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The Fractal Dimension of Sets Derived from Complex Bases

Published online by Cambridge University Press:  20 November 2018

William J. Gilbert
William J. Gilbert*
Affiliation:
Pure Mathematics Department University of Waterloo Waterloo, Ontario N2L 3GL Canada
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Abstract

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For each positive integer n, the radix representation of the complex numbers in the base —n + i gives rise to a tiling of the plane. Each tile consists of all the complex numbers representable in the base -n + i with a fixed integer part. We show that the fractal dimension of the boundary of each tile is 2 log λn/log(n2 + 1), where λn is the positive root of λ3 - (2n - 1) λ2 - (n - 1) 2λ - (n2 + 1).

Keywords

11K5551M20

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 4 , 01 December 1986 , pp. 495 - 500
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Dekking, F. M., Replicating superfigures and endomorphisms of free groups, J. Combinatorial Theory Ser. A 32 (1982), pp. 315–320.Google Scholar
2. Giles, J., Superfigures replicating with polar symmetry, J. Combinatorial Theory Ser. A 26 (1979), pp. 335–337.Google Scholar
3. Gilbert, W. J., The fractal dimension of snowflake spirals, Notices Amer. Math. Soc. 25 (1978). A-641, Abstract 760-D1.Google Scholar
4. Gilbert, W. J., Fractal geometry derived from complex bases, Math. Intelligencer, 4 (1982) 78 — 86.Google Scholar
5. Kâtai, J. and Szabô, J., Canonical numbers systems for complex integers, Acta Sci. Math. (Szeged.) 37 (1975), pp. 255–260.Google Scholar
6. Mandelbrot, B. B., The Fractal Geometry of Nature, Freeman, San Francisco, 1982.Google Scholar