Geometrie, Proportion, und Vermessung in der Liebfrauenkirche

or, if I can give a subtitle,

Octagons everywhere. As we will see, geometries based on the octagon control many elements in the design of the Liebfrauenkirche, in its elevation as well as in its remarkable centralized plan.

Here we see that plan, based on the survey recently completed by Das Büro Leonhardt für Architektur & Denkmalpflege. I am most grateful to Michael Leonhard, to his colleague Kristian Kaffenberger, and to Hans Berthold Busse vom Amt für kirchliche Denkmalpflege for permission to use this new survey data. Before proceeding to analyze the plan of the Liebfrauenkirche in detail, I would like to make a few general observations about geometrical planning in the larger context of the Trier Cathedral complex.

In this older plan of the complex, you can clearly see the square core of the old cathedral, whose format goes back to late antiquity.

Here I have highlighted that part of the structure with a red square.

A square of the same size neatly frames the centralized core of the Liebfrauenkirche. It seems plausible to me, therefore, that the thirteenth-century builders of the Liebfrauenkirche may have been influenced by their late antique predecessors, not only in their choice of a centralized church format, but also in their choice of overall scale. It seems to me, moreover, that the old core of the cathedral already incorporates an octagon-based planning strategy that would later be used in the Liebfrauenkirche.

To see this, let us begin by zooming in on that part of the cathedral.

According to my understanding, the sides of the basic square measure 140 Roman feet, each of which measures .296m, so that each side comes to 41.4m. In any case, each side can be called 100%, just by definition.

If we inscribe an octagon within this square, we can see that its corners align quite closely with the arches that subdivide the square into nine vaulted bays. And, if we inscribe a circle within the square, then we find that the circle intersects the diagonals at 8 points that lie on the interior wall surfaces of the building.

The walls, which are here shown in yellow, thus frame an interior space that is smaller than the outer square by a factor of 0,924, which is the cosine of the 22.5-degree angle between the main axes and the diagonals of the octagon. I have seen this relationship, which I call octature, in many of the Gothic buildings and drawings that I have studied over the past decade. In recent months, moreover, I have seen that it also helps to set the proportions in the Romanesque abbey at Jumieges, so perhaps I should not have been surprised that it can be found in late antique buildings like the old cathedral of Trier, as well. As I noted, this relationship will also be seen in the Liebfrauenkirche. In seeking to explain the design of the Liebfrauenkirche, however, one must clearly appeal not only to local models, but also to Gothic precedents from France.

As scholars since the 19th century have recognized, the present plan of the Liebfrauenkirche relates closely to that of St. Yved in Braine, whose east end and diagonally planted chapels you now see at left. As Bruno Klein has demonstrated, though, the two designs actually differ in a number of important respects. The Trier plan incorporates polygonal rather than rounded chapels, for example, and the Trier elevation with its tall arcade owes more to Toul Cathedral than to St. Yved. As I will show presently, moreover, a variety of more subtle factors distinguish the geometrical planning strategies used at Trier from those seen at Braine.

At St. Yved, the crossing bay is a square measuring 10.06 meters per side. Identical red squares adjacent to the crossing do not quite fill the transept arms, since the transept bays together form rectangles that extend slightly further to the north and south.

However, if one creates an octagon framed by the red squares, one finds that its diagonal facets align perfectly with the diagonal rib along the baseline of the chapels.

Extending these diagonals until they meet the red verticals framing the crossing, moreover, one finds intersection points that define the glass plane in the transept windows, here shown in yellow.

These extended diagonals then serve as the baselines for the chapels, whose plans are semicircles centered exactly on the baselines. The ribs in the chapels all align at regular multiples of 45 degrees, forming a simple array that links naturally to the vault pattern in the adjacent square bays. The geometrical layout of the Liebfrauenkirche is rather different.

First of all, the dimensions of the crossing bay are larger, since its sides measure 10.87meters. Actually, the crossing bay is not quite a perfect square, since its western facet is about 8cm smaller than the other three, but the 10.87 figure seems to have been intended, as I will demonstrate with analyses of the church’s elevation as well as its plan.

It is interesting, at least, that the crossing square of Braine is smaller than that of the Liebfrauenkirche by a factor of 0.924, which is the ratio between the diameter of an octagon and that of the circle circumscribed around it. This octature relationship, which we already saw in the context of the old cathedral, can here be seen in the dotted lines around the Braine crossing. But even if the dimensions of the Liebfrauenkirche were based in this sense on those of St. Yved—and I am not convinced of this—the two designs have strikingly different geometrical systems.

