Designing Chartres Cathedral: A Geometrical Perspective

Other visitors with a more artistic focus appreciate the fact that the cathedral has survived better than many of its cousins, still preserving impressive ensembles of sculpture and stained glass, like the figures from its western portal, seen at left. Chartres thus has great prominence in art history textbooks, as well as in the specialized scholarly literature on French medieval art and architecture. Today, I will focus on the cathedral building itself, aiming to shed new light both on its evolving design, and on its place within the scholarly narratives of medieval architectural history.

More specifically, I will be exploring the cathedral’s geometry, building on the empirical foundation established by my late colleague Andrew Tallon, whose laser scanning campaigns shed new light on its structure in a more literal sense. At left, for instance, you see Andrew’s cross-section of the nave structure, overlaid with colored lines that I’ve drawn with the Vectorworks CAD software to show its proportions. By the time I’ve finished my talk, I hope that you’ll appreciate both the logic of this linear system, and its implications for our understanding of the cathedral.

To set the stage for this analysis, I should briefly sketch the history of construction seen at Chartres, which is well understood in its broad outlines. The Gothic cathedral we see today was constructed mostly in the twelfth and thirteenth centuries, but its form cannot be understood without reference to the earlier substructures that lie beneath it. As the plan on the left shows, the cathedral grew up on a site with some Gallo-Roman remains, shown in dark blue. The surrounding apse walls, shown in red, were built in the decades immediately after a Viking sack of the town in 858. It was during this period of reconstruction that Emperor Charles the Bald donated to the cathedral the purported veil of the Virgin Mary, which helped to make Chartres into an important center of pilgrimage. Then, following a major fire in 1020, Bishop Fulbert undertook the reconstruction of the whole cathedral, producing the structures shown here in light blue, including an ambulatory around the apse, a set of three radiating chapels, and two long corridors heading west. These, I should emphasize, were the substructures of Fulbert’s church, which still survive today.

At right, for example, you see a view into this crypt’s axial chapel, with its small windows and heavy barrel vaults. This chapel extends out from the semicircular ambulatory, as the red box in the plan at left indicates. I will keep using the box to indicate where we are in the plan.

Stepping back and looking to the right, we now see the view in the ambulatory, showing at right the curving semicircular wall built around the old apse. The size of this semicircle, as I will demonstrate, served as the fundamental dimension from which the rest of the cathedral’s design would be unfolded.

The length of Fulbert’s crypt corridors, for example, derived directly from the size of the ambulatory. More surprisingly, perhaps, geometries based on this dimension would continue to govern the building campaigns undertaken at Chartres in the following two centuries.

The first of these campaigns, begun following another fire in 1134, involved the construction of the cathedral’s west façade and its portals, which are shown at right.

More precisely, these campaigns produced the two towers and the area beneath the rose window, resulting in a façade something like the one now shown in the reconstruction at left. The form of the northern spire shown on the far left is purely hypothetical, since this wooden structure was remodeled at least once before being destroyed by lightning early in the sixteenth century. The spire on the façade’s right, though, was built of stone already in the middle decades of the twelfth century, and it still stands today as one of the oldest true church spires in the history of architecture. In 1194, yet another major fire destroyed virtually all of the cathedral behind the façade block, except for the crypt structures that we have already seen. The rose window between the towers was added in the subsequent reconstruction campaign, which significantly increased the interior height of the church.

This reconstruction effort produced, by the middle of the thirteenth century, the vast majority of the cathedral we see today. This work was fairly rapid by the standards of the period, but it was neither straightforward nor complete. Scholars have recognized various subtle changes of design within this main campaign, as I will explain in more detail shortly.

Most dramatically, the tower bases around the transept and east end of the cathedral were abandoned incomplete, without the upper stories and spires that were evidently planned to crown them. This development, to my mind, deserves a lot more scholarly attention than it has received to date, particularly since it coincided with a tense period in Chartres, which saw the cathedral clergy exiled from the city from 1253 to 1258. This crisis, and other unrest earlier in the century, thus contradicts the romanticized notion of total social cohesion that has often colored interpretation of Chartres. In any case, it was not until the sixteenth century that the cathedral received the taller and more elaborate of its two stone spires, following the destruction by lightning of its timber predecessor.

