Architecture – Page 2 – Geometries of Creation

I’m pleased to have this opportunity to address a session with such a strong geometrical component. I’m also happy to say that my geometrical work has given me a new perspective on Suger’s Saint-Denis, one of the most influential monuments in the history of medieval architecture. I believe that I now understand in reasonable detail the geometrical thinking that governed the east end of the church, whose ambulatory vaults are seen here. This pattern of vault ribs, of course, demands to be understood in the context of the structure’s overall plan.
Here you see that plan, as recorded by Sumner Crosby, who spent much of his career studying the building. At first glance, the plan appears regular, with a corona of seven similar chapels around a central apse. Crosby recognized, however, that the plan was less regular than it seems. In particular, the three chapels close to the building axis are slightly deeper than the other four, with the axial chapel being deepest of all.
Crosby thus concluded that the plan had two distinct geometrical centers: one governing the three central chapels and one governing the other four, as you can see here. The plan can be explained more satisfactorily without this construction, as I will show, but another rather different multi-centered construction actually does have a bearing on the design.
As this photomontage published by Marvin Trachtenberg indicates, a priest at the high altar of Saint-Denis would have an unencumbered view of the chapel windows, since the outer and inner columns align perfectly from this perspective. The chevet thus has an optical center, one that differs, as you will see, from its geometrical center. It is hard to understand these subtleties of the Saint-Denis plan by looking at the main floor alone.
Helpful evidence about the whole chevet design can be found by analyzing the geometry of the crypt below. Indeed, the famous passage from Suger’s “De Consacratione” that mentions “arithmetical and geometrical instruments” explicitly notes that the upper columns and arches were to be placed upon the lower ones built in the crypt.
Here you see a plan of the crypt, based on Crosby’s survey. In this image I’ve superposed the ground plan of the structure on the photogrammetric data about the shape of the vaults, because the vault structure helps to reveal the logic of the design. As this plan shows, the crypt has only a single ambulatory, since solid walls separate the chapels.
Here you can see those walls, and two sets of thick arches. Arches like the one in the left foreground run radially, continuing the line of the walls toward the chevet center. The other arches seen here run circumferentially, separating the ambulatory from the chapels.
In plan, these circumferential arches describe a thick arc, as the red shading here indicates. This plan of the ambulatory thus involves a strictly circular geometry.
The four western radiating chapels are also circular in plan, at least as far as their outer outlines are concerned.
This means that we can unambiguously locate the geometrical centers of these chapels, and of the chevet as a whole, as the small open circles now indicate. With these results in hand, we can now begin to grapple with the overall logic of the crypt zone.
First, I can draw a large semicircle through the centers of the four western chapels. This can be seen as the fundamental generating figure of the whole chevet geometry.
Next, I can circumscribe half of a rotated square around that semicircle. As you can see, the top corner of that figure corresponds very closely with the easternmost point on the axial chapel buttress. The left corner of the figure corresponds closely, albeit imperfectly, with the left corner of the large buttress between the chevet and the straight bays. The basic observations I have been discussing so far are my own, but in the next several minutes I will present a series of more subtle geometrical observations made decades ago by Crosby’s former assistant Richard Nash Gould. I am grateful to Bill Clark for bringing Gould’s unpublished work to my attention, and I am grateful to Gould himself for allowing me to use his brilliant ideas as a springboard to my own more comprehensive analysis.
Gould’s first crucial observation is that the geometry of the Saint-Denis crypt involves subdivision of the chevet into wedges of 27 degrees, each corresponding to a single radiating chapel.
Gould also realized that the size of the chapels could be found by constructing squares concentric with the chapel centers, and expanding them until their inner corners touch; these intersection points coincide with the inner edges of the buttressing walls between the chapels. In the four western chapels, these squares fit neatly within the orange shaded circles describing the outer wall surfaces.
Their inner wall surfaces, meanwhile, correspond with the yellow circles inscribed within the squares. The interior dimensions of these chapels are thus smaller than the exterior dimensions by a factor of root two. As noted previously, the three eastern chapels are slightly deeper than the rest, with the axial chapel being the deepest of all. Their geometry does not involve a separate arc, as Crosby had supposed based on his examination of the chevet’s main floor. Instead, Gould’s analysis of the crypt geometry convincingly demonstrates that the chapels were extended by a simple construction related to the Golden Section.
More specifically, Gould showed that the chapel geometry could be found by creating isosceles triangles, here shown in green, whose short bases correspond to the radii of the normal chapels, and whose long sides parallel the side of the squares framing the third and fifth chapel. Because of the subdivision of the chevet into 27-degree slices, the sharp angles in these triangles measure 36 degrees. Each green triangle thus resembles the tip of a pentagonal star, a figure whose proportions are defined by the Golden Section. I’d be remiss not to mention that well-known relationship, but I won’t explicate it further, since I have more specific arguments to makes about Saint-Denis. I should point out, for example, that the center of the axial chapel can be found by striking lines from the adjacent chapel centers at a pentagon-defined angle of 72 degrees, as the green lines near the top of the figure indicate. Everything I have said in the past few minutes comes from Gould’s analysis, which impressed me greatly. In the rest of my talk, I want to build on his foundation. To begin with, I want to locate certain important elements in the crypt that he did not discuss, starting with the red shaded arc that separates the ambulatory from the chapels.
