Robert Bork1 Geometry, Proportion, and Measurement in the Liebfrauenkirche (English version of an article forthcoming in the proceedings of the October 2012 conference Liebfrauen Trier – Ein Schlüsselbau der europäischen Gotik) The Liebfrauenkirche in Trier occupies a very special place in the history of medieval architecture, not only because it ranks as one of the first German buildings to manifest the high Gothic style developed in France around 1200, but also because of its strikingly unusual centralized ground plan. Geometrical analysis can prove useful in clarifying both the inherent formal logic of the Liebfrauenkirche design, and the building’s complex relationship to French and German traditions. More specifically, the following analysis will demonstrate three principal facts: first, that the proportions of the walls and piers in the Liebfrauenkirche were established by the modular subdivision of basic dimensions in the ground plan; second, that the thickness of the arcade wall played an important role in the groundplan design at the Liebfrauenkirche, introducing geometrical irregularities that are not seen in its likely French prototypes, such as the church of St-Yved in Braine; and third, that the elevation of the Liebfrauenkirche was determined by an elegant system of interlocking octagons closely related to analogous systems seen in prestigious Gothic buildings including the cathedrals of Reims and the Cistercian Church at Altenberg. In both geometrical and formal terms, therefore, the designers of the Liebfrauenkirche demonstrated an awareness of up-to-date French Gothic prototypes, even as they remained grounded in local traditions. To set the stage for this discussion, it is helpful to consider the ground plan of a building with obvious topographic, institutional, and historical connections to the Liebfrauenkirche, namely, the cathedral of Trier (Fig. 1a).2 The core of the cathedral, whose basic design was established in late antiquity, has a simple square plan. The proportions of the building within this square, though, seem to have been set with an octagon-based geometry that would later play a crucial role in the design of the Liebfrauenkirche. The framing square around the outer wall surfaces of the cathedral measures 140 Roman feet, or 41.43 meters, on a side. When an octagon is inscribed within this square, parallel lines between its corners align fairly closely with the interior walls that subdivide the space of the square into nine bays (Fig. 1b).3 When a circle is inscribed within this octagon, moreover, it cuts the radial lines between the octagon’s corners at points that lie on the inside surfaces of the exterior walls; these points are shown as green dots in Figure 1c, while the walls are shaded in blue.4 The free span of the building interior between

                                                                                                               1 I am grateful for the support of many friends and colleagues who have helped to facilitate my participation in the conference Liebfrauen Trier: Ein Schlüsselbau der europäischen Gotik. In particular, I wish to thank: Norbert Nussbaum for bringing the conference to my attention; Andreas Tacke and Stefan Heinz for their kind invitation; Michael Leonhardt for his permission to use his building survey data; and Emily Schaum for help with the translation of my original English texts for publication in German. 2 On the relationship between the two churches, see Nicola Borger-Keweloh, Die Liebfrauenkirche in Trier (Trier, 1986), along with the essays in this volume, especially those by Winfried Weber, Christian Freigang, Bernd Nicolai, Matthias Müller and Hauke Horn. 3 The column centers do not quite align with the parallel lines, but they are quite close, considering the slight irregularities in their placement. The parallel lines also coincide closely with the remains of walls seen in the color ground plan of the cathedral complex , but see color graphic of subsurface structures published in Ronig-Bereths, Der Trierer Dom, Neuss, 1980. 4 Throughout this article, the steps will be discussed in the spectral sequence of a rainbow, with early steps shown in red, followed by orange, yellow, green, blue and violet. The original medieval designers