Among architectural historians, at least, the lion’s share of the scholarly attention devoted to Chartres has focused on the major campaigns that produced the cathedral’s main body in the decades after 1194. The most strikingly innovative aspect of this work was certainly the provision for huge clerestory windows, seen now at upper left. These windows, counting their twin lancets and large upper rosettes, are roughly as tall as the open arcades that constitute the elevation’s lower level.

The windows, importantly, are also taller than the cathedral’s main vault. This means that the vault springers are stilted significantly above the base of the windows, as the red box at left indicates. To see just how revolutionary this format was, it can be helpful to compare Chartres to two slightly older cathedrals.

At upper right, therefore, I show you the elevations of Laon Cathedral, Notre-Dame in Paris, and Chartres. As you can clearly see from the size of the red boxes, the clerestory zone at Chartres stands out as by far the largest, in proportional as well as absolute terms.

This proportional shift was accomplished in part by eliminating the vaulted galleries that had been used at Laon and Paris, which now appear framed by the lower red boxes. This leaves a simpler three-story elevation of arcade, triforium, and clerestory, which would become typical of subsequent Gothic design. Because of this radical innovation, which is often presented as marking a Promethean creative leap, Chartres cathedral has been widely seen as leaving early Gothic precedent behind, and establishing the paradigm for a new era, commonly known as High Gothic. The term clearly uses “high” to connote excellence, but High Gothic buildings were typically high in a literal sense, as the comparison of Chartres with the shorter cathedrals of Laon and Paris demonstrates. In structural terms, the builders of Chartres were able to get away with this increased scale, and with the expansion of the clerestory, in particular, by deploying the most elaborate flying buttresses of the era.

You can see those flying buttresses clearly in this spectacular cross-section of the cathedral’s nave, which Andrew Tallon made based on his highly precise laser survey data. I am tremendously grateful to Andrew for kindly sharing so many fruits of his work with me before his untimely death in 2018, and to his widow Marie for blessing my continued work with that material in the years since then. The geometrical analyses that I will be presenting in the rest of my talk would not be viable without access to high-quality survey data of this kind. As an aside, I should note that I have just finished editing a volume of essays honoring Andrew and our late mentor Robert Mark, which will be published by Brill next month, so I encourage you to check it out if you are interested in such things.

Returning to Chartres, though, I hope you can see from the red lines here how the huge clerestory windows seen on the cathedral’s interior line up with the buttress structures seen in the cross-section. The middle of the clerestory zone is braced by a massive system of flying buttresses, whose upper and lower arcs are joined on each side by three radial columns supporting a miniature arcade. The lower arc meets the wall at precisely the level of the capital where the main vault begins to spring from the interior wall. Slightly below this level, solid walls with sloping top edges brace the beltlike triforium zone.

At the top of the structure, meanwhile, a final set of flying buttresses braces the upper wall, aligning closely with the crown of the main vault. Because these flyers are so much smaller than the massive double set below, and because they perch rather awkwardly atop the buttress uprights, they have often been dismissed as additions or afterthoughts, but close observation of their masonry shows that they were built together with the upper wall. I will return to consider these small flyers at the conclusion of my talk, since their development offers a surprising and important perspective on the origins of the High Gothic design mode.

Before I get back around to considering the cathedral’s nave, I will also discuss the early Gothic façade, paying particular attention to the design of the south spire, and to its relationship with the adjacent portals. I will show that all of these structures, and the cathedral as a whole, were designed using a geometrical system that had its roots in the design of Fulbert’s eleventh-century crypt.

So, to begin this analysis, I here show you the plan of the crypt, with the east end at the right, with Fulbert’s walls shaded grey, and with the later Gothic walls shown unshaded.

The starting point for the design, as I mentioned before, is the circle framing the old Carolingian apse, which serves as the inner wall of Fulbert’s ambulatory. This circle happens to measure 21.29m in diameter, as I have indicated with the numerical label at right, but the absolute dimension is less important than the way further forms can be geometrically unfolded from this starting figure. Circumscribing a square around this inner circle, and a circle around the square, we can find the outer wall of the ambulatory. This nesting of concentric circles and squares, called quadrature, was a common design strategy throughout the Middle Ages, and I ask you to keep it in mind, since I’ll return to it repeatedly later in my talk.

