The Design of Church Windows

摘要

Robert C. Wheeler of Lincoln points out in his letter on church windows (Mathematics Today, February 2012, page 41) that the arches in Lincoln cathedral are of the equilateral variety. They cannot then be of the golden variety. This is useful information because, following my conjecture about golden arches, I have returned many times to the original article by Hugh Williams (Mathematics Today, August 2011, pages 194198) and have been at a loss to determine using a Cartesian template the true nature of the Lincoln arches from the photographs, because of the lack of a true elevation. The template that I use is simple and can be produced either on tracing paper or on an overhead transparency and involves the usual Cartesian coordinate axis with two lines passing through the origin, one at 60° to the horizontal and the other at 57°. The former is the angle subtended with the horizontal (springing line) by a chord through one of the arch's feet and its apex, when the arch is of the equilateral variety. The latter is the angle subtended similarly when the arch is of the golden variety, being an approximation to the true angle, which is tan-(√5/(√6-1))≈57.05°. These angles are close together and this tests the faculties when trying to determine arch types. The preference here has been to use a protractor to reproduce these angles on a template, but it should be appreciated that they can be constructed precisely using a straight edge, a pencil and a pair of compasses.