویسنده
چکیده
The Apollonian tiling of the plane into circles is analyzed with respect to its group properties. The relevant group, which is non-compact and discrete, is found to be identical to the symmetry group of a particular geometric tree-graph in hyperbolic three-space. A linear recursive method to compute the radii is obtained. Certain modiications of the problem are investigated, and relations to other problems, such as the universal scaling of circle-maps, are pointed out.
منابع مشابه
Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions
This paper gives n-dimensional analogues of the Apollonian circle packings in parts I and II. Those papers considered circle packings described in terms of their Descartes configurations, which are sets of four mutually touching circles. They studied packings that had integrality properties in terms of the curvatures and centers of the circles. Here we consider collections of n-dimensional Desc...
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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...
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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...
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ورودعنوان ژورنال:
- Physical review. A, Atomic, molecular, and optical physics
دوره 46 4 شماره
صفحات -
تاریخ انتشار 1992