The overall width of the cathedral’s east end, meanwhile, can be found by establishing a center point in the middle of the ambulatory, as indicated by the red dot, and then striking a circle back through the center point of the apse. The diameter of this circle equals the radius of the big circle framing the whole composition. As you’ll note, this outer circle quite precisely frames the interior spaces of Fulbert’s three radiating chapels.

The length of Fulbert’s crypt equals twice its width, as I’ve shown here with the two large yellow octagons, the eastern one of which frames the outer circle I was just mentioning. I have chosen to draw the octagons, rather than just the circles, since the inner corners of the octagons locate the inner wall surfaces of the corridors in Fulbert’s crypt, identified here by the inner orange horizontal lines that continue off the screen to the left.

The green horizontals now seen framing the outermost wall surfaces, meanwhile, pass through the midpoints of the octagon facets.

Many other elements of Fulbert’s crypt can be readily located within this system, like the wall thicknesses that I’ve shown here with blue outlines, but I will spare you these details, unless you want to discuss them in the Q&A period.

For the moment, I will shift my focus to the twelfth-century campaigns that produced the tower bases and portals of the west façade block. Here I’ve shown the basic plan of the façade block in red, at left.

As I’ll show in more detail in a few minutes, each tower should theoretically have a square base measuring 13.67m per side, a dimension equaling twice the span between the orange and blue horizontal lines established in the layout of Fulbert’s crypt. The twelfth-century builders of the façade block, in other words, appear to have been well acquainted not only with Fulbert’s crypt itself, but also with the geometrical logic used by their eleventh-century predecessors. Such knowledge transfer could have been mediated by the use of architectural drawings, and although none have survived from this early period, I see no reason to doubt their use. Indeed, the sophisticated geometrical layout of Fulbert’s crypt would have been difficult to achieve without the use of drawings. The twelfth-century towers also had sophisticated plans, but the two towers differ subtly from each other even at ground level, suggesting that two different teams of builders were involved.

To see this, I have zoomed in here on the plan of the west façade, based again on a scan by Andrew Tallon, with east now at the top of the screen. As I just mentioned, the notional plan of each tower is a square 13.67m per side. The space between these squares should in theory measure 16.40m, based on the geometry of Fulbert’s crypt, but the actual distance is just a few centimeters larger, at 16.44m. So, while the continuity of intention between the centuries seems clear, slight discrepancies between the campaigns are measurable with modern instruments. I want to note the 16.44m span here, because it would be used to set the scale of the High Gothic church built after the fire of 1194, as I will demonstrate in due course. First, though, I want to walk you through the design of the early Gothic façade and its pioneering south spire.

In principle, the design of the tower bases seems to have involved quadrature, with the buttress salience equaling the radius of the small yellow circle shown here, which also equals the space between the larger yellow circle and its inscribing chord. The space between the towers, meanwhile, has a diagonal span of 23.57m. Half of this distance, or 11.79m, will be used as the intended span between the buttress axes.

That relationship can be seen here with the two prominent red lines. Putting this information together, we can determine the theoretical buttress width.

That width will be 1.88m, as the label at left now indicates. All this geometry has a certain satisfying clarity, at least to my eye.

In practice, however, the north tower was built with a somewhat irregular plan. Its north and south walls sit 53 centimeters and 13 centimeters too far to the south, respectively, resulting in an overall compression of 53-13, or 40 centimeters. The span between the buttress axes is therefore just 11.39m, instead of 11.79m.

On the south tower, by contrast, the distance between the equivalent axes really is the theoretically predicted 11.79m.

In this case, though, the tower seems to have been located by aligning its outer buttress axis with the outer side of the theoretical box, as indicated by the red line at right, instead of by aligning the inner buttress face, as shown for the north tower with the red line at left. The south tower thus differs from the north tower not only in its slightly greater size, but also in its slightly greater displacement away from the cathedral’s central axis.

