Category: Medieval

At right you now see an elevation of the north transept in its actual format, with pointed arches over the rose and tower windows. I’ll say more later about why I tweaked those features in my reconstruction, but my main goal today is to describe the geometry of the present transept. I should confess right up front that I’ll be passing in silence over some of the most strikingly peculiar features of the north transept, like the format of the portals, or the way that the blocky masonry around them obscures all but the tips of the triple lancets, whose analogs are fully glazed on the south side. Instead, I’ll speak more generally about the governing geometrical framework of the transept, which reveals far more about the whole cathedral’s design than I had initially supposed.

To put this into context, I’ll begin by looking at the plan of the crossing zone, seen here with the east at the left, so that the buttresses on the bottom of the image fall in the same order as those seen on the elevation.

As Nancy Wu has shown, the crossing and adjacent bays form a nearly perfect square, seen here in green.

The square measures 31.02m meters on a side, so that its edges coincide closely with the axes of the outer tabernacles on the upper transept façade. Every dimension that I’ll cite today will derive from the size of this square, in an unbroken chain of geometrical relationships. These ideal dimensions correspond very well to those of the building, based on all the data that I’ve found in Nancy’s work, and in the monographs by Richard Hamann-MacLean and Alain Villes.

Nancy also convincingly showed that the proportions of the crossing by were set by this simple construction, in which the half-diagonals of the green square were unfolded to its faces. This sets the east-west depth of the crossing bay at 12.85 meters, and the widths of the adjacent bays as 9.09 meters, measured between the pier axes.

Half the latter dimension sets the height of the left-hand transept portal, measured to the base of its tympanum.

Crucially, however, the 12.85-meter span between the pier axes of the central vessel does not match the distance between the axes of the inner tabernacles, which is 13.54 meters.

This dimension at first seems to correspond to nothing in the ground plan.

However, if one takes it as the face-to-face diameter of an octagon, then one finds that the diameter of the circle circumscribed about that octagon equals the 14.65-meter span between the nave pier axes. The nave span and the span between the inner tabernacles thus relate by the simple principle that I have taken to calling octature, which is just like quadrature, except with rotating octagons instead of squares. To understand the geometry of the transept, though, I now have to somehow relate the nave span to the 31.02-meter span of the great central square. Such a connection can readily be found, but it requires more global consideration of the cathedral’s plan.

Here you see the plan of the cathedral’s eastern half, this time oriented more conventionally with east at the right. Here I have retained the green-shaded square of the crossing for reference. Since construction work may well have started with the chevet, though, I’m now going to get rid of it, leaving only its span of 31.02 meters.

That dimension, and the placement of the hemicycle center, is all we need to start. You can see that a red semicircle of this diameter passes through the centers of the engaged ambulatory piers.

Since the chevet has decagonal symmetry, I next create an orange star decagon with line segments passing through these pier centers. Its corners fit into a circle of diameter 50.20 meters. As you can see, this dimension sets the width between the outer wall surfaces of the straight bays. Also, the facets of the star intersect at the geometrical centers of the chapels.

Here, in yellow, I’ve added half of a decagon whose facets pass through these centers. Each of these yellow facets measures 11.85 meters per side. Now, if I just expand this by one step of what you might call decature, I get the choir span, which is what I was looking for.

Here, in green, you can see what I mean. I just project the 11.85 dimension westward until I meet the red rays between the first and second pairs of chapels. Then, I draw the semicircle of the hemicycle through these intersection points, and I find that it has a diameter of 14.65 meters, which closely matches the choir span. So, starting from the single master dimension of 31.02 meters, I’ve derived not only the location of the chapel centers, but also the widths of the main vessel, and of the building as a whole. With these results in hand, it becomes surprisingly easy to derive the east-west dimensions of the whole east end and crossing.

If one simply extends the rays between the first and second chapel pairs until they hit the choir axes, for example, one finds that the intersections lie 5.32 meters west of the chevet baseline, thus describing the length of the first straight bay.

The overall scale of the whole crossing zone can be set by unfolding the diagonals of the yellow square adjacent to the hemicycle center, which has the familiar side length of 31.02 meters. These unfolding arcs span a total of 56.72 meters from north to south, creating a box that neatly frames the outer transept walls, and the outer buttress surfaces of the straight bays. In the east-west dimension, meanwhile, the yellow arcs reach to points 43.87 meters west of the hemicycle center, thus defining the axes of the piers just west of the crossing. This matches the actual dimension in the building, as measured by Hamann-MacLean, literally to the centimeter.

It is from this yellow baseline that one can insert the familiar green square of the crossing zone. As you can see along the bottom margin, the leftover space between this square and the eastern straight bay measures 7.53 meters, which is the depth of the second straight bay. The depths of the tower bays and main transept bay, which are 9.09 and 12.85 meters respectively, were already determined, you will recall, from the simple half-diagonal constructions discovered by Nancy Wu, which you see here now in green.

