# The Dimension of a Point: Computability Meets Fractal Geometry

Conference paper

## Abstract

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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