Cutting cakes and kissing circles

TL;DR

This paper introduces precise geometric constructions, using only a knife and no compass, to solve cake-cutting problems exactly, including bisecting circles and dividing cakes into equal parts, based on classical geometric methods.

Contribution

It presents novel geometric constructions for cake-cutting problems that do not rely on traditional tools like a straightedge and compass, extending classical methods to practical cutting scenarios.

Findings

Method for bisecting circular cakes with unknown centers using three cakes.

Procedures for dividing a cake with a known center into 3, 4, and 6 equal parts.

Based on classical Steiner and Cauer constructions from the 19th and early 20th centuries.

Abstract

To divide a cake into equal sized pieces most people use a knife and a mixture of luck and dexterity. These attempts are often met with varying success. Through precise geometric constructions performed with the knife replacing Euclid's straightedge and without using a compass we find methods for solving certain cake-cutting problems exactly. Since it is impossible to exactly bisect a circular cake when its center is not known, our constructions need to use multiple cakes. Using three circular cakes we present a simple method for bisecting each of them or to find their centers. Moreover, given a cake with marked center we present methods to cut it into n pieces of equal size for n=3,4 and 6. Our methods are based upon constructions by Steiner and Cauer from the 19th and early 20th century.