Geometry and Planning

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Geometry and Planning

Architecture, Beauty, and Geometry.

Modern observers are often filled with awe when they consider the combination of engineering and aesthetic sophistication required to achieve the complex structures that were evolving even in the early Middle Ages. Of the many stages involved in the construction of a medieval building, the first step, the planning phase, remains the most elusive. This was the moment when thoughts and conversations began to take physical form as architects translated functional requirements and a patron's vision into structure and space. What strategies did they employ to ensure that an edifice was both stable and beautiful? Geometry provided the means by which masons designed and laid out plans, the common language of communication between craftsman and cleric, and the vehicle by which architectural form could be invested with meaning. Medieval thinkers, such as St. Augustine, followed Vitruvius, a Roman architect who wrote On Architecture at the time of Augustus in the late first century b.c.e., by defining beauty as a harmony of parts that arose out of geometrical regularity. Constantly cited by writers from the ninth century on, Vitruvius's ideas on architectural planning, along with other classical works, were well known in the Middle Ages. In 1321, the monks of the abbey of Saint-Ouen in Rouen wrote, as they planned for the reconstruction of their church, that "we have decided to build in accordance with former learned treatises" by "industrious and accomplished artisans who are recognized to possess proven and wellknown experience in such matters." Whether or not medieval masons actually read these "treatises," this document suggests that they were able to put into practice the architectural theories familiar to their scholarly patrons.

Squares, Circles, and Triangles.

According to Vitruvius, plans of buildings require geometry that "teaches the use of straight lines and the compass." The designs of most medieval buildings can be understood in terms of skillful combinations of squares, circles, and triangles. An architect might take a "modular" approach based on the repetition of a standard unit, as illustrated by the layout of Charlemagne's palace at Aachen which used a 12-foot length, organized into larger squares of 84- and 360-feet, or the plan of Saint-Gall that was governed by a grid of 40-foot squares. However, after the fifth century and the collapse of centralized Roman authority, different regions of Europe and even cities within the same region adopted their own standards of measurement that might vary by as much as four inches per foot. The potential for chaos if an architect designed a building according to one foot-length while the craftsmen constructed it using another standard of measure was great. To avoid such problems, builders turned to ratios generated by the manipulation of basic geometric figures as the means of organizing and coordinating the complex of parts and spaces of the church. One of the most common techniques was based on the ratio of the side of a square to its diagonal, that is, one to the square root of two (or mathematically 1: 1.414). For example, imagine that a person laid out a square then drew its diagonal. Then the person swung the diagonal until it lined up with the side of the square. That person has just designed most of the twelfth-century Cistercian abbey of Jerpoint in Ireland: its cloister is a 100-foot square while the nave, created by the diagonal, is 142 feet long. Swinging the diagonal in the other direction establishes the width of the east range of the cloister buildings. A similar square root of two ratios ran through Norwich Cathedral, begun in the late eleventh century, and controlled the relation of width to length of the papal church of Saint-Urbain in Troyes which was begun in 1262.

Other Ratios.

A second ratio that can be found frequently in medieval architecture is the "golden section" (1: 1.618), easily produced from the diagonal of half a square. Used as far back as ancient Egypt and continued by the Romans in such important Christian structures as Old St. Peter's in the fourth century, the golden section was applied in the designs of Amiens Cathedral, and again at Saint-Urbain in Troyes and Saint-Ouen in Rouen. Despite their differences, each plan reveals a golden section ratio between the wide central space and the flanking aisles. Other ratios detected in medieval monuments include the square root of three (1: 1.732), taken from an equilateral triangle and found at the abbey church of Fontenay, and the square root of five (1: 2.236). All of these ratios involve irrational rather than whole numbers and are derived from working directly from the geometrical figures of the plan. These same ratios were projected upwards into the superstructure. The square root of two controls the composition of the elevation at Durham Cathedral. At Milan, arguments broke out during the 1390s over whether the cathedral should be designed on the basis of a square (ad quadratum) or a triangle (ad triangulatum). Debate raged over which solution would be more beautiful and structurally sound, and advocates of the various solutions supported their positions by citing the Bible as well as Aristotle.


