The Jordan curve theorem is non-trivial

Abstract

The formal mathematical definition of a Jordan curve (a non-self-intersecting continuous loop in the plane) is so simple that one is often lead to the unimaginative view that a Jordan curve is nothing more than a circle or an ellipse. In this paper, we pursue the theme that a Jordan curve can be quite fantastical in the sense that there are some bizarre properties such a curve might have (jagged at every point, space filling, etc.) or that such a curve can have a difficult to discover inside and outside as promised by the celebrated Jordan Curve Theorem (JCT). We explore the JCT theorem through its history and some hand drawings which not only challenge the viewer's preconceived notions of interior and exterior or that the JCT is a trivial result, but also challenge the reader's notion that a curve is a cold boring object, incapable of telling an interesting story.