The Cathedral Alphabet

      In these four sheets there are figures of the art of geometry, but whoever wants to know which one he should use, should have great regard to know them.

Villard de Honnecourt

It's time to discover the geometric figures used by medieval architects. They are surprisingly simple and few in number.
To do this, I'm going to start with a saying from the Bauhütte, the organization that federated the stonemasons' lodges of the Holy Roman Empire.

A point in the circle,
And which is placed in the square and the triangle.
Do you know this point? All is well,
Don't you know it? All is vain.

This charade seems to contain, in esoteric form, a tracing trick. The main figures of Gothic tracings can be found here. So that all is not "in vain", let's start with the first line of this poem and plant the point of our compass. With a gesture, let's create a circle, a spiral in time.

I've just traced the primordial geometric figure, the genesis of all sacred architecture. It's impossible to summarize its scope, so vast is it. From the dome to the semicircular arch, from the ambulatory to the Gothic arch, the circle is embodied in every technique. Likewise, all regular figures are inscribed within its circumference.

Let's move on to the second line, "The point is placed in the triangle", a figure which, for Viollet-le-Duc, is "entirely satisfactory, perfect, in that it gives the most exact idea of stability."

When it forms a right angle, it is a rectangle and is associated with the square. On the building site, journeymen obtain it using a rope divided by knots. This rope, also known as Druid rope or Egyptian rope, is divided into twelve equal spaces by thirteen knots, each the value of a cubit. Think of it as a surveyor's chain.

Take the Pythagorean triangle, for example. With a side of 3, another of 4 and a hypotenuse of 5, it forms a right-angled triangle, the first to be generated by an arithmetic series. We can see that it enables us to lay down axes and create a grid. This operation remains at the origin of all work and legitimizes the square's privileged place in the symbols used by masons and journeymen. The ruler and compass can be substituted for this instrument. For example, a simple straight line can be used to draw two symmetrical circular arcs to create a perpendicular bisector.

Knotted rope is a builder's Swiss Army knife. It can be used to take measurements and perform simple calculations, like an abacus. It can also be used to construct geometric figures, such as the equilateral triangle with three equal sides. This triangle, familiar to all schoolchildren, will enable me to introduce the tracing of Gothic arches. The equilateral triangle not only sits at the center of cathedrals, but also allows us to draw the ogives known as "tiers-points".


To do this, we need to study the different arch shapes used by medieval builders. In his notebook, Villard de Honnecourt shows us how to draw them, using a single compass opening (fig. 3a).

 

Fig. 3a - Traces of pointed arches (Villard de Honnecourt - folio 41)

The first thing you see is a semicircular arch (fig. 3b). It is formed by a simple semicircle (the point marks the origin of the arc). This figure is self-explanatory. Secondly (fig. 3c), we find a broken arch based on a line separated into three parts. In fig. 3d, we see the famous "third point" I've just mentioned. Connected, the points form an equilateral triangle.

Fig. 3b - Round arch

Fig. 3c - Broken arch

Fig. 3d - Third-point arch

Note: points mark the origins of arcs.

On the same principle, the line could have been divided by three, four or five points to define arches with different openings. The geometry appendix (1) provides further information on vault layouts.

The triangle, like all polygons, gives rise to a regulating rectangle, i.e. the initial shape of a geometrical layout, its nature. Let's take an example we'll be coming back to shortly, the transept crossing of Rheims Cathedral (fig. 4a). If we draw a circular arc between the two lower columns, we see that the arc intersects both the vertical median axis and the line drawn by the double arch (fig. 4b). If we connect these three points, we obtain an equilateral triangle (fig. 4c). Finally, we can construct a rectangle around its vertices (fig. 4d).

Fig. 4a

Fig. 4b

Fig. 4c

Fig. 4d

Here we're guided by the drawing of the plan, but if we were to reproduce this construction directly on a blank sheet of paper, we'd get the same result, the same proportion. This is what I call tracing.

"The point is placed in the square". With its stable shape, the square can indicate the cardinal points, the four elements and the four seasons. Similarly, the Bible describes the earth as a flat, square surface. And so, the cathedral rises from the square of the earth, itself a product of the square of the sky. The relationship between this figure and architecture is clear and omnipresent.

Let's take the example of the previous chapter, dedicated to the analysis of a nave. The result was a construction based on the diagonal of a square. Instead of a square, I could have chosen a triangle or any other combination of these shapes (fig. 5). In the end, the combination of these figures forms a polygon on which one of the vault designs we've just studied is directly built.

 
Fig. 5 - Elevations based on squares and triangles

On the right-hand side of Fig. 5, you'll notice the procedure for transferring two-thirds of the width of a line (Fig. 6a). This is a proportion equivalent to that of a triangle where the catheters would be three and two units respectively (fig. 6b).
Now, if we double this transfer by two-thirds, we obtain a side of 4 on a base that remains 3, i.e. the proportions of the Pythagorean triangle.

Fig. 6a - Transferring 2/3 of a width

Fig. 6b - Equivalent 3/2 triangle

In the same vein, other triangles can be used to construct a regulating polygon. As you can see, elevation is always governed by simple geometric relationships.