As a result of all this, we can find that the space between the buttresses of the two towers measures 17.37m, as shown here in purple. You’ll notice that this space is divided into three bays, each corresponding to one doorway of the western portal, with the middle portal being wider than the lateral ones.

More specifically, we can find the width of those portals by inscribing a half-octagon and a semicircle into the space between the towers. I find it somewhat amusing, from the standpoint of numerical symbolism, that the interaxial width of the central portal is 6.66m. Rather than attributing this to diabolical influence, I would just remind you that these numbers are unimportant in themselves, since they are just counting out geometrically established dimensions, using a modern measurement unit that was unknown in the Middle Ages. The geometry is what matters to me, and I believe that it is what mattered to the medieval designers, as well.

In any case, let us now move from the ground plan of the façade block to its elevation.

Here I have just listed for reference the interaxial and face-to-face widths of the two towers, reminding you that the south tower is slightly wider than the north.

And here, once again, is the octagon-based construction giving the width of the central portal.

Now, I have projected the axes of the two towers upwards, using different colors to signal their slightly different widths.

The first story of each tower, measured to the first set of small capitals, is given by a square framed by the buttress axes, although the details of the arrangement differ subtly between the two towers.

The first corbeled cornice on each tower falls at height 16.11m, which can be found by inscribing an equilateral triangle between the axes of the south tower as indicated by the more steeply inclined red lines seen at right. This alignment suggests to me that the dimensions of the more regularly constructed south tower had come to guide both towers by the time this level was reached.

Scooting the octagon construction upwards until its center reaches this level, we can find both the total height of the triple-window story, and the capitals of the two lateral windows.

Now, scooting our point of view upward, we can see several new details in this zone, and we can also see, more importantly, that the height to the second corbeled cornice in the south tower is 32.22m, or exactly twice the height to the first such cornice. Using diagonals starting on this level to construct an approximate heart shape in the upper right, we can also find the upper capitals at 38.12m, and the top of the square-planned tower story at height 41.06m, a height that was carried over to the north tower, also.

Within the basic frame that I have already introduced, we can also observe several other details that I, at least, find interesting, because of what they reveal about the tower designer’s working methods.

In the lowest tower story, for example, I’ve shown in blue that the arch widths and socle height were set by quadrature, with the blue square inscribed within the blue circle framed by the red buttress axes. I’ve also constructed small blue circles between the blue square and the red verticals. Then, to subdivide those circles, I’ve drawn blue verticals rising from their centers, which go on to describe the edges of the buttress strips in the upper tower stories. Intersections between these lines and other 45- and 60-degree lines give many of the other heights in the tower, some of which I’ve indicated here, though I can again spare you the details.

Moving upwards now into the zone where the tower meets the spire base, we can see an interesting and important geometrical transition, as the plan of the structure shifts from square to octagonal. As you can see here, I’ve drawn a large circle resting on the same horizontal line at height 38.12m that located the upper capitals in the square tower story. The diameter of the circle exceeds the interaxial width of the tower by a simple quadrature factor, as you can see from the way the circle circumscribes the highlighted red square. The equator of the circle locates the springline of many small arches surrounding the main tower core, and the top of the circle locates the baseline of the spire proper, where the structure begins to taper.

Again, I can add further details into this framework, which can readily explain subtle asymmetries such as the fact that the lefthand pinnacle is slightly taller than the right one, being located by a 60-degree line instead of a 45-degree line.

There’s a further subtlety that really surprised me when I undertook this analysis, even though I’ve thought about the Chartres spires literally for decades. The south spire, seen at the far right in the photo, has been widely praised for its streamlined silhouette, in which the spire appears to emerge smoothly from the tower below, with no awkward setbacks or visual disjunctions.

This transition works so well in part because the corner pinnacles flanking the main spire seem to continue the upward lines of the corner buttress below, and in part because the spire base fills so much of the available space atop the square-planned tower story. Somehow this all just seems to work better at Chartres than it does at most comparable spires, but I couldn’t put my finger on why until recently.