these shaded walls is thus approximately 38.28m, i.e. 92.4% of the 41.43m span across the outer wall surfaces. In modern mathematical parlance, this factor of .924 can be described as the cosine of the 22.5-degree angle between the main axes of the octagon and the lines connecting its corners. Medieval architects, however, needed no trigonometric training to arrive at these proportions. Instead, they could use elementary geometrical operations with the compass and straightedge to create nested circles and octagons. Because this design strategy was so common, both in Trier and in medieval architecture more generally, it makes sense to describe it by the short-hand term “octature,” by analogy to the better-known strategy of “quadrature” described in late medieval design handbooks.5 As the bottom of Figure 1c shows, the size of the Liebfrauenkirche relates in an interesting way to the size of the cathedral. The centralized groundplan of the Liebfrauenkirche fits rather neatly into the same orange octagon, with diameter 41.43m, seen in the plan of the so-called Quadratbau. In light of the close liturgical and institutional relationships between the two buildings, it is tempting to imagine that the overall scale of the Liebfrauenkirche as well as its centralized format were influenced by the design of its venerable northern neighbor. This case cannot be made convincingly, however, without a more careful consideration of the Liebfrauenkirche design. Fortunately, the completion of a new and highly precise survey of the building makes it possible to analyze its design with a new level of geometrical rigor.6 As the following discussion will demonstrate, such an analysis can help to clarify the relationship of the Liebfrauenkirche to both local and international precedents. Scholars have long recognized the Liebfrauenkirche and the roughly contemporary church of St-Elizabeth in Marburg as the first two buildings in the German-speaking world to incorporate the so-called high Gothic style seen at French buildings such as the cathedrals of Reims and Toul.7 The traceried windows and cantonnated piers of the two German buildings attest clearly to this influence, and the two-storied elevation of the Liebfrauenkirche, with its tall arcades, recalls the format of Toul particularly closely. The unusual ground plan of the Liebfrauenkirche, meanwhile, has long been associated with the smaller and older church of St-Yved in Braine, because both designs feature pairs of diagonally-planted chapels between the transept and the choir. In Trier, of course, this same basic arrangement also occurs on the western side of the church, producing its unique centralized plan. As Bruno Klein has observed, many important differences of detail separate St-Yved from the Liebfrauenkirche, demonstrating that the same designers and builders cannot have been responsible for both buildings.8 The chapels at St-Yved have round rather than polygonal plans, for example, and elevation at St-Yved has three stories rather than just two. The striking similarities between the basic outlines of the ground plans, though, strongly suggest that the builders in Trier may have been influenced by ideas from Braine, communicated perhaps through a simple schematic drawing.9                                                                                                                                                                                                                                                                                                                                                                      may have worked in a slightly different sequence, but the color-coding should at least illustrate how a sequence of simple steps can produce the complex forms seen in Gothic architecture. 5 On octature, see Robert Bork, The Geometry of Creation, Farnham, 2011, esp. pp. 25-26. 6 This survey was carried out by Das Büro Leonhardt für Architektur & Denkmalpflege. 7 On the connections between Champagne, Lorraine, and the Empire, see Marc Carel Schurr Gotische Architektur in mittleren Europa 1220-1340: von Metz bis Wien, Berlin, 2007, and Christoph Brachmann, Um 1300: Vorparlerische Architektur im Elsass, in Lothringen und Südwestdeutschland, Korb, 2008. 8 See Bruno Klein, St-Yved in Braine, Berlin, 1984, esp. 230-234. 9 It is worth noting that the groundplan that appears as a Ritzzeichnung in the stair turret of the Liebfrauenkirche differs in many important respects from the groundplan of the actual building. The Ritzzeichnung shows a groundplan in which the chapels are very tiny compared to the main vessel, which appears to include only a single full bay on each side of the crossing instead of the two seen in the present

Geometrical analysis of the Braine and Trier ground plans helps to clarify the relationship between the two designs, while also helping to explain the origin of many subtle irregularities seen in the Liebfrauenkirche. As Figures 2C and 2D show, the crossings of St-Yved and the Liebfrauenkirche are squares measuring 9.89 and 10.87 meters per side, respectively.10 In the Liebfrauenkirche, the two bays flanking each side of the crossing together fill a space identical to the crossing square. At St-Yved, on the other hand, the two analogous bays protrude slightly further out from the crossing, as the red squares in Figure 2C show. More specifically, as Jeoralgdean McClain showed in a short but elegant article on the geometry of St-Yved, each wing of the transept has the proportions of a so-called Golden Rectangle.11 The span of each wing measured from the building centerline to the window plane of the outer wall, in other words, exceeds the span of the crossing by factor of ϕ, the ratio (1+√5)/2 = 1.618 that satisfies the equation ϕ=1/ (ϕ-1), which relates a whole harmonically to the sum of its parts. Geometrically, this ratio can be constructed at Braine by unfolding the half-diagonal of the crossing square, as the orange lines in Figure 2E indicate. Diagonals struck back from the eastern corners of the transepts then define the geometrical baselines of the diagonally planted chapels. The yellow lines in Figure 2G then subdivide those diagonals in half, establishing the size of the rectangular bays flanking the crossing, and locating the point of tangency between the two diagonally-planted chapels on each side of the building. The chapels have half-circular apses, centered one quarter and three quarters of the way along the orange diagonals.12 The ribs

