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Abstract
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.