This summer I finally got access to a highly precise elevation drawing of the spire, as you see at upper right. This drawing, which comes from a French photogrammetric survey, matches well with the older horizontal cross-section of the tower shown below. What these drawings show, which I had never noticed before, is that the spire base at Chartres is not a regular octagon. Instead, the cardinal facets are slightly larger than the diagonal facets, which means that the corner pinnacles have less empty space to fill than they otherwise would. From this perspective, the striking smoothness of the spire silhouette can be seen to result from an unexpected geometrical adjustment. Since there are no God-given rules about how spires must be designed, one cannot call this cheating, but it certainly does introduce unusual subtleties into the plan of the spire.

These diagrams show the first stages in the geometrical development of the spire base. This pattern already looks fairly complex, but these first stages actually just involve a few very basic permutations on quadrature.

The outer wall strips here shown in yellow, for example, are generated by the same principle of subdivision used to create the buttress strips lower on the tower.

Here, in green I have added the inner wall strips, while in grey-blue, I have added the plans of the corner pinnacles flanking the spire, which start out as perfect squares.

Those squares, however, are partly occluded by the spire base, which I show here with red shading. It’s very close to being a regular octagon, but if you look closely at the corners of the figure, you’ll see a subtle distinction between the yellow rays to its corners and the orange rays to the corners of an equally sized regular octagon. The geometry in this zone really is complex, and I suspect that you won’t mind my sparing you the details yet again.

Things become a lot more tractable when we move into the elevation of the spire proper. Here I have just begun to construct a stack of squares, each with a side length equal to the diameter of the circle used to frame the transitional zone.

As you can see at left, the tip of the spire coincides with the tip of another box, the third in the stack. This basic stacking strategy would be used in spire design for centuries to come, as I have demonstrated in previous geometrical analyses of late Gothic spires from the German world, so I was excited to find its application at Chartres, in the first great Gothic spire of them all. The blue constructions, finally, describe the geometry of the metal cross and weathervane mounted atop the tip of the spire proper. You’ll note that this all works out with gratifying precision, based on a geometrical framework that I have developed step by step from the ambulatory radius established in eleventh-century crypt. Fulbert’s builders, of course, could hardly have imagined the spire built by their twelfth-century successors, but they were all working within the same broad tradition of geometrical design.

In the remainder of my talk, I will demonstrate that this chain of geometrical development can be further extended to explain the main body of the cathedral as it was rebuilt after the fire of 1194. This analysis will reveal why Chartres was built with an enormous clerestory, thereby offering a valuable new perspective on the origins of the High Gothic style. Before we consider the clerestory, however, we have to work our way upward from crypt level.

Here I once again show you the plan of the crypt, with Fulbert’s eleventh-century walls shaded, and with the west façade block that we have just been discussing at left.

After the fire of 1194, the plan of the cathedral was dramatically expanded by the construction of a wide transept, here framed in red. The center of the transept, indicated by the red dot, lies at the point where the cathedral’s axis intersects the diagonal continuing the side of the big yellow octagon that I drew when originally discussing the geometry of Fulbert’s crypt. The eastern corridors in the transept substructures, moreover, align with the western facet of that octagon. This already shows that the builders of the transept, like the builders of the façade block half a century earlier, understood and appreciated the geometrical logic of Fulbert’s crypt.

The east-west length of the transept extensions, moreover, equals the span between the inner wall surfaces of Fulbert’s crypt corridors, as the red circle now indicates.

More interesting, and more surprising, is the geometry that established the salience of the transept beyond Fulbert’s corridors. The highlighted red diagonal that reaches from the outer wall of Fulbert’s crypt to the corner of the transept has a slope of 36 degrees, or one tenth of a full circle. This slope thus hints at the introduction into the plan of geometries based on the decagon and the pentagon, which we have not had occasion to discuss before. Most of the geometries we have discussed so far have been based on squares and octagons, with a few equilateral triangles creeping into the elevation of the west façade. As we will see, the High Gothic builders at Chartres after 1194 worked hard to combine all of these figures into their design, which thus involved plays on the numbers 3, 4, and 5.

To see this, I now show you the plan of the rebuilt cathedral’s main floor, with the geometries established in the crypt still superposed for reference. The physical structure of the upper cathedral obviously must agree fairly closely with the footprint of the crypt, but many aspects of the cathedral design, such as its buttress and pier locations, were not fixed by the design of the crypt.

