Mathematika

Research Article

The residual set dimension of the Apollonian packing

David W. Boyda1

a1 Department of Mathematics, The University of British Columbia, Vancouver, Canada

In this paper we show that, for the Apollonian or osculatory packing C0 of a curvilinear triangle T, the dimension d(C0, T) of the residual set is equal to the exponent of the packing e(Co, T) = S. Since we have [5, 6] exhibited constructible sequences λ(K) and μ(K) such that λ(K) < S < μ(K), and μ(K)–λ(K) → 0 as κ → 0, we have thus effectively determined d(C0, T). In practical terms it is thus now known that 1·300197 < d(C0, T) < 1·314534.

(Received May 08 1973)

Key Words:

  • 52A45: CONVEX SETS AND GEOMETRIC INEQUALITIES; Packing