Conjecturally Optimal Coverings of an Equilateral Triangle with Up to 36 Equal Circles
- 1 January 2000
- journal article
- research article
- Published by Informa UK Limited in Experimental Mathematics
- Vol. 9 (2), 241-250
- https://doi.org/10.1080/10586458.2000.10504649
Abstract
This paper presents a computational method to find good, conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles. The algorithm consists of two nested levels: on the inner level the uncovered area of the triangle is minimized by a local optimization routine while the radius of the circles is kept constant. The radius is adapted on the outer level to find a locally optimal covering. Good coverings are obtained byapplying the algorithm repeatedly to random initial configurations. The structures of the coverings are determined and the coordinates of each circle are calculated with high precision using a mathematical model for an idealized physical structure consisting of tensioned bars and frictionless pin joints. Best found coverings of an equilateral triangle with up to 36 circles are displayed, 19 of which are either new or improve on earlier published coverings.Keywords
This publication has 11 references indexed in Scilit:
- Covering a rectangle with six and seven circlesDiscrete Applied Mathematics, 2000
- Dense packings of congruent circles in a circleDiscrete Mathematics, 1998
- Packing up to 50 Equal Circles in a SquareDiscrete & Computational Geometry, 1997
- Loosest Circle Coverings of an Equilateral TriangleMathematics Magazine, 1997
- Covering a Rectangle With Equal CirclesPeriodica Mathematica Hungarica, 1997
- Improved Coverings of a Square with Six and Eight Equal CirclesThe Electronic Journal of Combinatorics, 1996
- Densest Packings of Congruent Circles in an Equilateral TriangleThe American Mathematical Monthly, 1993
- Unsolved Problems in GeometryPublished by Springer Science and Business Media LLC ,1991
- Algorithm 611: Subroutines for Unconstrained Minimization Using a Model/Trust-Region ApproachACM Transactions on Mathematical Software, 1983
- A Newton-Raphson method for the solution of systems of equationsJournal of Mathematical Analysis and Applications, 1966