Elevation of Beauvais Cathedral

      My civilization is rooted in Christian values.
I will reflect on the construction of the cathedral to better understand its architecture.

Saint-Exupéry

Let’s continue with an examination of the elevation of Beauvais. Professor Stephen Murray of Columbia University examined this cathedral in a fascinating article titled *Plotting Gothic: A Paradox*. Figure 1 shows an example of his work.
 

Fig. 1 - Elevation of Beauvais Cathedral according to S. Murray

Let’s try our hand at this elevation (Fig. 2). As in the previous examples, the main nave contains two equilateral triangles positioned vertically. Here, in order to increase the vaulted height of the nave, this ratio is supplemented by a polygon based on a 2/3 scale.

Fig. 2 - Elevation of Beauvais Cathedral (Step 1)

As for the profile layout, it consists of a double square (Fig. 3). This long square is also complemented by one of the first polygons encountered in this book: the Pythagorean triangle. We will find this triangle again in the side aisles, positioned vertically above a square topped by an equilateral triangle. In the latter case, the Pythagorean triangle doubles the vertical proportion of a 2/3 scale.

I will conclude by noting that the ogive of the main nave is derived from a division of the base by two points, whereas that of the side aisle is derived from off-center points.
At this stage, it is fascinating to note that cathedrals as different as Bourges and Beauvais share the same basic layout for the main nave, and that all examples employ common geometric methods.

Fig. 3 - Elevation of Beauvais Cathedral (Step 2)
 

Let us conclude this analysis by combining the two previous diagrams (Fig. 4). As in Bourges, the lines defined by the two systems coincide exactly. In the nave, the squares merge with the triangles, and the vertex constructions intersect to establish a single height in unison.
The master builder of Beauvais was a genius of geometry. The complexity and boldness of the designs go hand in hand with astonishing coherence. This design is a masterpiece. 
 

Fig. 4 - Elevation of Beauvais Cathedral (Step 3)

Credits: The scan used here comes from a document provided by Andrew Tallon to Professor Robert Bork for his article “The Geometry of Bourges Cathedral” under a CC BY license. The document has been converted to negative to improve readability.