Since the upper church has geometrical logic of its own, it makes sense to sweep aside the lines established in the crypt, leaving only the crypt center as a point of reference.

The crossing bay at Chartres, which was likely one of the first parts of the rebuilt church to be laid out, has a rectangular plan, measuring 13.99m East-West and 16.44m North-South. The latter dimension, as you may recall, already appeared in the west façade, and it almost perfectly matches the 16.40m width established by the inner walls of Fulbert’s crypt. The relationship between the two sides of the crossing bay is quite interesting.

If you create a pentagon measuring 16.44m per side, 13.99m will be the length that a perpendicular to one side rises before it intersects the ray from the pentagon center to the adjacent corner, as shown here. This relationship was already noted in print by Otto von Simson in the 1950s, but since he provided no illustration, I initially missed his point, and I suspect that many of his other readers did, also. In any case, the proportions of the crossing confirm that the pentagon would be fundamental to the building campaigns undertaken after 1194. Starting from this figure, it is gratifyingly easy to derive all the main widths governing the Chartres nave.

The 32.88m span between the interior wall surfaces is twice the width of the crossing, as the red arcs here indicate.

The 44.41m span between the buttress surfaces can be found by unfolding the sides of the crossing bay, as the larger orange arcs now show.

The 33.72m span between the aisle windows can be found by first circumscribing a decagon around the pentagon, and then unfolding its facets.

The 36.50m span between the outer wall surfaces, finally, can be found by creating an octagon framed by the windows, and then circumscribing a circle around that octagon, as shown here in yellow. So, the main widths can be found with pentagons, decagons, and octagons. But what of the bay lengths? The nave bays at Chartres vary in length, with the western ones becoming shorter and shorter, presumably so that the new structure could be wedged in behind the existing façade block. The four eastern bays of the nave, though, are quite regular, each measuring close to 7.11m in length.

This dimension is just the altitude of an equilateral triangle whose base equals the half-span of the crossing, as the green lines here show.

The bay lengths in the choir are even less regular than in the nave, but the same set of four stacked green triangles closely approximates the total length of the choir, as you can see at right. Next, I want to consider the salience of the transepts. These are particularly interesting because some scholars have doubted that the sculpted porches were planned together with the rest of the transept. As I showed you, however, their substructures were carefully planned based on decagon-derived rectangles of 36-degree slope, which already suggests to me that they were planned together.

When I place blue pentagons on the sides of the bays adjacent to the crossing, moreover, I find that their tips align quite precisely with the free-standing supports of the porches, providing further evidence for this unitary planning.

When I circumscribe purple decagons around those pentagons, finally, I find the width of the transept bays, which at 7.35m are 22 centimeters longer than the nave bays. At least, that’s the case on the west side of the transept, which appears to have been built with the nave, and in which everything lines up nicely. Things get a bit messier to the east of the crossing, which seems to have been built a few years later and by a different team. For sake of completeness, I want to briefly consider these remaining aspects of the plan, before I go on to consider the cathedral’s elevation.

As you can see with the lines I’ve just added at right, the outer walls of the choir aisles skew inwards slightly, so that the bays at the east are a bit narrower than at the transept.

The same skewing occurs in the intermediate piers, especially on the north side.

Still, we know from the transept geometry where these axes should stand, as indicated by the labels I’ve just added at right.

This means that we can find the notionally intended radii of the circles defining the chevet. The radius of the apsidal chapels, including their substructures, seems to have been set equal to the distance between the two main arcs, as shown here in orange. There is also a black arc in this diagram, corresponding to the curved step leading from the outer ambulatory into the chapels.

This arc closely matches a circle, shown here in yellow, whose radius equals the length of three choir bays together. All of this works well, in terms of matching the fabric of the building, because two errors essentially cancel each other out. On the one hand, as we have seen, the aisles at the east end are a bit narrower than they should be, so that their span would give shorter radii.

On the other hand, the geometrical center inherited from Fulbert’s crypt, shown here in blue, lies a bit to the east of the choir endpoint predicted by the green triangles.

The actual geometrical center of the east end superstructure lies between these points, as indicated in purple, far enough to the east that the shortened purple arc serves well to locate the columns between the ambulatories. In this case, therefore, two wrongs pretty much do make a right.

