Dimension of some fractal sets on hexagonal lattices
P.S. Bogdanov, V.M. Chernov

PDF, 391 kB

Full text of article: Russian language.

DOI: 10.18287/0134-2452-2014-38-2-330-334

Pages: 330-334.

Abstract:
In this paper fractal dimension of fundamental domain boundary for all possible ternary quasicanonical numerical system in the ring of Eisenstein integers is calculated. Modified method that was used by W. Gilbert and J. Thuswaldner for computing of fundamental domain boundary fractal dimension for canonical numerical system is considered.

Key words:
canonical numerical system, quasicanonical numerical system, fundamental domain, fractal dimension.

References:

  1. Mandelbrot, B.B. How long is the coast of Britain // Science. – 1967. – V. 155. – P. 636-638.
  2. Mandelbrot, B.B. A fast fractional Gaussian noise generator // Water Resources Research. – 1971. – V. 7. – P. 543-553.
  3. Mandelbrot, B.B. Fractals: Form, Chance, and Dimension / B.B. Mandelbrot. – San Francisco: W.H. Freeman and Company, 1977. – 365 p.
  4. Mandelbrot, B.B. The Fractal Geometry of Nature / B.B. Mandelbrot. – New York: W.H. Freeman and Company, 1982. – 468 p.
  5. Feder, J.E. Fractals / J.E. Feder. – New York: Plenum Press, 1988. – 283 p.
  6. Crownover, R.M. Introduction to fractals and chaos / R.M. Crownover. – Boston; London: Jones and Bartlett, 1995. – 306 p.
  7. Schroeder, M.R. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise / M.R. Schroeder. – New York: W.H. Freeman, 1990. – 429 p.
  8. Soifer, V.A. Analysis and recognition of the nanoscale images: conventional approach and novel problem statement / V.A. Soifer, A.V. Kupriyanov // Computer Optics. – 2011. – V. 35(2). – P. 136-144. – (In Russian).
  9. Gilbert, W.J. The Fractal Dimension of Sets derived from Complex Bases // Canadian Mathematical Bulletin. – 1986. – V. 29. – P. 495-500.
  10. Gilbert, W.J. Complex bases and fractal similarity // Annales des Sciences Mathématiques du Québec. – 1987. – V. 11(1). – P. 65-77.
  11. Thuswaldner, J.M. Fractal dimension of sets induced by bases of imaginary quadratic fields // Mathematica Slovaca. – 1998. – V. 48. – P. 365-371.
  12. Thuswaldner, J.M. Fractal Properties of Number Systems / J.M. Thuswaldner, W. Müller, R.F. Tichy // Periodica Mathematica Hungarica. – 2001. – V. 42. – P. 51-68.
  13. Nye, J.F. Physical properties of crystals: their representation by tensors and matrices / J.F. Nye. – Oxford: Clarendon Press, 1972. – 322 p.
  14. Conway, J. Sphere Packings, Lattices and Groups / J. Conway, N. Sloane. – New York: Springer-Verlag, 1999. – 703 p.
  15. Hernandez, Y. Aberration-corrected HRTEM image of a graphene monolayer obtained by exfoliation of graphite in liquid phase / Y. Hernandez, V. Nicolosi, M. Lotya, F.M. Blighe // Nature Nanotechnology. – 2008. – V. 3(9). – P. 563-568.
  16. Borevich, Z.I. Number theory / Z.I. Borevich, I.R. Shafarevich. – Academic Press, 1986. – 434 p.
  17. Chernov, V.M. Arithmetical methods of synthesis of fast algorithms of Discrete orthogonal Transforms / V.M. Chernov. – Moscow: “Fizmatlit” Publisher, 2007. – 264 p. – (In Russian).
  18. Bogdanov, P.S. Classification of ternary quasicanonical number systems in imaginary quadratic fields and their application / P.S. Bogdanov, V.M. Chernov // Computer Optics. – 2014. – V. 38(1). – P. 139-147.

© 2009, IPSI RAS
151, Molodogvardeiskaya str., Samara, 443001, Russia; E-mail: journal@computeroptics.ru ; Tel: +7 (846) 242-41-24 (Executive secretary), +7 (846) 332-56-22 (Issuing editor), Fax: +7 (846) 332-56-20