The compass and straightedge in the teaching of Euclidean geometry with applications in Gothic architecture

Abstract

Construction with compass and straightedge have been widely used in art and design across many cultures. In this way, Greek geometry (primarily through Euclid's "Elements") has influenced art and architecture across the world. As translation of Euclid's "Elements" arose in various cultures, it inspired artists and gave them a theory of the idealized shapes that were their basic tools; artists repaid their debt to mathematics by making creative advances in the techniques that begged mathematicians to examine the underlying geometry. Using a compass and straightedge to recreate geometric designs found in artistic masterpieces are wonderful problems of Euclidean geometry for high school students; they are visually interesting and complex, but based on relatively simply underlying principles. The designs found in Gothic cathedrals are especially well suited for this purpose. Constructions with straightedge and compass provide limitless opportunities for students to apply the results of Euclidean geometry, they allow students to participate in the mathematical process of discovery and justification, and they are a link between geometry and art, architecture, and a variety of trades. A survey of constructions with compass and straightedge and an examination of how they can be used to enrich the teaching of high school geometry are provided within the framework of applications to Gothic architecture.