Self-Similarity in Geometry, Algebra and Arithmetic

TL;DR

This paper introduces the concept of self-similarity across geometry, algebra, and arithmetic, providing a unified framework, examples, and a categorical approach to understanding these objects and their dimensions.

Contribution

It formalizes self-similarity using endomorphisms, unites various mathematical results, and develops a categorical framework for these objects.

Findings

Examples of self-similar objects in different fields

A notion of dimension for self-similar objects

Categorical treatment of morphisms between self-similar objects

Abstract

We define the concept of self-similarity of an object by considering endomorphisms of the object as `similarity' maps. A variety of interesting examples of self-similar objects in geometry, algebra and arithmetic are introduced. Self-similar objects provide a framework in which, one can unite some results and conjectures in different mathematical frameworks. In some general situations, one can define a well-behaved notion of dimension for self-similar objects. Morphisms between self-similar objects are also defined and a categorical treatment of this concept is provided. We conclude by some philosophical remarks.