At Trier, the regular straight bays are exactly half as long as the crossing bay, so that the pairs of bays in each arm fit precisely into red squares like those shown in the crossing, instead of extending beyond them as they do at Braine.

At Trier, moreover, the baseline of the chapels is kinked in the middle rather than straight, as the admittedly exaggerated yellow lines at right suggest. The chapels thus rotate subtly outwards from the crossing bay, like pairs of gears that are about to grind up the stairway turrets planted between them.

The reason for this, as I will demonstrate more carefully in a few moments, is that the ribs converge to points on the outside of the thick arcade walls, rather than to points on the arcade axes. For related reasons, the buttresses flanking the turrets are pulled in toward them, instead of aligning with the light green axes from the pier centers as they would if the logic of the Braine scheme had been followed. Leaving Braine aside, then, let us consider the proportions of the Liebfrauenkirche in more detail.

Here we see the western portion of the church, with its basic armature of square and double-square bays.

And here we see a set of five circles framed by the main arcade axes. I have drawn these in because the design of the piers and arcade walls seems to have been based on this subdivision.

To see this, I have here added a sequence of identical circles on the western side of the crossing, this time arranged so that the first and last are concentric with the crossing piers, with four other circles between the piers. As you can see, the crossing piers neatly fill the circles, at least at this level of precision. In fact, they are about 2cm wider than they should be based on a perfect five-fold subdivision of the main vessel.

I am willing to overlook this small error, which might even be attributed to thickness in the mortar joints, because the design of the crossing piers is also based on subdivision into fifths. As you can see here, the full diameter of the pier can be found by constructing a sequence of five circles, each equal in diameter to its secondary shafts.

The diameter of the upper torus molding in the pier base equals four shaft diameters.

The diameter of the pier core itself, meanwhile, equals 3.5 shaft diameters.

The rectangular blocks beneath the shafts are framed by an octagon whose faces are 5.5 shaft diameters apart.

And the sides of those blocks align with the corner of a smaller octagon, shown here in dark green, that circumscribes the core of the pier.

The outside edges of the plinth, finally, can be found by extending small diagonals from the front faces of these blocks, and from the corners of the larger framing octagon. I find the intersection between modular and geometrical design strategies in this pier design to be intrinsically interesting, and I think that it offers some valuable hints about the larger design strategies at work in the plan of the Liebfrauenkirche.

As I noted a few moments ago, a crucial point about the pier is the way its diameter divides into five shaft diameters, or ten shaft radii.

So, when we zoom back outwards to consider the plan as a whole, it is not surprising to find that the arcade walls are half as wide as the crossing piers, or one tenth the span between pier axes.

Here I have indicated those wall thicknesses with yellow shading.

And now, we can begin to see why the chapel geometry at Trier is so much more convoluted than at Braine. Here, in light green, I have drawn in diagonal axes like those at Braine, departing from the pier centers.

The actual ribs and buttress axes at Trier, which I show here in darker green, are significantly offset from these, because they depart from the edge of the arcade wall rather than from the pier centers.

The ribs that form the geometrical baseline of the chapel pairs, similarly, are slightly kinked, because their centerpoint is on the main grid of pier centerlines, while their endpoints correspond to the corners of the yellow-shaded rectangle framing the arcade walls. Here I have shown the segments in dark blue, with their endpoints identified as light blue circles. This kinking of the chapel baselines introduces awkward distortions in the placement of the vault keystones, and in the alignment of the ribs that support them.

The keystones seem to have been located empirically, about halfway between the lighter and darker green axes that I have drawn. The ribs define vault cells of slightly different widths. The first rib, which appears nearly vertical in my graphic, is offset by some 45 degrees from the chapel baseline. But since the baseline is rotated some 2.5 degrees, the rib and its associated buttress are also slightly offset from the cardinal axis of the church. The second rib is set at an almost perfect diagonal, so that the vault cell between the first and second ribs is pinched down to some 42.5 degrees. The third vault cell spans roughly 45 degrees, while the fourth and final one is wider than the rest, with a span of some 47.5 degrees. These values are not absolutely precise, but the qualitative pattern of distortions is the same in all eight chapels. Since no such distortions occur in the simpler geometry of St. Yved at Braine, this analysis corroborates Klein’s point about the separateness of the workshop traditions that produced the two buildings.