As it turns out, the inner margin of that arc can easily be found by drawing 30-60-90 triangles whose outer corners lie on the semicircle through the chapel centers, as you can see here in green. The proportions of the crypt, therefore, involve the geometry of the equilateral triangle as well as those of the square and pentagon.
The outer margin of the arc, now shaded in blue, falls halfway between the inner margin and the semicircle through the chapel centers. The thickness of the arc defines an interval, indicated by the small blue triangles, that will be important for the geometries of both the crypt and main floor.
A smaller arc of the same width, which frames the columns of the hemicycle, can be found by striking diagonals from the endpoints of the outer arc until they reach the 30-degree sides of the large green-shaded triangles.
The centers of the large columns at the base of the hemicycle stand two arc widths to the west of the chevet’s geometrical center. The eastern face of the salient buttress just to its north begins one half of an arc width west of that center point, though its surface steps back slightly further out from the building, as you can see in the lower left section of the graphic. The salience of the buttress can be found by extending the line through its east face and the diagonal of the great isosceles triangle until they intersect. This intersection defines the overall width of the chevet, an important dimension to which I will return at the conclusion of my talk. First, though, I want to show how the results that I have illustrated in the crypt go on to inform the geometry of the Saint-Denis chevet on its main floor.
Here you again see the familiar plan of that upper zone. The most significant difference from the crypt plan, of course, is that the main floor has open spaces rather than walls between the chapels, creating a second ambulatory.
In this slide you can now see the ghostly image of the crypt plan bleeding through into the plan of main floor, to facilitate comparison between the geometries of the two levels.
Here you see the same isosceles triangle I had introduced before, along with the 27-degree wedges identified by Gould.
Here once again are the seven touching squares, four of which are concentric with the western crypt chapels.
And here again are the yellow circles inscribed within them, which describe the interior walls of those crypt chapels.
The green shaded triangles here show Gould’s clever scheme to determine the extra lengths of the three central chapels, while the unshaded green sloping lines show the 30-degree construction that I used in the crypt to locate the arches separating the chapels from the ambulatory. The green verticals dropping through the resultant intersection points locate the axes of the intermediate aisle columns in the straight choir bays, seen here at the bottom of the graphic. Since the columns at the base of the ambulatory stand slightly further in towards the building centerline, in accord with the geometry of the chevet, the arches connecting these columns diverge slightly from east to west.
Here, in blue, I’ve added the same construction used in the crypt to determine the radius of the hemicycle.
And here, in violet, I’ve again shown the constructions that govern the buttress depth and the location of the two large columns at the base of the hemicycle. All of that should be familiar from my discussion of the crypt. Now we can begin to see in some new ways in which the geometry of the crypt affects that of the main level.
The bold red semicircles here define the inside margins of the thick arcs seen in the crypt.
As you can see here, tangents to those semicircles defined the axes of the main arcades, as well as those of the intermediate columns in the aisles.
More surprisingly, perhaps, a yellow circle through the tips of the green shaded triangles does more than locate the center points of the intermediate columns between the ambulatories; it also determines the length of the first straight bay in the choir.
At this bay division, the columns between the aisles align with those of the main arcade, as the green horizontals here indicate. At the base of the chevet, however, things are slightly more complicated.
Here, the arches defining the bay divisions in the inner aisle slope slightly, since the centerline of the salient buttress, seen at left, does not align with the centerline of the main arcade columns. The buttress thickness is five times the half-span of the arcs defined in the crypt, as the five small green circles indicate.
The centers of the hemicycle piers lie on the midline of the crypt arc, as the green semicircle between the red and blue ones shows. You’ll notice here that the boss of the 13th century vault over the hemicycle coincides with the geometrical center of the chevet. The columns of the hemicycle, though, are not perfectly aligned with the red rays converging to this center.
Instead, they are aligned with these blue rays, which converge on a point halfway between the geometrical center of the chevet and the bay division separating the chevet from the straight bays.
From the high altar located at the convergence of the blue lines, therefore, a priest would see the famous vista of all the chevet windows, with the outer columns of the ambulatory perfectly hidden by the inner columns of the hemicycle. This alignment, with which I began today’s presentation, was clearly contrived deliberately by the Saint-Denis designer, using an optical center displaced subtly westward from the chevet’s geometrical center. I see this as a beautiful result, and I confess to being thrilled about the way this whole geometrical analysis of Saint-Denis has come together. I imagine, however, that some of you may harbor doubts about elements of this analysis, so in the time that remains to me, I’d like to briefly put this material into a broader context, using evidence from several related buildings to buttress some of my assertions that may seem far-fetched. You may find it strange, for example, that I’ve used the geometry of the equilateral triangle to help establish the proportions at Saint-Denis, since the forms of the building do not express this geometry in any overt way.
