The Geometry of Beauvais Cathedral: The Thirteenth-Century Campaigns

Here we see the plan of the cathedral choir, based on a laser survey commissioned by Murray, and published in 2014.* This graphic usefully summarizes interaxial dimensions measured in the real building, which can then be compared to the “ideal” values proposed in the geometrical model. All the “ideal” dimensions will be derived from a single ”seed dimension,” which will here be chosen as the interaxial span of the choir vessel. Since the northeast crossing pier had to be reconstructed in the sixteenth century, the most reliable dimension would seem to be the 15.34m interaxial span of the adjacent bay.

Stephen Murray, “Plotting Gothic: A Paradox,” in Architectural Histories, vol. 2, issue 1 (2014), fig. 1.

Here we see another version of the plan, now shown without the dimensions, but showing the angular division of the chevet.* To begin the analysis, establish the interaxial width of the planned choir, 15.34m. It will be convenient to measure from the centers of the eastern crossing piers, which therefore set level 0 in the east-west dimension. The crossing should notionally be a square, so its center will lie 15.34m/2 = 7.67m west of the pier centers. The arcade axes are very subtly sloped compared to the exterior wall axes, for reasons to be discussed below. Although the walls were built first, the ideal geometrical figures described here will be aligned with the arcade axes, so that they agree with the pier extensions above them.

* Stephen Murray, “Plotting Gothic: A Paradox,” in Architectural Histories, vol. 2, issue 1 (2014), fig. 6.

As Murray noted, the plan of the cathedral can be understood as three great corridors of space, each equal in width. The central one corresponds to the choir, while each of the side ones corresponds to a pair of aisles and the thickness of the exterior wall. To find the width of the side aisles, construct an octagon whose facial radius equals the 15.34m choir span, and whose center lies on the southern edge of the crossing square. Its southeast diagonal facet will cut the 0 line 14.02m south of the crossing square. Each of the two aisles measures 7.01m, or half this span. (Murray gives 6.96m and 7.04m in this zone; total error 2cm out of 14m, or .999 accuracy). (The south transept buttress axis is clearly to the east of the predicted line; Murray gives the e-w bay length as 7.82m there, instead of the predicted 7.67m. This, however, results from the previously mentioned skewing of the interior pier compared to the exterior wall; on the north the dimension is 7.74m, for a good match.)

Murray noted in 2014 that the east-west length of the choir closely matches the altitude of an equilateral triangle framed by the intermediate arcade axes, but he thought that this might be coincidence. The match should not be dismissed, because it is so precise. With the baseline of the yellow triangle equaling 15.34m+14.01m=29.35m, its altitude will be 25.41m. Murray’s previous figure gives the bay lengths on the south side at 7.82m, 8.82m, and 8.71m, which sum to 23.35m. The match here is a difference of 6cm over 25m, or .997 accuracy. There is good reason to believe that the two eastern bays of the choir were meant to have equal lengths of 8.87m. The second bay measures 8.85m on the north and 8.82m on the south, agreeing well with this hypothesis. The eastern bay measures 9.17m on the north and 8.71m on the south, due to rotation of the chevet to be discussed below. These value average to 8.94m, still matching the hypothesis to .992 accuracy.

The buttresses of the choir protrude by 2.81m from the exterior wall, so that their outer surfaces are 25.81m from the building centerline. By comparison, the wall surfaces are 3/2 x (15.34m) = 23.01m from the centerline. The ratio between these is 23.01/25.81=.891, which is the cosine of 27 degrees. This is a very interesting result, since Richard Nash Gould convincingly demonstrated that the choir of Saint-Denis was laid out in 27-degree slices. The 27-degree angle can be easily constructed by subtracting the 45-degree angle seen in the square from the 72-degree angle seen in the pentagon, as the green figures in the apse here indicate. It should be noted that these relationships of buttress width involve the structure build in Murray’s phase a, the episcopate of Miles of Nanteuil. Before going on to consider how this planning may have been adopted and modified in the following phases, it makes sense to consider the heights established in phase a.

Here is the laser-scanned cross-section of the first bay division east of the crossing, again based on Murray’s survey.* Much of the superstructure shown here was reconstructed, but the lower stories evidently date to phase a. The choir and aisle widths have been indicated at the bottom of the graphic.