In sum, then, you can see that the entire plan of the cathedral unfolds from the geometry of the crossing bay, which was based on the geometry of the pentagon. Now, let’s consider how the same principle plays out in elevation.

Here at right I show you the crossing bay and its surrounding pentagon and decagon, now flipped into the vertical plane, and floating above the laser-scanned cross-section of the lower nave structure. All the widths listed across the bottom of the slide result from the unfolding of the polygons shown in the diagram, which I already described when discussing the development of the plan. In this slide, therefore, I am not showing you anything geometrically new.

Now, at left, I have made visible the upper nave walls and the massive double battery of flying buttress. What can we say about the proportions in the vertical dimension?

Well, for one thing, the height of the buttress uprights excluding their pyramidal caps precisely matches the height of an equilateral triangle framed by the red verticals aligned with the inner aisle wall surfaces. These verticals also align with the width of the buttress uprights below their caps, confirming that this relationship was intended and significant.

We can get a different perspective on this relationship by drawing the triangle upside down, so that its baseline stands on the nave floor and its apex at height 28.47m.

This apex point clearly mattered even to the builders of the nave’s lowest levels, as one can tell by examining the sloping wall surface or glacis just beneath the aisle windows, which I have here highlighted with the red box. If one extrapolates a line along this sloping surface, it will intersect the building centerline at the apex of the red triangle.

Here I have shown that relationship with the green lines.

The really noteworthy thing about these lines is that they have 54-degree slope, which is the same slope that one finds in the rays of a regular pentagon, as I now show at right. Putting all this together, one can infer that the builders of the aisle walls were already thinking about applying pentagon-based geometries in elevation. Furthermore, they were linking them to geometries based on the equilateral triangle, as one can tell from the fact that green and red lines converge to the same apex. I should note, finally, that the intersection of the green 54-degree lines with the aisle glass planes sets the bottom edge of the windows, which are at a height of 5.26m. In conceptual terms, therefore, it seems to me that the design of the aisle walls was determined with the apex of the triangle at height 28.47m very much in mind. To put all this into context, let me remind you of some issues that I mentioned near the beginning of my talk.

At right you now see the cathedral’s full cross-section, with its suspiciously spindly upper flying buttresses perched awkwardly above the pyramidal caps of the main buttress uprights. To my eye, they really do look like afterthoughts. But, why would they have been considered necessary? I propose that they were added because the cathedral’s upper vault was built higher than originally planned. When I first began to entertain this notion, while looking at these geometrical overlays, I was tempted to imagine that the vault apex was originally meant to coincide with the triangle apex at height 28.47m. Such elevations based on equilateral triangles framed by exterior walls were common in early Gothic design, as I have shown in exploring the geometries of buildings including Sens Cathedral, and Saint-Germain-des-Prés and Notre-Dame in Paris. The more I explored this idea, though, the more problematic it became; there just isn’t quite enough space to fit main components of the Chartres elevation under that red horizontal. Late this summer, though, I realized that there was a compelling alternative scenario.

At left I’ve now added blue lines with pentagon-derived 54-degree slope that are framed by the exterior buttress surfaces at ground level. These lines converge at height 30.56m, closely aligned with the tips of the pyramids capping the buttress uprights. In some slightly earlier Gothic buildings, such as the cathedral of Laon, a comparable construction was used to set the height of the vaults.

I strongly suspect that the same basic scheme was originally planned for Chartres, which would have given the format seen at left. In this design, the apex of the vaults would have been exactly twice as high as the main arcades, which terminate at height 15.28m. Such proportions were typical of many Gothic designs both before and after the construction of the Chartres nave, so it would make sense in this context, as well. At risk of sounding too subjective, I will say that this reconstruction of the original design just looks right, with all the elements correlated into a robust and well-balanced structure. More concretely, I hope you can see that the blue 54-degree lines descend from the vault apex precisely to the upper capitals where the vault springs from the wall.

