Geometric control of the building site and of the edifice as it rose from the ground were absolutely essential for three overlapping reasons:
1. Ease and Security of Construction An accurately-conceived edifice allows work to proceed more easily--Villard de Honnecourt, the 13th-century student of Gothic tells us as much: pour legierement ouvrer. The templates necessary for manufacturing the elements of multiple piers, windows and vaults were designed with straight-edge and compass. The skeletal structure of Gothic relies upon the correct alignment of the thrust of an arch with the support of a buttress--there is little room for error. This was practical geometry, derived from the continuing practice of the land-surveyors or agrimensores whose simple surveying instruments had been already deployed by the Romans. Plotting the site began with the stretching of ropes pegged to the ground. Spatial control was achieved through the formation of squares whose perfect right-angled corners were achieved through the application of a Pythagorean triangle or through an optical surveying instrument.
2. Authority It was essential that the vision of the initiators should have the authority to command the respect of the builders and users during the decades of construction and beyond. The elevated status of the medieval master mason was built partly upon the claim to have preserved the theoretical secrets of geometry from the Ancients. The cathedral makes many references to Greek and Roman architecture: the use of the classical order, Attic molding profiles, Greek meander patterns in the pavement and a grid of squares and double squares similar to the street grid of a Roman city. The authority of Biblical prototypes is achieved through shapes and numbers: heaven is square (Revelation 21): square schematism is everywhere in the cathedral. Heaven is 144 cubit high: the cathedral 144 feet. Noah's Ark is 50 cubits wide: the cathedral 50 feet.
3. Dynamic Affect The geometric matrix of the cathedral plan was based upon the stability of the square animated by the rotation of diagonals to create rectangles whose proportions were fixed by the square root of two and the Golden Section. This dynamic geometry was generated from a great central double square whose proportions matched those of contemporary images of the Celestial City. This scheme produced a memorable plan where spaces are generated from the center outwards, and where the user moving from west to east will discover a spatial explosion at the dramaturgical heart of the cathedral, the crossing space, with attention directed forward to the center of the great circle which fixed the eastern hemicycle. The hidden center point marks the transformative spot where the bread and wine of the Eucharist turn into the body and blood of Christ and where the precious relics of the saints were elevated in gold and silver containers on the relic tribune. The overall transverse section of the cathedral is fixed by a great square and the levels of the elevation planned around square schematism.
The sharply linear quality of the interior forms of articulation enhances the impression of the architectural projection of pure geometry.
A little book compiled in the 1220s and 1230s provides precious images and words allowing us to better understand the phenomenon we call "Gothic." An introductory inscription gives the name of the alleged author: "Villard de Honnecourt," though it is clear that the book was the work of more than one contributor. We are provided with information and images of the engiens--the devices or tricks--that lay behind architectural production, including templates, plans and sections. Folio 20r. provides an image of the key geometric devices of Gothic: the use of a square with a second square turned at 45 degrees inscribed inside. Vitruvius, in his Ten Books on Architecture, mentions the procedure as a means of doubling the space of a square plot. In Gothic design the procedure is often called Quadrature and applied to the articulation of pinnacles and towers producing the effect of dynamic growth. The rotated square provides the key device in the generation of the Amiens Cathedral plan.