Geometry of the tracery of the gothic windows

Abstract

Window tracery takes a special place in the Gothic architecture. One can observe some problems with the conjugations on such windows – inscribed circle touches two arches of different diameters. Modern computer technology makes it easy to determine the points and radii of such conjugates, but the aim of the work is to understand the general pattern of the construction of such windows. There are three types of the most common Gothic windows: lowered arch, lancet arch and equilateral arch. Exactly the different combinations of such arches and the circumferences inscribed in them give a great variety of window tracery. Studying the geography and various configurations of such windows, one can pay attention to the fact that in most cases such windows are designed according to the same algorithm. This algorithm, according to the authors, ancient architects used window tracery in their designs. The essence of it is as follows: 1. Two rectangular triangles are made; 2. In accordance with their configuration, their legs and hypotenuse are calculated; 3. The radii of inscribed circles are determined with the Pythagorean theorem. The examples of authentic Gothic windows of different buildings were considered to confirm the proposed algorithm: the Cathedral in Salisbury, the church of St. Matthias in Budapest, the Cathedral in Poitiers, the arcade on the Westminster Abbey, the Heidelberg Castle. The authors also suggest using of this method in the restoration of old windows.