The Dimension of a Point: Computability Meets Fractal Geometry

  • Jack H. Lutz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)


Recent developments in the theory of computing give a canonical way of assigning a dimension to each point of n-dimensional Euclidean space. Computable points have dimension 0, random points have dimension n, and every real number in [0,n] is the dimension of uncountably many points. If X is a reasonably simple subset of n-dimensional Euclidean space (a union of computably closed sets), then the classical Hausdorff dimension of X is just the supremum of the dimensions of the points in X. In this talk I will discuss the meaning of these developments, their implications for both the theory of computing and fractal geometry, and directions for future research.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jack H. Lutz
    • 1
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA

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