Self-similar carpets over finite fields

  • Authors:

  • Mihai Prunescu

  • Affiliations:

  • Brain Products, Freiburg, Germany and Institute of Mathematics of the Romanian Academy, Bucharest, Romania

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2009

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Abstract

In [Dann E. Passoja, Akhlesh Lakhtakia, Carpets and rugs: An exercise in numbers, Leonardo 25 (1) (1992) 69-71] an informal algorithm 'to display interesting numeric patterns' is described without any proof. We generalize this algorithm over arbitrary finite fields 𝔽q of characteristic p and we prove that it always generates some self-similar sets. For the prime fields 𝔽p the generalized algorithm produces p-1 different self-similar sets. These sets are classified according to their arithmetic and their groups of symmetry.