Abbott Dimension, Mathematics Inspired by Flatland

TL;DR

This paper introduces Abbott dimension, a geometrically intuitive way to define dimension inspired by Flatland, and compares it with classical definitions, revealing differences and raising questions about the 'right' way to measure dimension.

Contribution

The paper proposes Abbott dimension, a novel geometric definition inspired by Flatland, and analyzes its relationship with classical dimension concepts.

Findings

Abbott dimension often differs from classical definitions.

It provides an intuitive geometric perspective on dimension.

The paper raises questions about the nature of the 'right' dimension definition.

Abstract

What is the "right way" to define dimension? Mathematicians working in the early and middle $20$th-century formalized three intuitive definitions of dimension that all turned out to be equivalent on separable metric spaces. But were these definitions the "right" ones? What would it mean to have the "right" definition of dimension? In this paper we attempt to inspire thought about these questions by introducing Abbott dimension, a geometrically intuitive definition of dimension based on Edwin Abbott's $1884$ novella \emph{Flatland}. We show that while Abbott dimension has intuitive appeal, it does not always agree with the classical definitions of dimension on separable metric spaces.