The parbelos, a parabolic analog of the arbelos

TL;DR

This paper introduces the parbelos, a parabolic analog of the classical arbelos shape, and explores its properties, drawing parallels to the arbelos and revealing new geometric insights using classical theorems.

Contribution

It presents the first detailed study of the parbelos, establishing its properties and connections to classical geometry, including a construction from the arbelos and related constants.

Findings

Seven properties of the parbelos are demonstrated.

A construction of the parbelos from an arbelos via a locus is provided.

Connections to the Universal Parabolic Constant and origami are discussed.

Abstract

The arbelos is a classical geometric shape bounded by three mutually tangent semicircles with collinear diameters. We introduce a parabolic analog, the parbelos. After a review of the parabola, we use theorems of Archimedes and Lambert to demonstrate seven properties of the parbelos, drawing analogies to similar properties of the arbelos, some of which may be new. The seventh constructs a parbelos directly from an arbelos via a locus. Along the way we mention the Universal Parabolic Constant (an analog of pi) and an origami fold.