As my first witness for the defense, I’d like to call Notre-Dame in Paris, whose east end was designed roughly two decades after than of Saint Denis. At Notre-Dame, as at Saint-Denis, the equilateral triangle plays a major but covert role in shaping the ground plan.
Notre-Dame is famous for having double ambulatories, around which a belt of chapel was later added between the main buttresses. All three of these belts have the same width, which is slightly less than half the span of the main vessel. More specifically, the ratio in question is simply .866, the cosine of 30 degrees. This means that the half-span of the main vessel relates to the width of each aisle as the side of an equilateral triangle relates to its altitude. I will show you what I mean in a moment. First, though, I want to observe that the radial subdivisions of the Notre-Dame chevet are really peculiar. The apse is not subdivided into five equal slices, despite the five-fold subdivision of the hemicycle. Instead, the wedges flanking the straight bays are narrower than the three in the middle, corresponding to two bay divisions in the outer wall rather than three. I think, however, I think I have figured out the logic of this plan.
Here you see a schematic showing likely steps in laying out the east end of Notre-Dame. First, make a red circle of radius 6.65m to describe the hemicycle. Then, inscribe 30-degree wedges within that circle, just as I did at Saint-Denis. Those wedges intersect the circle at points whose distance from the building axis, 5.76 meters, is smaller than the hemicycle radius by the aforementioned cosine of 30 degrees. This 5.76-meter interval corresponds to the width of the each ambulatory belt. As the blue lines indicate, the central slice of the chevet could in theory be found by drawing lines from the hemicycle center to the points where the main arcade axes intersect the inner green circle, which once defined the outer wall of the building. This construction gives an angle of 42.94 degrees, which could be cloned to give the two adjacent wedges, as the purple arcs show.
In practice, the builders of Notre-Dame appear to have made an error. Instead of using the intersection between the arcade axis and the green circle, they evidently used the intersection between the arcade axis and the tangent to the circle, as the red wedge indicates. As a consequence, the whole chevet geometry was rotated slightly counterclockwise, dragging the southern buttresses east and pushing the northern buttresses west.
To convince you that this isn’t random speculation, I’ve here superimposed my schematic on an extremely precise laser scan of the east end made by Andrew Tallon, whose sharing this data with me I greatly appreciate. As I hope you can see, everything fits beautifully, even the rotated elements. All of this Notre-Dame geometry, I remind you, depends on the establishment of a simple equilateral triangle construction like the one I introduced at Saint-Denis.
The cathedral of Reims, designed half a century after Notre-Dame, can also be seen as a descendant of Saint-Denis, not least in terms of its east end design.
Reims is famous for having five deep eastern chapels, as you can see in this plan that I have created based on surveys made by Henri Deneux. Although this format at first appears very different than that of Saint-Denis, where we saw seven shallow chapels and two ambulatories on the upper floor, the Reims design has much in common with the Saint-Denis crypt. Indeed, the ground plan geometries of the Reims chapels closely follow those of Suger’s building, although I don’t have the time today to discuss the subtleties of this relationship.
Even more striking evidence for the impact of the Saint-Denis chevet on thirteenth-century designers comes from the Cistercian church of Altenberg, begun in 1259. Here, as in Suger’s church, seven chapels cluster around the east end.
The plan of Altenberg, like that of Saint-Denis, develops around a rotated half-square that links the salient buttresses at the base of the chevet with the axial chapel at its apex. As Norbert Nussbaum has demonstrated, these were the first three parts of the chevet to be constructed.
The diameter of the hemicycle equals half the side length of the half-square, and the diameter of the ambulatory can be readily found within this framework, through a series of steps that I have no time to trace. Two points about the relationship between Altenberg and Saint-Denis deserve particular note, however.
First, the columns of both chevets are carefully arranged to create optical centers slightly to the west of their geometrical centers, so that a priest standing at the high altar would enjoy an unimpeded view of the windows in the chapels. Second, and even more remarkably, the two buildings share a common scale. In fact, the radii from their geometrical centers to their outer buttress faces match literally to the centimeter. It thus seems clear that the designer of the Altenberg chevet had gained an intimate familiarity with the Saint-Denis design, either through first-hand observation and measurement, or through the medium of scaled drawings.
In conclusion, therefore, I feel that geometrical investigation has given me new insight not only into the design of Saint-Denis itself, but also into the reception of its lessons at sites including Paris, Reims, and Altenberg. I hope that the convergence of this evidence has convinced you of the value of this approach to Gothic buildings, which surely rank among the most spectacular medieval examples of “geometrized substance.”