The Triple Enclosure

      We are dwarfs standing on the shoulders of giants. If we see more and farther than they did, it is not because of the sharpness of our vision or our stature, but because they have lifted us up.

Look at the floor plan of Reims Cathedral. Try to identify the center from which a master plan might emerge. Instinctively, the crossing of the transepts may catch your eye. It was at the center of this crossing that “the column” once stood—the gnomon whose shadow, during the solstices, separates the cardo from the decumanus, that is, the north-south axis from the east-west axis, respectively.

Here, the crossing is built around an equilateral triangle shown in dotted lines (Fig. 2). We encountered this example on a previous page. Technically, it is a polygon, itself surrounded by two other polygons. This arrangement is common to all cathedrals. We can therefore consider that each of them is defined vertically and horizontally in relation to the one before it.

Fig. 2 - Apse of Reims Cathedral

All we have to do is imagine that a template could help us identify the spaces separating them—and that it is right before our eyes. Redrawing a floor plan would then pose no difficulty at all...

But it’s time to introduce a bit of symbolism into my discussion. Don’t forget that I’m talking about cathedrals, not train station concourses. That is why the polygon at the crossing of the transepts will be called the “Table”—a term that is symbolic if ever there was one.

In architecture, an enclosure is a structure that surrounds or protects a space. Symbolically, the pillars that define spaces constitute enclosures. That is why the outer polygons will be called the “second enclosure” and the “third enclosure,” respectively (Fig. 3). I refer to the whole as the “triple-enclosure system.”

Fig. 3 – Generic layout of the enclosures

But where might this magical template, this abacus I was referring to, be found? To answer this question, we must identify the element of the plan capable of containing the various deviations we are seeking. Once we understand that each enclosure gives rise to the next, it seems logical to start with the one from which everything stems: the Table. Thus, each Gothic cathedral could be defined at its center by the crossing of the transept. It would implicitly contain the deviations we are seeking. It’s a fascinating hypothesis.

Let’s verify this deduction by drawing the Table of Reims. As I just mentioned, it is constructed on an equilateral triangle (Fig. 4), which defines a right-angled quadrilateral. Let’s remove the triangle and draw the axes. In this new figure, we can draw three line segments.
The first can be drawn by tracing an arc of a circle with radius [0,1]. By dividing the top of the Table into two parts, it defines segment A (Fig. 5a). The second, segment B, is simply the radius we just used, namely the segment [0,1] (Fig. 5b). The last, segment C, is directly given by half the width of the Table. Of course, these distances are not chosen at random. They correspond exactly to those found between the cathedral’s various naves. By plotting them on a blank page, we should be able to reconstruct the building’s floor plan.
 

Fig. 4 - Reims Table 

Fig. 5a - Plot of segment A

Fig. 5b - Plot of segments B and C

Fig. 6a – Outline of the second enclosure

Fig. 6b – Outline of the third enclosure

Note that one of the segments is common to both structures. For reference, the height of the table, when projected vertically from the second enclosure, indicates the starting point of the apse. As for the bays of the nave, they are all defined by segment C.

Thus, we have identified the main proportions of Reims. The width of the main nave, the side aisles, the apse, the length and width of the transept, the apse—which is an arc of a circle drawn in continuity with the table—and the arch of the ambulatory, which is determined by the second enclosure. The entire plan of the cathedral is contained, to scale, within a single figure: the Table. Its name thus takes on its full meaning. As the origin of the cathedral, it constitutes both its geometric soundboard and its vibrational soundboard, much like a musical instrument.

But what is the true nature of these lines inscribed in the Tables? We will see later that they share a common logic, but their intrinsic meanings—or the criteria by which they were designed—remain unclear. Are they simply a mnemonic device, a way to encode the proportions to be applied to a building, or are they the geometric consequence of a higher-order design? To be honest, I am still pondering this question, but given the empirical approach of the early master builders, it seems more than likely to me that they sought to record the proven proportional relationships in order to remember them and pass them on in a simple way. What could be more natural for these architect-surveyors than to prepare for construction by drawing them as layouts at the center of the sanctuaries?
If you have reservations on this point, simply consider the system of segments as a universal means of measuring and redrawing a Gothic plan, setting aside all other considerations. Certain questions remain. Why does the cathedral feature such varied volumes? Wouldn’t it have been simpler for the architect to define a single module for the entire building?

There are several answers to this. First, a technical one. Indeed, the cathedral is built of walls and vaults. The vaults require complex masonry work to be erected. These structures take up space, creating a gap that deviates from an ideal layout. Furthermore, the cathedral embodies numerical, geometric, and aesthetic symbolism that is an integral part of its design and function.

All of this raises a single question: how did the builders manage to reconcile these different requirements? In fact, the answer, in my view, lies in the principle of segments, which resolve—through a simple geometric process—a problem that is far from simple. But let us make no mistake: the segments are, above all, the result of a practical architectural principle, and the triple enclosure is the expression of a symbolic message.