8 papers · 1 filter
Repeated Patterns of Dense Packings of Equal Disks in a Square
Ronald L. Graham, Boris D. Lubachevsky
We examine sequences of dense packings of n congruent non-overlapping disks inside a square which follow specific patterns as n increases along certain values, n = n(1), n(2),... n…
Dense Packings of Equal Disks in an Equilateral Triangle: From 22 to 34 and Beyond
R. L. Graham B. D. Lubachevsky
Previously published packings of equal disks in an equilateral triangle have dealt with up to 21 disks. We use a new discrete-event simulation algorithm to produce packings for up…
Curved Hexagonal Packings of Equal Disks in a Circle
B. D. Lubachevsky, R. L. Graham
For each k >= 1 and corresponding hexagonal number h(k) = 3k(k+1)+1, we introduce m(k) = max[(k-1)!/ 2, 1] packings of h(k) equal disks inside a circle which we call "the curved he…
Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio
Boris D. Lubachevsky, Ronald Graham
We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the…
Improving Dense Packings of Equal Disks in a Square
David W. Boll, Jerry Donovan, Ronald L. Graham +1
We describe a new numerical procedure for generating dense packings of disks and spheres inside various geometric shapes. We believe that in some of the smaller cases, these packin…
Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions
R. L. Graham, J. C. Lagarias, C. L. Mallows +2
This paper gives -dimensional analogues of the Apollonian circle packings in parts I and II. We work in the space $\sM_{\dd}^n$ of all -dimensional oriented Descartes configu…