Properties of regular Tangles

TL;DR

This paper introduces new types of Tangles made from circle arcs of thirds and sixths, relating them to regular tilings, and explores their properties including link count and enclosed area.

Contribution

It extends the study of Tangles by defining new families based on circle arc fractions and connects them to regular tilings to analyze their properties.

Findings

Introduced Tangles with thirds and sixths of circles.

Established relationships between Tangles and regular tilings.

Proved results on link count and enclosed area.

Abstract

A Tangle is a smooth simple closed curve formed from arcs (or ``links'') of circles with fixed radius. Most previous study of Tangles has dealt with the case where these arcs are quarter-circles, but Tangles comprised of thirds and sixths of circles are introduced. Together, these three families of Tangles are related to the three regular tilings of the plane by squares, regular hexagons, and equilateral triangles. This relationship is harnessed to prove results about the number of links comprising a Tangle and the area that it encloses.