The section of the Liebfrauenkirche is, in my opinion, simpler and more lucid than the plan. Here I show you the interior elevation of the building, based once again on the new survey data that Michael Leonhardt and his colleagues were kind enough to share. Before I introduce any new lines, I want to call your attention to the prominent horizontal shelf where the Obergadenmauer ends.

I hope you will agree with me that it is logical to consider this level as one of the most important in the elevation, along with the floor level. The axes of the crossing piers, meanwhile, provide important verticals. Together these lines describe the box you see here, whose proportions are quite special.

This box fits precisely between the horizontal facets of a regular octagon. The Liebfrauenkirche was by no means the only church to have such proportions.

The Cistercian church at Altenberg, begun in 1259, follows a very similar scheme, as I discovered while working alongside Norbert Nussbaum.

The same octagon-based proportions can also be seen in the Parlerian drawings for the section of the Veitsdom in Prag, and in the masonry of the cathedral itself.

In Trier, the details of the elevation develop naturally within a similar octagonal framework.

The equator of the octagon, for example, aligns closely with the centers of the tracery sexfoils in the lower windows. This relationship is not absolutely precise, because the shapes of the window couronnements change slightly depending on the widths of the bays, but I doubt that the alignment was coincidental.

And, I feel 100% confident in stating that the arcade capitals were deliberately placed at a height equal to the span of the main vessel, as you can see here. The midpoint of the octagon falls halfway between these capitals and the base of the clerestory.

The upper capitals, similarly, were placed at a geometrically fundamental level, although a few intermediate steps are required to see the scheme. First, it is necessary to evenly subdivide the space between the eastern crossing pier and the eastern vertical facet of the octagon, as the dotted yellow construction indicates. Then, one strikes a circle concentric with the large octagon, such that its eastern edge aligns with the dotted vertical. Finally, one strikes a diagonal up from the center of the octagon until it intersects the circle. This intersection point defines the height of the upper capitals.

The prominent horizontal molding lower on the interior walls and pillars can also be determined by simple geometrical means. If one draws a circle circumscribed by an octagon, with their center on the ground line and their sides frames by the crossing pier axes, then the molding falls at the level where the circle crosses line from the center of the figure to the upper corner of the octagon. The proportions of the Liebfrauenkirche thus involve not only octagons per se, but also the octature relationship that I described at the beginning of my talk in the context of the old cathedral. This principle also helps to set the proportions of the tower at the Liebfrauenkirche, the last component of the building that I will discuss.

The first story of the lantern tower fits neatly into a square, shown here in green, with sides equal in length to the 10.87m span between the axes of the crossing. For reasons that probably involve the difficulty of constructing the tower base, the baseline of the square is shifted about 10cm down and to the west compared to the top facet of the large red octagon, but I strongly suspect that they were meant to coincide perfectly.

The upper margin of the masonry in the tower, shown here as a black horizontal, can be found by striking diagonals inward from the corners of the green square. The capitals in the lantern windows, similarly, can be found by striking diagonals in from the corners and midpoint of the square. If we call the side of the square one unit, therefore, the capitals fall at height one quarter unit, and the masonry terminates at height one and a half units.

The width of the tower can be found by inscribing an octagon in the basic green square, and then constructing a circle around that octagon. Here again, therefore, we see an octature relationship.

The height of the whole tower, finally, can be found by carrying the wall lines upward until they hit the previously defined top edge of the masonry, and then drawing diagonals in until they converge at the tip of the roof, as you see here in violet.

In sum, therefore, I feel that I have developed a reasonably good understanding for the geometrical logic of the Liebfrauenkirche. As one of the first unequivocally Gothic buildings in the German-speaking world, it clearly owes much to French influence, but its centralized plan is highly unusual, and its details distinguish it clearly from one frequently mentioned source, St. Yved in Braine. The Trier ground plan is more convoluted than that of Braine, but the elevation is quite lucid, with both the overall scheme and its details set by interlocking octagons like those seen at Altenberg and Prag. This geometrical investigation, therefore, has enhanced my appreciation of the Liebfrauenkirche, and its pivotal place in the early history of German Gothic design practice. I am grateful for the opportunity to investigate this “Schlüsselbau” with such good data, and I thank all of you for your time and attention.