The plan of a building could be laid out at full scale on the ground using ropes and stakes to mark its outlines. But as medieval architecture developed in complexity, drawing became an increasingly important tool in the process of design and construction. Few drawings are known before the twelfth century: the famous plan of Saint-Gall, created in the early ninth century, is less a construction blueprint than a conceptual diagram that sets out the scheme of an ideal monastery. As late as 1178 when William of Sens, architect of the new choir at Canterbury Cathedral, was seriously injured after falling from scaffolding, he attempted to supervise construction by having himself carried to the building site on a stretcher. Although William had supplied templates or patterns for architectural details, there apparently were no drawings that could keep the project on course in his absence. Numerous full-scale drawings survive from the thirteenth century onwards. Incised into floors, on walls, or on plaster surfaces, the extant engravings encompass preliminary sketches and finished representations of building parts such as windows, piers, gables, portals, and flying buttresses. They offered the master mason a chance to refine his formal ideas, to evaluate the merits of a design before it was executed, and to check the accuracy of the cutting before stones were mortared into place. Finally, in the Gothic period, drawings on parchment evolved as structures were composed of precisely coordinated networks of shafts and moldings, and intricate and finely scaled vault and window patterns. Drawings not only became indispensable to construction, but they allowed an architect to supervise multiple projects simultaneously. In 1434, Pierre Robin, a Parisian master, was hired to design the parish church of Saint-Maclou in Rouen. He stayed in the city for five months, delivered a complete set of plans, then left the building entirely to local contractors. Although construction at Saint-Maclou lasted for ninety years, Robin's plans were followed scrupulously, and the church displays a remarkable unity despite the participation of many hands. The most lavish drawings of the later Middle Ages, embellished by color washes, might be shown to patrons to confirm the genius of the design, sent to potential donors, or exhibited in public to raise funds for construction. An example of such geometrical drawing occurs in the portfolio of Villard de Honnecourt in the 1230s.


introduction: So popular were pilgrimages throughout the Middle Ages that various guidebooks were written to help pilgrims negotiate the voyage and understand the significance of the route and the buildings along the way. Among the most famous of these was the Liber Sancti Jacobi (the Book of Saint James), a compilation made between 1139 and 1173 of five books, treating such subjects as the four French roads leading to the major pilgrimage routes, the cities and stations along the way in Spain (treated very negatively), hospices, road maintenance, and rivers to be crossed. The fifth book of the compilation, perhaps written by Aimery Picaud from nearby Poitiers in France, is entitled The Pilgrim's Guide to Santiago de Compostela. In the book's eleven chapters, appear a liturgy to serve the cult of Saint James, twenty-two miracles of Saint James, glorification of the cult of Saint James, the story of the legendary Bishop Turpin (who appears as a character in the Song of Roland), and three long chapters on relics, reliquaries, peoples, and lands. The section below, describing the cathedral itself, contains details on measurements and proportions.

The Measurements of the Church

The basilica of Saint James measures in length fifty-three times a man's stature, that is to say, from the west portal to the altar of the San Salvador [the eastern chapel]; and in width, thirty-nine times, that is to say, from the French [north transept portal] to the south portal. Its elevation on the inside is fourteen times a man's stature. Nobody is really able to tell its length and height on the outside.

The church has nine aisles on the lower level and six on the upper, as well as a head, plainly a major one, in which the altar of the San Salvador is located, a laurel crown, a body, two members, and further eight small heads [chapels] in each of which there is an altar. …

In the largest nave there are twenty-nine piers, fourteen to the right and as many to the left, and one is on the inside between the two portals against the north, separating the vaulted bays. In the naves of the transepts of this church, that is to say from the French to the south portal, there are twenty-six piers, twelve on the right and as many on the left, as well as two more placed before the doors on the inside, separating the entrance arches and the portals.