                                                                                                                                                                                                                                                                                                                                                                     building. The angles in the drawing are also badly distorted, exhibiting irregularities different than those seen in the building itself. In the drawing, the facets of the western, northern, and southern apses are angled more strongly outwards than the 45 degrees that they would be in a regular 5/8 octagonal termination, but in the actual building the angle is slightly less than 45 degrees, not more. The likely reason for this is that the geometrical center of the double-square bay to which the apse ribs converge in the real building is closer to the main crossing than they would be in a 5/8 termination. In light of such distinctions, it is hard to credit the idea that the drawing faithfully records and early phase in the planning of the Liebfrauenkirche. Instead, it should probably be interpreted as the work of a dilettante who was trying to record the work on the building as it was under construction, as Leonhard Helten argues in his essay for this volume. 10 The figure of 9.89m for Braine comes from Jeoragldean McClain, “Observations on the Geometric Design of Saint-Yved at Braine,” Zeitschrift für Kunstgeschichte, 49. Bd., H. 1 (1986), pp. 92-95. For Trier, the data comes from the new building survey undertaken in 2012 by Das Büro Leonhardt für Architektur & Denkmalpflege. Three sides of the crossing measure 10.87m, while its western facet is smaller by about 8cm. Analysis of the elevation confirms that the 10.87m dimension was very close to that intended by the designers. 11 McClain’s article “Observations on the Geometric Design of Saint-Yved at Braine,” provides a powerful and plausible geometrical explanation for the proportions of the church, demonstrating that its elevation recapitulates its floorplan at half scale. McClain’s geometrical analysis decisively supersedes the module-based explanation for the building’s groundplan proposed by E.E. Viollet-le-Duc in the article “Symétrie” from the Dictionnaire raisonné de l'architecture française du XIe au XVIe siècle. Viollet-le-Duc’s modular analysis was based on careful measurement, and it deserves to be taken seriously, in the sense that it may describe some aspects of work practice at Braine, but its emphasis on numerical sequences, with particular attention to the number 7, has less visual explanatory power than McClain’s scheme. In many instances the module-based dimensions seen in medieval buildings seem to have been chosen to approximate geometrically-determined schemes; they can thus be called “pseudo-modular,” to adopt the term introduced by Franklin Toker in “Gothic Architecture by Remote Control: An Illustrated Building Contract of 1340,” Art Bulletin , 67 (1985): 67-95. 12 The length of the extended diagonal, in other words, equals the sum of the two chapel diameters. The constructions shown in Figure 2 are precisely equivalent to those illustrated by McLain, of which they are