Stephen Murray, “Plotting Gothic: A Paradox,” in Architectural Histories, vol. 2, issue 1 (2014), fig. 6.

As Murray recognized, the 15.34m height up to the base of the arcade capitals equals the interaxial width of the choir. Crucially, this height expressed itself on the outer aisle wall belonging to phase a, and not just on the arcade columns constructed in phase b. The lip of the upper aisle wall, and the springing of a prominent gargoyle, can be found at height 22.34m, which is just the sum of the choir and aisle widths, as the red and orange diagonals here show.

The base of the aisle triforium lies at height 14.02m, equaling two interaxial aisle widths, and the vertices of the transverse aisle arches are at height 21.03m, or three aisle widths, as the yellow diagonals here show.

The anomalous low flying buttress on the north side may provide a clue about a preliminary phase in the design of the cathedral’s superstructure. Since the flyer has 30-degree slope, is seems tantalizing that the line along its upper surface intersects with the building centerline precisely at the tip of an equilateral triangle whose baseline spans between the vertices of the transverse aisle arches. It is tempting to imagine that this might have been the vault height planned when the flyer was built, but this scenario obviously must remain hypothetical. It is important to note, moreover, that this flyer was built at part of phase b, corresponding to the episcopate of Robert de Cressonac. Murray has plausibly suggested that the original designer active in phase a might have envisioned an even lower building, incorporating flyers without intermediate uprights. Now, we can turn to consider the ground plan of the the cathedral’s east end…

The chevet has seven chapels. The pair at the base of the chevet are bent slightly inwards compared to the aisle axes; this would not be the case in a perfect 7/12 geometry like that seen at Cologne. Murray therefore proposed that the Beauvais east end was instead supposed to be 7/13, i.e., seven thirteenths of a 13-gon. This would give wedges of 27.7 degrees. Since a perfect 27-degree wedge was used at Saint-Denis, and apparently in Beauvais phase a to set the buttress widths, it is tempting to imagine that the phase a designer intended to use 27-degree wedges in the Beauvais hemicycle as well. The difference between these options is small. There are two problems with these hypotheses, however, as the blue constructions here show. First, a regular figure of this sort would have its geometrical center about a meter to the west of the center Murray identified. Second, the angles of the first chapel pair would have been much more than we see.

The violet constructions here show an alternative construction that matches the building much better. Since Murray identifies the radiating chapels as some of the earliest work in phase b, it seems likely that misunderstandings of the plan geometry could have crept in at that point. In particular, it would have been easy for the phase b designer to think that the chapels were supposed to pinch in by 9 degrees in total, instead of the 9 degrees per side that they would in the construction with 27-degree wedges, or the 13-gon. If the south chapel was angled at 4.5 degrees instead of 9 degrees, and if its opening were taken as 7.01m, equal to the aisle span, then the perpendicular bisector of that segment would intersect the building centerline 2.36m east of the straight bays, agreeing superbly with the geometrical center Murray located. This center also gives an arc large enough to pass through the engaged pier centers.

If construction began on the south, as seems likely, any errors in layout would accumulate as work progressed north. The red arcs shown here all have radii of 7.22m, which is interaxial width of each chapel. This distance can be found by starting at the southern engaged pier, and then measuring along the violet line describing the southern chapel opening until it is intersected by the red line from the chevet center, which is tangential to the violet arc giving the first approximation to the chapel opening. It this 7.22m dimension, which closely matches Murray’s measurements, is reproduced from south to north as the red arcs indicate, it will fail to cover the full arc of the chevet, leaving a gap of .36m, as shown by the small red box at the north. This error explains the torsion of the chevet geometry (from phase b) compared to the aisle walls (from phase a). This also explains why the east bay of the choir appears longer than it should be, measured between pier axes, on the north side only.

Moving back into the vertical dimension, we here see the scanned section of the choir at the base of the chevet.* As before, the notional choir and aisle widths are indicated at the bottom, and the wall and buttress widths have also been translated from the ideal plan construction.

* Stephen Murray, “Plotting Gothic: A Paradox,” in Architectural Histories, vol. 2, issue 1 (2014), fig. 7.