In the crown of the church there are eight single columns around the altar of the Blessed James. …

In this church, in truth, one cannot find a single crack or defect: it is admirably built, large, spacious, luminous, of becoming dimensions, well proportioned in width, length and height, of incredibly marvelous workmanship and even built on two levels as a royal palace.

He who walks through the aisles of the triforium [gallery] above, if he ascended in a sad mood, having seen the superior beauty of this temple, will leave happy and contented.

Concerning the Towers of the Basilica

There are nine towers in this church, that is to say, two above the portal of the fountain, two above the south portal, two above the west portal, two above each of the corkscrew staircases, and another, the largest one, above the crossing, in the middle of the basilica. By these and by the many other utmost precious works the basilica of the Blessed James shines in magnificent glory. All of it is built in massive bright and brown stone which is hard as marble. The interior is decorated with various paintings and the exterior is perfectly covered with tiles and lead.

source: The Pilgrim's Guide to Santiago de Compostela. Trans. and ed. William Melczer (New York: Italica Press, 1993): 120–121; 126.

Making and Meaning.

While medieval architecture was becoming increasingly technological in its methods, it also maintained symbolic and religious significance. The medieval church building was likened by theologians to the Temple of Solomon as well as to the celestial Jerusalem. The Bible describes these sacred edifices in detail, including their dimensions: the temple was 60 cubits long, 20 cubits wide, and 30 cubits high; the heavenly city measured 12,000 stadia with a wall of 144 cubits (cubits were measurements of about 17 to 21 inches and stadia were about 607 feet). In addition, Christian theology interpreted numbers in symbolic terms. Four was equated with the gospels, five was the number of divinity, evoking the five wounds of Jesus, and six was the perfect number, according to St. Augustine, because it was both the sum and product of its factors, 1, 2, and 3. Thus although plan designing was a process that involved the mechanical manipulation of geometric shapes by secular craftsmen, it was possible to endow a building with symbolic meaning by basing the design on a resonant module or by encoding a significant number into its geometry. The chapel at Aachen, for example, was based on a twelve-foot module and its total length was 144 feet, both dimensions clearly intended to connect it with the image of heaven. The royal Sainte-Chapelle in Paris quotes the proportions and dimensions of the Palace of Solomon, described in I Kings 7, perhaps as a way to associate the contemporary French monarch with Solomon, the paragon of the wise ruler. And the design of the papal church of Saint-Urbain in Troyes unfolds out of a central square that is 36 feet per side using the square root of two and golden section operations. Six is, of course, a factor of 36, so that the perfect number literally lies at the heart of the plan. Its height, 72 feet, is determined by two of these squares, but is also a factor of the number 144 associated with heaven. In sum, medieval masons used geometry as the practical means to ensure the harmony of their designs while patrons found in number and shape potent symbols that linked the present with the past and connected their earthly projects to divinely inspired models.


Anne Berthelot, "Numerology," in Dictionary of the Middle Ages: Supplement 1. Ed. William Chester Jordan (New York: Charles Scribner's Sons, 2004): 427–428.

Theodore Bowie, ed., The Sketchbook of Villard de Honnecourt (Bloomington, Ind.: Indiana University Press, 1959).

J. H. Harvey, The Medieval Architect (London: Wayland, 1972).

Robert Mark, ed., Architectural Technology up to the Scientific Revolution (Cambridge, Mass.: MIT Press, 1991).

Roland Recht, ed., Les Bâtisseurs des cathédrales gothiques (Strasbourg, France: Editions les Musées de la Ville de Strasbourg, 1989).

Vitruvius: Ten Books of Architecture. Trans. Ingrid D. Rowland (New York: Cambridge University Press, 1999).