in the chapels, which are shown in green in the figure, are all oriented at precise multiples of 45 degrees. It is noteworthy that the preceding discussion, which suffices to locate most of the relevant elements in the ground plan of St-Yved, makes no mention of wall thickness. All of the elements have been described simply as points and lines, arranged into a single coherent geometry that produces a regular array of chapels. The geometry of the Liebfrauenkirche’s ground plan is far more convoluted than that of St-Yved, in part because wall thickness plays a prominent role that it had not in the French building.13 As the dotted green lines in Figure 2d indicate, diagonals launched from the centers of the arcade piers do not align precisely with the middle buttress axes of the chapels, as they do at Braine; this misalignment can also be seen at a larger scale in Figure 4. Instead, the buttress axes correspond with the blue diagonals introduced in Figure 2f, which are displaced from pier centers by the thickness of the arcade walls, which are shown with yellow shading. The thickness of the walls also introduces a slight twist into the orientation of the chapels, in a manner that will be explained more fully below. To understand the magnitude of these displacements, one must first know that the thickness of these walls is roughly 1.10m, which is almost exactly one tenth of the 10.87m span between the arcade axes.14 Consideration of the crossing pier geometry suggests that this tenfold subdivision was intentional. 15 The crossing piers measure 2.20m, or two wall thicknesses, across their outside surfaces. This means that five of them could fit side by side between the arcade axes, as the small yellow circles in Figure 2f indicate. The format of the crossing piers at the Liebfrauenkirche provides valuable clues about the design of the building in general, and about the origins of the wall thickness, in particular. Figure 3a shows the plan of the pier, with the central core and its four flanking shafts shaded with cross-hatching, and with the circular and polygonal forms of the pier base below shown in outline. The mixture of centralized and cruciform elements in the pier plan recalls the plan of the whole church, at least in a very general sense. As the red circles in Figure 3b show, moreover, the total width of the pier equals precisely five shaft diameters. Since this five-fold subdivision was clearly intentional, it seems likely that the total pier width was deliberately set to one fifth of the

                                                                                                                                                                                                                                                                                                                                                                     effectively a subset. The yellow lines in Figure 2g, in particular, correspond to the 21-m square described by McClain in “Observations on the Geometric Design of Saint-Yved at Braine.” 13 This is not to say, of course, that wall thicknesses were completely irrelevant in French Gothic design. Even in France, diagonal ribs do not consistently converge to the centers of piers. Instead, they often converge to small shafts that rise from the edges of the piers to flank the wall surface. In most French buildings, including Braine, these effects are rather subtle, because the walls are slender. In Trier, by contrast, the arcade walls are far wider than even the transverse ribs, so that the displacement of the diagonal ribs becomes an important factor in the geometrical layout of the building. This may reflect the lingering influence of Romanesque building practices, in which continuous walls played a greater role than the point supports emphasized in the Gothic era. In a certain respect, therefore, the thickness of the arcade walls at the Liebfrauenkirche can perhaps be seen as the structural analog of its formal murality, which allows it to harmonize visually with the neighboring cathedral complex. 14 It is conceivable that the slight discrepancy between the ideal and actual values, which amounts to just a centimeter, might simply reflect the presence of mortar joints between stones that had been cut precisely to 1.087m, but this remains to be investigated. 15 It is interesting in this connection that the 10:1 ratio between span and wall thickness would be recommended centuries later in the Unterweisung of Lorenz Lechler, from 1516. See Ulrich Coenen, Die spätgotischen Werkmeisterbücher in Deutschland: Untersuchung und Edition der Lehrschriften für Entwurf und Ausführung von Sakralbauten (Munich, 1990), pp. 15-25 and 146-52. McClain notes, meanwhile, that the pier cores of Saint-Yved in Braine measure .99m on a side, or one tenth of the crossing span. See “Observations on the Geometric Design of Saint-Yved at Braine,” p.92.