To get the total width of the transept arms, finally, it suffices to extend the central green square into a double square, as the blue extensions here show. This construction works, more specifically, when one measures to the outside faces of the southern transept buttresses, and those of the northwest transept tower, rather than to the wall between the north transept portals, which sits slightly further back. Because these geometrical steps are so simple and elegant, and because they produce forms that match the fabric of the building so precisely, I feel reasonably confident that they were actually intended by the cathedral’s first designer, who evidently conceived at least the whole eastern half of the cathedral according to a single coherent ground plan. So, while I admire many aspects of Alain Villes’s recent monograph, I am unconvinced by his suggestion that the cathedral was originally planned to be more like Chartres, with four equally-sized choir bays and three narrow bays per transept wing. My analysis also has some possible implications for the relative dating of the chevet, transept, and nave campaigns, but I do not pretend that I have thought through all of these angles, which I’d be happy to discuss in the Q+A period. For the moment, I want to extend my analysis into the vertical dimension.

Here you see the cross-section of the nave and the elevation of the north transept, set to the same scale.

By way of reminder, the crucial dimensions in the transept plan were the 31.02 meters between the outer tabernacle axes, the 13.54 meters between the inner tabernacle axes, and the 12.85 meters between the crossing pier axes.

The span of the nave meanwhile, was already shown to be 14.65 meters, which was greater than the 13.54- meter inner tabernacle span by a single octature factor. As it turns out, octagons hold the key to the entire cathedral elevation. Look what happens when you make an octagon whose base facet equals the base of the nave.

Its equator comes exactly to the top of the arcade zone, at the triforium base.

When one superposes the same exact octagon on the transept elevation, one finds that its equator aligns with the prominent molding at the triforium base. The octagon’s upper facet, meanwhile, passes almost exactly over the top of the rose window, as if it were meant to frame it. This observation raises the tantalizing possibility that the transept rose was originally meant to sit under a round rather than a pointed arch. There are good formal reasons to suspect that this may have been the case, too. As I pointed out to a student when I first began thinking about this project, a round rose frame would harmonize nicely with the round-headed arches over the three rosettes in the triforium. A round arch over the rose would also make sense, given that such rose frames were also seen at the cathedrals of Laon, Chartres, Paris, and Amiens, to name just a few familiar comparanda.

I soon found out that Alain Villes had been coming to very similar conclusions, as you can see in this illustration from his monograph. He provides two alternatives here, beyond the fact that he’s showing variants on the north and south transepts, respectively. In the first, he keeps the single large lancets in the towers flanking the rose, and he gives the tabernacles proportions similar to those seen today. In the second, at right, he imposes a single horizontal terminal molding across the whole façade, which forces him to shorten the tabernacles and eliminate the framing arches in the towers.

This is basically the format that I adopted in my Photoshopped version of the north transept. In making this image, I left the proportions of the twin lancets in the towers unaltered. Their height works well, I think, with that of the rose window in its round frame, creating an ensemble rather similar to the transepts of Chartres, or the west façade of Amiens. The reasons for choosing the round-headed arch are not just formal.

As Deneux recognized decades ago, the cathedral’s present vaults are steeper and more sharply pointed than those originally planned. If you continue the curve of the original springers, as Deneux did in the image at right, you find that the original keystones would have been about 1.70 meters lower than those seen today. This is one reason why both Peter Kurmann and Alain Villes began to suspect that a round arch had originally been planned over the transept rose. And it’s also a reason why I believe that a single cornice line governed the whole composition. Geometrical analysis greatly strengthens this argument.

When you superpose Deneux’s drawing of the original vault curvature on the overall nave elevation, as I’ve done at left, you find that the originally planned interior height would have precisely equaled the height of the great octagon.

Other formal details in both the nave section and the transept elevation underscore the importance of this governing figure.

For starters, I’ll note the small shafts flanking the transept tabernacles, whose bases stand at the same level as the top of the octagon, which I now show within a yellow-shaded framing square.

This square is 35.36 meters on a side.

If we now start to trace horizontals within the octagonal frame, we begin to find many key levels in the cathedral’s elevation. At left, for example, the green shading shows that the lower lateral corners of the octagon align with the bases of the capitals in the wall piers of the nave, and with the equators of the arcade capitals. On the right, you see that the same level corresponds to a prominent horizontal molding over the western portal of the north transept. The transept triforium, also shown in green, fills the interval between the octagon’s equator and the level where the rays rising to its upper lateral corners intersect the framing axes of the outer tabernacles.

The height of the little shed roof in the western transept tower, seen here in blue, seems to have been set similarly, by the intersection of the descending ray with the blue vertical of the pier centerline. On the eastern tower, a similar construction involving the orange upright of the octagon frame locates the prominent horizontal over the portals, whose height thus equals the 14.65-meter span of the nave. The horizontal molding under the rose window terminates at the same height as the octagon’s upper lateral corners, as the subtle orange shading in that zone shows. Now, let’s go upwards to consider the geometry of the towers flanking the rose.