In the scheme shown at left, these capitals would be significantly lower than we see in the photo at right. Indeed, the vault would spring directly from the top of the triforium, as the red arrows indicate. If this were the case, of course, the clerestory windows would necessarily have been much smaller than they are today. There would have been no room to insert the big rosettes above the double lancets. By itself, however, that is no argument against the scheme I am proposing.

We need only walk into the east end of Chartres Cathedral itself to find a window format that would have fit perfectly beneath the lower vaults that I hypothesize, as shown at bottom right. Here in the choir aisle windows we see only a small rosette tucked between the tips of the lancets, beneath broad round arches very much like those framing the nave clerestory. Taken together, these geometrical and formal arguments have convinced me that the Chartres clerestory was originally planned to be significantly shorter than we see today, and that the current format was improvised in the course of the cathedral’s construction. To put such a revision into context, we must remember that Gothic builders in the decades around 1200 were building higher and higher, engaged in a competitive dialog that can be compared to an arms race. When the rebuilding of Chartres began in 1194, the new cathedral was designed to rival or outshine the largest churches yet built, including even Notre-Dame in Paris.

Less than a year after the fire at Chartres, though, work had begun on the cathedral of Bourges, seen at right, which set new standards for height and grandeur. The windows at Bourges were not particularly large, remaining confined within the space of the vault, as was typical in early Gothic construction. The arcades, however, were enormous, rising tall enough to include clerestory windows of their own.

This arrangement results from the fact that Bourges has an unusual spatial organization, with five aisles of staggered height, as shown at left. As in many earlier Gothic churches, the proportions were set by an equilateral triangle, but since the five aisles give Bourges such a wide plan, its vault height of 35.79m was record-breaking.

Measured against Bourges, the 30.56m vault height likely planned for Chartres would have been underwhelming. The exterior roofline probably would have been much lower than at Bourges, also. At Bourges, the height of the whole structure exactly matches the width, so that the whole composition would fit into a perfect square, and I strongly suspect that the same was true in the original Chartres scheme, as well, as I show on the right. In that graphic, I have drawn the Chartres roof with a 54-degree slope, agreeing with the slope used to determine the vault height. This seems plausible not only because of the intrinsic geometrical logic of Chartres, but also because the closely related cathedral of Soissons has a roof of this pitch. The key point, in any case, is that the original design for Chartres probably foresaw a building that would have been significantly shorter than Bourges both on the interior and on the exterior.

It was in response to this challenge, I believe, that the leaders of the Chartres workshop adjusted their design, moving the vault apex upward from 30.56m to the current height of 33.72m, a dimension matching the span between the aisle windows. This was still shorter than the Bourges vaults, but not by much. This modification was structurally daring, producing windows of unprecedented scale, and requiring the addition of the small supplementary flyers to brace the upper wall. The crest of the roof now reaches the impressive height of 49.91m, which is about a meter taller than at Bourges. As the orange arc up to that level shows, this height is also equal to the diagonal of the great square, measuring 35.29m on a side, framed laterally by the aisle walls and capped vertically by the floor of the walkway along the base of the roof. The cross-section of Chartres Cathedral thus has a great deal of geometrical coherence, even though its current form almost certainly results from a dramatic revision to the design in the course of construction.

In closing, my larger point is that geometrical analysis can give scholars a valuable new perspective both on the history of individual buildings like Chartres Cathedral, and on the larger art-historical narratives that involve them. In the particular case of Chartres, I have traced a continuous series of geometrical steps from Fulbert’s 11th-century crypt through the construction of the 12th-century façade block and spire, through the replanning of the cathedral after the fire of 1194, and to the subsequent revision of the master plan in the 13th century. Along the way I have been able to explain not only the forms and dimensions of the building components in question, but also the origin of subtle anomalies such as the asymmetry between the western towers at ground level. Most important, perhaps, I have argued that the tall clerestory and stilted vaults of Chartres were designed in a competitive midstream response to developments at Bourges, rather than being conceived by a single designer making a spontaneous Promethean leap into the High Gothic mode. I know that geometrical analysis will not be every art historian’s cup of tea, but I see great merit in the approach, and I hope that it will gain more currency as tools such as CAD systems and laser scanners become more widely available. In more personal terms, I find this work to be very rewarding, and I have enjoyed sharing it with you today. Thanks very much for your time and attention.