The triple enclosure

we are dwarfs sitting on the shoulders of giants. If we see more and farther than they do, it's not because of our keen eyesight, or our greatness, it's because we are elevated by them.

Look at the floor plan of Reims Cathedral. Look for the center from which a master plan can emerge. Instinctively, the transept crossing is likely to catch your eye. It was at the center of this crossing that "the column" was once erected, the gnomon whose shadow at solstices separates the cardo from the decumanus, i.e. the north-south and east-west axes respectively.

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

Fig. 2 - Chevet of Reims Cathedral

All we have to do now is imagine that a template would enable us to find the spaces between them, and that it would be right in front of our eyes. Tracing a floor plan would no longer present any difficulty...

But now it's time to introduce a little symbolism into what I'm saying. Remember, I'm talking about cathedrals, not station halls. That's why the polygon of the transept crossing will be called a "Table", a symbolic term 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. This is why the outer polygons will be called "second enclosure" and "third enclosure" respectively (fig. 3). I call the whole thing a "triple enclosure system".

Fig. 3 - Generic speaker layout

But where can we find this magic template, this abacus I was talking about? To answer this question, we need to determine which element of the plan is likely to contain the various gaps we're looking for. Once we understand that each enclosure gives rise to the next, it seems logical to start with the one from which everything flows, the Table. Thus, every Gothic cathedral could be defined at its center by the transept crossing. The discrepancies we're looking for would be found in the watermark. A fascinating hypothesis.

Let's verify this deduction by drawing the Table de Reims. As I just mentioned, the Table de Reims is built on an equilateral triangle (fig. 4), which determines a right-angled quadrilateral. Let's remove the triangle and draw the axes. In this new figure, it will be possible to draw three segments.
The first can be drawn by tracing a circular arc with radius [0,1]. Cutting the top of the Table in 2, it defines segment A (fig. 5a). The second, segment B, is simply the radius we've just used, i.e. segment [0,1] (fig. 5b). The last, segment C, is directly given by the half-width of the Table. Of course, these distances are not chosen by chance. They correspond exactly to those found between the cathedral's various enclosures and, by transferring them to a blank page, should enable us to retrace the building's floor plan.

Fig. 4 - Reims table

Fig. 5a - Plot of segment A

Fig. 5b - Tracing segments B and C

Fig. 6a - Layout of the second enclosure

Fig. 6b - Drawing of the third enclosure

Note that one of these segments is common to both constructions. For the record, the height of the table, if transferred vertically from the second enclosure, indicates the birth of the apse. As for the nave bays, they are all regulated by segment C.

In this way, we've rediscovered the main proportions of Reims. The width of the main nave, the aisles, the chevet, the length and width of the transept, the apse, which is a circular arc traced in the continuity of the table, and the arch of the ambulatory, which is regulated by the second enclosure. The entire plan of the cathedral is contained, to scale, in a simple figure known as the Table. Its name thus takes on its full meaning. The origin of the cathedral, it is both its geometric and vibratory sounding boards, like a musical instrument.

But what is the real nature of the traces inscribed on the Tables? We'll see later that they share a common logic, but it's hard to understand their intrinsic meanings or the criteria by which they were conceived. Are they simply a mnemonic device, a means of encoding the proportions to be given to a building, or are they the geometric consequence of a superior layout? To tell the truth, I'm still pondering the question, but knowing the empirical approach of the first master builders, it seems more than likely that they sought to record the ratios of proven proportions, so as to remember them and pass them on in a simple manner. What could be more natural, for these geometric architects, than to prepare for construction by drawing them as tracings in the center of the sanctuaries?
If you have any reservations on this point, it will suffice to consider the system of segments as a universal means of measuring and tracing a Gothic plan, to the exclusion of all other considerations.

Some questions remain. Why does the cathedral have such different volumes? Wouldn't it have been simpler for the architect to define a single module for the entire building?

There are several answers. Technical, first of all. The cathedral is built of walls and vaults. The latter require complex equipment in order to be installed. The structures take up space, a discrepancy that adds up to an ideal layout. At the same time, the cathedral expresses a numerical, geometric and aesthetic symbolism that is an integral part of its design and function.

All of which raises a unique question: how did the builders manage to marry these different imperatives? For me, the answer lies in the principle of segments, which, through a simple geometrical procedure, resolve a question that is far from simple. But make no mistake, the segments are first and foremost the consequence of an operative architectural principle, and the triple enclosure the expression of a symbolic message.