The Geometry of the Nave of Saint-Denis
Here one sees the transverse section of the thirteenth-century nave at Saint-Denis abbey, based on a laser scan by Andrew Tallon. The view is to the east, with north at the left. This slice shows the piers and buttresses between the sixth and seventh nave bays measured from the west, as one can tell from the presence of the small sculpted crucifix on the right-hand arcade pier.
The interaxial width of the nave is 13.30m, which is the same as that of the choir at Notre-Dame in Paris. Here at Saint-Denis, the aisles are each half as wide as the nave, so that the distance from the building axis to the wall surfaces at ground level is also 13.30m. The height of the aisles up to their keystones is 11.53m, which is the altitude of an equilateral triangle with a base measuring 13.30m.
The height to the capitals, and to the horizontal molding above the wall passage, is .7.68m, a dimension that can be found by striking lines of 30-degree slope up from the middle of the nave floor until they intersect the aisle wall plane A circle centered on the nave floor and struck through these intersection points will have a radius of 15.36m. Its top point thus aligns with the top edges of the openings in the exterior wall passage at triforium level.
The overall organization of the nave section depends crucially on a stack of five equilateral triangles, each of which has height 13.30m and base 15.36m The interior height of the nave beneath the vaults 26.60m, corresponding to two such units, while the tip of the roof reaches to 39.90m, corresponding to three of them. The 60-degree slope of the roof independently confirms the relevance of this geometrical system.
The main nave space fits precisely into a double square measuring 13.30m by 26.60m. The floors of the wall passages are at heigh 3.84m, halfway up to the capitals. The arcade axes intersect the large yellow triangles at height 28,38m, thus locating the base of the roof structure.
Octagons also figure into the geometry of the Saint-Denis nave. An octagon inscribed within the previously defined circle of radius 15.36m will have a facial radius of 14.19m; this is the distance from the building centerline inner surface of the wall at the level of the aisle windows. An equivalent construction within the original circle of radius 13.30m locates the inner faces of the engaged wall piers 12.29m from the building centerline. The radii of the octagon intersect the arcade axes at height 16.05m, which will play a crucial role in governing the elevation.
The clerestory window bases lie on the octagon-determined height 16.05m, but the lip defining the top of the triforium lies at height 15.71m, halfway between the previously determined heights 15.36m and 16.05m. Red verticals drawn upwards from the tips of the large yellow triangles define the axes of the slender columns supporting the heads of the lower flying buttresses.
The centers of the lower flyers lie at height 16.05m. A circle centered at this level and reaching down to height 11.53m will reach up to height 20.58m, thus locating the springing of the main vault above its capitals. Circumscribing a hexagon around that circle, and another larger circle around the hexagon, one finds the arc of the lower flyer intrados, which meet the clerestory wall at height 21.28m.
The inner margins of the buttress uprights are verticals tangent to the lower flyer intrados. The main body of the buttress thickens slightly futher out, with the interval equaling the .34m difference between height 15.71m and 16.05m, as the small yellow circle centered at the latter level indicates. Lines of 30-degree slope rising from the buttress centers intersect these verticals at heights 19.07m and 19.27m, thus locating the bottom and lip of the top horizontal molding on the buttress upright; a similar falling line intersects the inner vertical at height 13.04m, locating the roof slab over the aisle and the floor of the passage through the buttress uprights.
The upper flying buttress arcs are centered at height 19.07m on the main arcade axes. Their radii reach out to the yellow vertical defining the thick part of the buttress upright, and the arcs thus sweep up to meet the wall at height 25.33m.
The upper chords of the upper flyers descend with 30-degree slope from the triangle tip at height 26.60m to meet the buttress upright at height 23.58m. The analogous chords of the lower flyers intersect the arcade axes at height 23.04m, aligned with the tip of a great equilateral triangle framed by the aisle walls 13.30 offset from the building centerline. The lower flyers are slightly less steep than 30 degrees, however, likely because their design was awkwardly adjusted to meet the thick body of the buttress upright rather than its inner flange. The outer surfaces of the buttresses, finally, are 16.60m from the building centerline, so that lines of 60-degree slope rising from their base points pass directly through the centers of the lower flyer arcs.