Improved bounds for the disk-packing constant

@article{Boyd1973ImprovedBF,
  title={Improved bounds for the disk-packing constant},
  author={D. W. Boyd},
  journal={aequationes mathematicae},
  year={1973},
  volume={9},
  pages={99-106}
}
  • D. W. Boyd
  • Published 1973
  • Mathematics
  • aequationes mathematicae
This result lends considerable weight to the heuristic estimate S ~ 1.306951, obtained by Melzak [6]. The improvement is a result of some new inequalities involving the disk-packing function M(a, b, c; t). Our principal new result is that M is a strictly convex function of (a, b, c). This, combined with the fact that M is a symmetric function of these three variables and an auxiliary lemma, allows us to show that 
The interval of disk packing exponents
The set of disk packing exponents is an interval equal to (E, 2] or [E, 2]. The set of triangle packing exponents is [log2 3, 2]. The analogy strongly suggests that E is attained and that E=S, theExpand
The Hausdorf dimension of the Apollonian packing of circles
We formulate the problem of determining the Hausdorf dimension, df, of the Apollonian packing of circles as an eigenvalue problem of a linear integral equation. We show that solving aExpand
The sequence of radii of the Apollonian packing
We consider the distribution function N(x) of the curvatures of the disks in the Apollonian packing of a curvilinear triangle. That is, N(x) counts the number of disks in the packing whose curvaturesExpand
New Results in the Theory of Packing and Covering
Let J be a system of sets. If all members of J are contained in a given set C and each point of C belongs to at most one member of J then J is said to be a packing into C. If, on the other hand, eachExpand
A new class of infinite sphere packings
The oscillatory or Apollonian packing in two dimensions is well known and is described for example in [13]. Recently we investigated the three dimensional osculatory packing of a sphere [4] However,Expand
The residual set dimension of the Apollonian packing
In this paper we show that, for the Apollonian or osculatory packing C 0 of a curvilinear triangle T , the dimension d ( C 0 , T ) of the residual set is equal to the exponent of the packing e ( C oExpand
An asymptotic formula for integer points on Markoff-Hurwitz varieties
We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation x21+x22+⋯+x2n=ax1x2⋯xn+k. When n≥4, the previous best result is by Baragar (1998) that givesExpand
The exponent for the Markoff–Hurwitz equations
In this paper, we study the Marko-Hurwitz equations x 2 +::: +x 2 = ax0 xn. The variety V dened by this equation admits a group of automorphismsA =Z=2 Z= 2(an n+1 fold free product). For a solution PExpand
An asymptotic formula for integer points on Markoff-Hurwitz surfaces
We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation \[ x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=ax_{1}x_{2}\ldots x_{n}+k. \] When $n\geq4$ the previousExpand
Geometric Sequences Of Discs In The Apollonian Packing
This study began innocently enough with a search for extremal conngurations of circles in the Rodin and Sullivan \Ring Lemma". This is an elementary geometric lemma which nevertheless plays a keyExpand
...
1
2
3
4
...

References

SHOWING 1-6 OF 6 REFERENCES
The disk-packing constant
The lower bound was subsequently improved by Wilker [8] to 1.059, and by the author [2] to 1.28467. An improved upper bound of 1.5403 . . . . (9+x/41)/10 was proved in [3], but the arguments there,Expand
Lower Bounds for the Disk Packing Constant
An osculatory packing of a disk, U, is an infinite sequence of disjoint disks, f Un }, contained in U, chosen so that, for n _ 2, U,, has the largest possible radius, r,,, of all disks fitting inExpand
On the Solid-Packing Constant for Circles
A solid packing of a circular disk U is a sequence of disjoint open circular subdisks Ul, U2, . whose total area equals that of U. The Mergelyan- Wesler theorem asserts that the sum of radiiExpand
Open disk packings of a disk
The Apollonian Packing of Circles