Each tower is a perfect double square, framed by the outer green tabernacle axes and the inner blue pier axes. As you can see, the double lancets begin to spring at the top of the first of these squares, while the tower cornice terminates atop the second. This cornice is thus somewhat higher than the one over the rose, creating an awkward step format that was later partially filled in by a row of statues. Such stepped moldings were certainly seen in other facades, including the west façade of Laon, but in light of the geometrical consistency evident in most of Reims, I tend to agree with Alain Villes that the present tower format, like the pointed format of the arch over the rose, probably reflect revisions to the original transept design. Before I conclude with several comparanda to set the planning for Reims into context, I’ll say just a few more words about the geometry of Reims itself. As I mentioned early in my talk, the relative spans of nave arcade and inner tabernacle axes seem to have been set by octature, i.e. by the diameter of a circle compared to that of an octagon inscribed within it.

As you can now see at right, the same relationship sets the width of the transept buttresses, which frame a red circle circumscribing the main orange generating octagon. And as you can see in both of these images, the top of the circle coincides with the top of the current sharply pointed vaults. The adjustment of the vault height, in other words, was not arbitry. Instead, it marked a logical extension of the octature-based design scheme already evident in the building plan.

Adding another of these intervals, one arrives at the top point of the upper flyer, which thus intersects the nave wall at a height equal to the diameter of the red governing circle.

Adding another pair of these intervals, one arrives at the top edge of the nave wall, where the timber roof begins. This is very close to, but not exactly equal to, the level where the transept towers terminate. Moving higher up, though, even the height of the present roof seems to have been carefully planned.

In fact, the height of the roof exceeds the originally planned height of the vaults by a perfect factor of the Golden Section, as the arc at left indicates. I imagine that the roof was meant to be shorter in the original design phase, with its lower vaults and presumably round arches over the transept roses. But this coordination of the roof height with the building’s overall geometrical framework shows that proportional coherence was seen as important in the Reims workshop, even as changes were made in successive phases of the construction process. In closing, I want to briefly demonstrate the impact of this Remois design system, which helps to set what I’ve just shown you into context.

As this image shows, for example, the Liebfrauenkirche in Trier has an octagon-based elevation very like the one that appears to have been planned for Reims. This is not surprising, in light of the strong Remois influence evident in the column and tracery designs at Trier, where construction likely began around 1227. This cannot be taken as a terminus post quem for the vault modifications at Reims, though, since a number of later buildings also adopt the strictly octagon-based elevation format.

At left, for example, you see the cross section of the Clermont-Ferrand Cathedral, begun in 1248, while at right you again see Reims Cathedral, with its more pointed vaults. In these graphics I have equated the size of the octagons, rather than adopting a common scale for the buildings themselves; the relative size of Reims is greater than this slide implies.

At left you now see the choir section of the Cistercian Church at Altenberg, begun in 1259. As at Trier and Clermont, its vaults fit within the framework of the octagon itself. As the green horizontal line indicates, though, its geometrical structure involved octature rather than just the octagon. The height of the main vault capitals thus matches the level where the rays to the corners of the octagon cut the circle inscribed within it.

Here, now, you see the first of the two drawings from the so-called Reims Palimpsest, which has even taller relative proportions than Reims Cathedral itself, involving expansion of the octagonal frame not only by the circumscribing circle, but also by the extension of the octagon’s upper diagonal facets.

This latter principle appears in simpler form in the second drawing from the Reims Palimpsest, now seen at right.

And, last but not least, we see the same principle of the extended octagon in the section of the Cologne Cathedral choir, begun in 1248.

On geometrical as well as formal levels, therefore, I think it makes sense to see the Cologne Cathedral design as an elaborated response to the stretching of the Reims Cathedral vaults, which was probably decided upon around a decade earlier.

The rather awkward layout of the Reims north transept in its current form provides valuable evidence for this transformation of the elevation.

The original design likely featured a round-headed arch over the rose, which would have harmonized with the rosettes in the triforium, while fitting neatly into the octagonal frame that governs the cathedral’s elevation. While I cannot claim to have worked out all of the puzzles regarding the north transept portals, I do believe that I have made good sense of the axis shifts between the upper and lower portions of the transept, or more specifically between the pier axes and the inner tabernacle axes. Their relationship, as I showed in the first part of my talk, actually hints at the larger relationship between the geometry of the crossing zone and the plan of the cathedral as a whole. The north transept has never been my favorite part of Reims Cathedral, but I’m very pleased that this session has at last given me the occasion to consider in detail the geometry of the building where I first fell in love with Gothic architecture. It’s been fun for me to work on this project, and I hope it’s been fun for you to hear about it, even at this early morning hour. Thanks so much for your time and attention.

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.”