Norton, Anderson Hassell
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This paper examines some of the historical problems that mathematicians have faced in defining dimension. Several definitions are considered and several mathematical “monsters” are introduced, such as fractals and space-filling curves, which challenge previously developed definitions and suggest new definitions. Space-filling curves introduce particularly interesting challenges because, counter to intuition, they actually fill too much of the target space to establish homeomorphisms. The paper picks up with a comparison of two classes of constructions for space-filling curves and the question of one-to-oneness: How much too much do space-filling curves fill? Both types space-filling constructions relate to fractals, which, in turn, lead us to consider fractal dimensions: self-similarity dimension, Hausdorff dimension, box-counting dimension.