The Gothic Altarpieces

      That was when I saw the Pendulum. The sphere, suspended at the end of a long thread attached to the choir vault, traced its sweeping oscillations with isochronous majesty.

Umberto Eco (Foucault’s Pendulum)

It is interesting to study the construction of the polygons that form the apses. In many monuments, you will notice that the diagonal of the choir’s last bay directly determines the angle of the polygon. By virtue of its design and the choice of an initial proportion, the final layout is predetermined. For those who might have doubts, they will notice that Villard de Honnecourt drew these lines in a single stroke (Fig. 1),

Fig. 1 – Plans of apses (Villard de Honnecourt – folio 29) D.P.

 
I have developed various methods for drawing this type of layout. Here is one that is particularly interesting (Fig. 2). On the last bay of the choir, we define a square whose diagonal is equal to half the width of the nave (as indicated by the dotted arc).

The height of this square allows us to trace the boundary of the apse. At the upper corners of the rectangle thus formed, I mark the pillars and draw the semicircle of the apse with center O. From the lower corners, I then draw two lines that intersect at O, defining two additional pillars where they intersect the circle.
The distance between the pillars we have just defined determines two of the sides we are looking for. It will therefore suffice to extend them symmetrically to complete the division of the circle.

Fig. 2 – Division of an apse into five parts

The method we have just examined governs the layout of the last bay of the choir and the apse of many Gothic buildings. It can be found in Reims, as well as at the Abbey of Saint-Martin-aux-Bois and the Royal Collegiate Church of Saint-Quentin.

As for apses crowned by seven chapels (Fig. 3), their construction follows the same principle. The width of the central nave is divided into three parts, allowing a rectangle measuring three by one to be drawn inside the apse. The intersection of the diagonals (point O) serves as the origin of the circles, while their extensions define the starting angle and determine the position of the columns.

Fig. 3 – Division of an apse into seven parts

Since half the length of the line segment is equal to the value that must be transferred to the circle to form the angles, nothing could be simpler than drawing the pseudo-heptagon. These are, in fact, 1-by-2 triangles arranged in a fan shape. This method can be found, among other places, in Beauvais and Amiens.
These two apse designs are generic systems. It should be noted that some cathedrals have their own unique layouts, such as Chartres.

The Abbey Church of Saint-Martin-aux-Bois in the Oise department will allow me to illustrate this point (Fig. 4). The building features a nave without a transept, enclosed by an apse similar to that of Reims.
 

Fig. 4 - Choir of Saint-Martin-aux-Bois

We have just seen that the side of the square determines the width of the last bay, while its diagonal is equal to half the length of that same bay. To determine the height of the nave bays, the architect simply used this half-width to form equilateral triangles. There are therefore two per bay. This segment (the red lines) is common to the entire building. It alone explains its harmonious balance.

In this example, it is difficult to tell where the layout of the choir begins and that of the nave ends. Here, they share the same fundamental nature. You can view the laser scan of this building on the website saint-martin-aux-bois.org