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We define the concept of self-similarity of an object by consideringendomorphisms of the object as `similarity' maps. A variety of interestingexamples of self-similar objects in geometry, algebra and arithmetic areintroduced. Self-similar objects provide a framework in which, one can unitesome results and conjectures in different mathematical frameworks. In somegeneral situations, one can define a well-behaved notion of dimension forself-similar objects. Morphisms between self-similar objects are also definedand a categorical treatment of this concept is provided. We conclude by somephilosophical remarks.

Self-Similarity in Geometry, Algebra and Arithmetic