As seen previously in the section of the transept-adjacent bay, the aisle capitals begin at height 15.34m, the aisle arch tips at height 21.03m, and the aisle wall lip at height 22.34m. The interval between the latter two dimensions, 1.31m, equals the width of the wall as established in phase a. If we add another such interval to the height, 22.34m + 1.31m = 23.65m, locating the height of the uppermost horizontal molding on the exterior buttress, which was among the last parts to be built in phase b.

As at Reims and Amiens, the proportions of the triforium are set by an octagon centered at the top of the aisle level, here meaning 22.34m. The facial radius of the octagon is 7.01m, matching the span of the aisles. Its diameter is thus 14.02m, setting the span between the arcade walls constructed in phase b. The top lip of the triforium, at height 28.82m, corresponds not to the top of the octagon, but to the points where the rays to its corners cut the circle inscribed within it. A prominent molding on the intermediate pier upright occurs at the same level. This upright, the triforium, and everything above evidently belong to phase c.

Here the overall logic of the phase c cross-section design comes clearly into view. The section is a large square, in several senses. Arguably the most important square is the one filling the space from floor level to height 45.99m, which is framed in the horizontal dimension by the outer wall surfaces; its diagonals are shown in blue. If these diagonals are further extended to meet the rising green verticals framing the outer buttress surfaces, though, one reaches a level 48.80m above floor level, corresponding to the base of the timber roof. The ridge of the roof, 59.49m above the floor, corresponds to the top point of an arc swung from the intersection the blue diagonals through the corners at 48.80m. All told, this suggests that the composition should be read as a great square measuring 51.62m per side (twice the 25.81m distance from the building centerline to the buttress surfaces established in phase a). The baseline of this square would lie 2.81m below the cathedral’s floor level (i.e., the same dimension as the buttress salience), thus corresponding closely to the ground level on the south side.

Within this frame, many secondary levels can be easily found. These include: –the molding on the intermediate upright at height 37.67m, where the blue diagonals cut the intermediate pier axis; –the gargoyle and the main capital tops at height 38.33m, which is 7.67m or half a choir span below the top of the “master” square are heigh 45.99m; –the top of the tiny pinnacles on the buttress uprights at height 44.68m, where the blue diagonals cut the axes rising from the interior wall surfaces; –the top of the intermediate pier uprights below the pinnacle, at height 45.99m, aligned with the top of the “master” square. It is interesting that the top of the main choir vault, and the tops of the upper flyers, lie between the “master” square in orange and the larger framing square shown in green, but the reasons for this soon become apparent…

The centers of curvature for the transverse ribs of the high vault lie at level 37.67m, aligned with the intersections of the blue diagonals and the intermediate pier axes. These centers are located directly above the points where the rays of the green octagon in the triforium cut the circle within it. The interior surfaces of the pier extensions, shown here in violet, are halfway between the walls of the green octagon and the analogous lateral intersection points. When arcs with the previously described centers are struck tangent to these violet verticals, they intersect 46.70m above the floor (not the 46.02m that would be 3 x 15.34m). When similar arcs are struck through the red pier axes, their tops rise to height 48.01, locating the top edge of the clerestory walls. From that level the upper flyers descend with 30-degree slope, intersecting the blue diagonals at height 41.66m, locating the tops of the en-delit shafts on the culées. The lower set of main flyers also has 30-degree slope, passing through the points at height 37.67m where the blue diagonals cut the intermediate pier axes.

For a final perspective on how this geometry matches the building, the exact same set of lines just shown has here been superposed over the familiar section of the transept-adjacent bay. The roof and vault forms here still match, since they are essentially identical to those seen in the eastern section. The interesting new results in this graphic involve the buttress uprights, parts of which were rebuilt after the partial collapse of 1284. Most notably, it can be seen that these uprights all terminate at height 45.99m, aligned with the top of the orange “master” square. The northern buttress upright of the choir (i.e., discounting the transept) survived the partial collapse, and its articulated shaft terminates at height 41.66m, aligned with the top of the en-delit shafts seen in the eastern culées. Despite the cathedral’s checkered history, its geometrical logic still shines through…