Orange Peels and Fresnel Integrals

TL;DR

This paper explores the geometric properties of spiral orange peels, deriving a formula for their shape, and demonstrates that these spirals tend to an Euler spiral upon rescaling, which has diverse scientific applications.

Contribution

It introduces a novel geometric analysis of spiral orange peels and shows their convergence to the Euler spiral, linking everyday observations to mathematical and scientific concepts.

Findings

Derived a formula for the spiral shape of orange peels.

Proved that rescaled spirals tend to the Euler spiral.

Highlighted applications of the Euler spiral in science and engineering.

Abstract

There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when unfolded on a table. We derive a formula that describes the corresponding flattened-out spiral. Cutting the peel with progressively thinner strip widths, we obtain a sequence of increasingly long spirals. We show that, after rescaling, these spirals tends to a definite shape, known as the Euler spiral. The Euler spiral has applications in many fields of science. In optics, the illumination intensity at a point behind a slit is computed from the distance between two points on the Euler spiral. The Euler spiral also provides optimal curvature for train tracks between a straight run and an upcoming bend. It is striking that it can be also obtained with an orange and a kitchen knife.