Topic
Circle packing in an equilateral triangle
About: Circle packing in an equilateral triangle is a research topic. Over the lifetime, 141 publications have been published within this topic receiving 2125 citations.
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01 Oct 1948
TL;DR: In this article, it was shown that it is possible to dissect a triangle into unequal equilateral triangles but not necessarily into triangles and rhombuses so that no two of these figures have equal sides.
Abstract: In a previous joint paper (‘The dissection of rectangles into squares’, by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, Duke Math . J. 7 (1940), 312–40), hereafter referred to as (A) for brevity, it was shown that it is possible to dissect a square into smaller unequal squares in an infinite number of ways. The basis of the theory was the association with any rectangle or square dissected into squares of an electrical network obeying Kirchhoff's laws. The present paper is concerned with the similar problem of dissecting a figure into equilateral triangles. We make use of an analogue of the electrical network in which the ‘currents’ obey laws similar to but not identical with those of Kirchhoff. As a generalization of topological duality in the sphere we find that these networks occur in triplets of ‘trial networks’ N 1 , N 2 , N 3 . We find that it is impossible to dissect a triangle into unequal equilateral triangles but that a dissection is possible into triangles and rhombuses so that no two of these figures have equal sides. Most of the theorems of paper (A) are special cases of those proved below.
261 citations
TL;DR: The problem of finding packings of congruent circles in a circle, or, equivalently, of spreading points in acircle, is considered.
214 citations
TL;DR: An efficient implementation of circle packing is described, a central role is played by new and subtle monotonicity results for "flowers" of circles, and recent applications are illustrated.
Abstract: A circle packing is a configuration P of circles realizing a specified pattern of tangencies. Radii of packings in the euclidean and hyperbolic planes may be computed using an iterative process suggested by William Thurston. We describe an efficient implementation, discuss its performance, and illustrate recent applications. A central role is played by new and subtle monotonicity results for "flowers" of circles.
190 citations
Posted Content•
TL;DR: A simple proof of Thue theorem on circle packing is given in this paper based on density analysis of Delaunay triangulation for the set of points that are centers of circles in a saturated circle configuration.
Abstract: A simple proof of Thue theorem on Circle Packing is given. The proof is only based on density analysis of Delaunay triangulation for the set of points that are centers of circles in a saturated circle configuration.
92 citations
TL;DR: The Packing of Equal Circles in a Square as mentioned in this paper is a seminal work in the field of equal circle packing and it is based on the idea of equal circles in a square.
Abstract: (1970). The Packing of Equal Circles in a Square. Mathematics Magazine: Vol. 43, No. 1, pp. 24-30.
74 citations