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Finding Fibonacci in a Fractal

In: Applications of Fibonacci Numbers

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  • Nathan C. Blecke
  • Kirsten Fleming
  • George William Grossman

Abstract

The focus of this paper is to further investigate properties of a two-dimensional fractal that involves counting and Fibonacci numbers. We determine the fractal dimension using the Box Counting Theorem and also the concept of similitude. We find affine transformations that generate some of the set of points that are in the fractal, which have the form Ax + b for a pair of two matrices A 1 and A 2 and some vectors x, b 1 and b 2. We denote these transformations S 1 and S 2. We find examples of the limit points generated, by taking repeated applications of the operators on some starting points (which are vertices of triangles) in some prescribed order. The fractal, denoted G, is the countable intersection of the countable union of a set of triangles. The fractal is shown to be a totally disconnected set.

Suggested Citation

  • Nathan C. Blecke & Kirsten Fleming & George William Grossman, 2004. "Finding Fibonacci in a Fractal," Springer Books, in: Frederic T. Howard (ed.), Applications of Fibonacci Numbers, pages 43-62, Springer.
    • Handle: RePEc:spr:sprchp:978-0-306-48517-6_7
    DOI: 10.1007/978-0-306-48517-6_7
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