Elevation of Beauvais Cathedral

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

Saint-Exupéry

Let's continue with the study of Beauvais's elevation. Professor Stephen Murray of Columbia University has studied this cathedral in a fascinating article entitled Plotting Gothic: À Paradox. Figure 1 shows an example of his work.
 

Fig. 1 - Elevation of Beauvais Cathedral, after S. Murray. Murray

Let's try our hand at this elevation (fig. 2). As in previous examples, the main nave houses two equilateral triangles set vertically. Here, in order to increase the nave's vaulted height, this ratio is complemented by a polygon based on a 2/3 ratio.

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

For its part, the profile layout consists of a double square (fig. 3). This long square is complemented by one of the first polygons encountered in this book, the Pythagorean triangle. This triangle is found again in the collaterals, set vertically above a square topped by an equilateral triangle. In the latter case, the Pythagorean triangle doubles the vertical proportion by 2/3.

I'll end by pointing out that the main nave's ogive is derived from a division of the base by two points, whereas that of the side aisle is derived from eccentric points.
At this point, it's fascinating to note that cathedrals as different as Bourges and Beauvais have the same basic layout for the main nave, and that all the examples are based on common geometric processes.

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

Let's finalize this analysis by merging the two previous diagrams (fig. 4). As in Bourges, the lines given by the two systems overlap exactly. In the nave, the squares merge with the triangles, and the summit constructions intersect to set the same height in unison.
The master of Beauvais was a geometrician of genius. The complexity and audacity of his layouts rhymes with astonishing coherence. This layout is an apotheosis.
 

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

Credits: The scan in figure 1 comes from a document provided by Andrew Tallon to Professor Stephen Murray for his article Plotting Gothic: À Paradox licensed under CC BY 3.0. Here, the scan has been turned negative and I've shown the projections of the triple speaker system in elevation in green.
 
N.B. The reader will find other examples in the book, including a study of the elevation of Chartres Cathedral.