span between the arcade axes, as well. Such resonances between the microcosm and the macrocosm, after all, were typical of Gothic design practice. The rest of the pier design can be easily established using a mixture of modular and geometrical strategies. The largest torus of the pier base, for example, has a diameter equal to four shaft diameters, as the orange circle in Figure 3b shows. The diameter of the pier core equals 3.5 shaft diameters, as the yellow circles in Figure 3c show. The diagonal facets of the pier base coincide with the diagonal facets of an octagon with face-to-face diameter of 5.5 shaft diameters, which is shown in green in the graphic. Turning to a more geometrical approach, the width of the rectangular bases under each shaft equals the facet width of an octagon circumscribed around the pier core, as the blue constructions in Figure 3d indicate. The cardinally-oriented facets of the main pier base, finally, can be found by extending small diagonals from the corners of these sub-bases until they meet the extensions of green octagon’s diagonal facets; these last steps are shown in violet in Figure 3d. Since this analysis shows that the designer of the Liebfrauenkirche used subdivision into fifths together with geometric unfolding to set the design of the pier, it is hardly surprising that similar principles seem to have governed the design of the church as a whole. And, since the pier diameter is twice the wall thickness, this analysis reinforces the idea that the wall thickness really was meant to be one tenth of the span between the pier axes. The impact of the wall thickness on the chapel geometry of the Liebfrauenkirche becomes apparent in Figures 2h and 4. Instead of having simple, regular geometries like those seen at Braine, the chapels at Trier are subtly rotated with respect to the building axes. This rotation results from a conflict of two geometrical systems; one based on the linear grid of the pier axes, and the other based on the wall thicknesses.16 The geometrical baseline of each chapel pair is kinked in the middle because its midpoint relates to the former system, while its endpoints relate to the latter. These points are shown as light blue dots in the illustrations, while the ribs of the baseline are shown in violet. The midpoint sits on the corners of the pier axis grid, which is shown in orange, while the endpoints sit on the corners of the yellow shaded rectangles describing the walls. 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 green diagonal axes through the pier centers and the blue ones that define the actual buttress axes. The ribs define vault cells of slightly different widths. The first rib, which appears nearly vertical in Fig. 4, 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

                                                                                                               16 The grid described here, which is based on the 10.87m crossing span, is subtly distinct from the related one identified by Leonhard Helten, whose essay appears also in this volume. Helten’s grid is based on the size of the vault in the crossing tower, measured between the interior wall surfaces; as the present article shows, this area measures .90 x 10.87m = 9.78m per side, and each of the square boxes in his grid measures half as large, or 4.89m per side. When squares of this dimension are clustered around a center that aligns with the axes of the cylindrical columns in the Liebfrauenkirche, their corners will coincide closely with the engaged shafts that separate the two middle windows in each chapel. In view of the rotations in the chapel geometry described here, though, these approximate alignments seem unlikely to have been established early in the layout of the building, since the geometry would have been more regular if the grid had been used to locate all the points in the chapels.

precise, but the qualitative pattern of distortions is the same in all eight chapels. Thus, while the chapels of the Liebfrauenkirche are articulated as if they were meant to be octagonal, close examination of its ground plan reveals that they were not laid out as perfect octagons.17 The irregularity of the chapel layout at the Liebfrauenkirche suggests that the plan was developed from the interior outwards, rather than from the exterior inwards. This point deserves particular emphasis because it bears on the geometrical relationship between the Liebfrauenkirche and the old Quadratbau of Trier Cathedral. As noted previously, and as illustrated in Figure 1c, the Liebfrauenkirche fits fairly neatly into the same octagon that evidently governed the proportions of the Quadratbau. A quick glance at Figure 1c might suggest, moreover, that the axes of the chapel buttresses in the Liebfrauenkirche were constructed by connecting the corners of the octagon with pairs of parallel lines, like those that describe the interior walls of the Quadratbau. As consideration of Figure 4 demonstrates, though, this cannot have been the case, since the buttress axes are slightly skewed by the rotation of the chapels. It is still possible, and even likely, that the size of the Liebfrauenkirche was based on that of the cathedral. Figure 2h shows the same orange octagon as in Figure 1c, juxtaposed this time with the plan based on the recent building survey, showing that the western octagon facet coincides quite closely with the plane of the front wall flanking the main portal. The precise relationship between the dimensions of the cathedral and the geometrical structure of the Liebfrauenkirche, however, has yet to be clarified.18 The irregularities in the chapel layout at the Liebfrauenkirche distinguish its geometry from that of St-Yved in Braine. This analysis thus corroborates Klein’s point about the separateness of the workshop traditions that produced the two buildings, but the similarities between the overall plans of the two buildings nevertheless remain striking. In fact, the distortions seen in the Trier plan are precisely those that one might expect to arise if a schematic plan based on Braine had been used by a building crew used to taking wall thicknesses as primary.19

                                                                                                               17 The presentation on which the present essay was based had the subtitle “Octagons Everywhere,” but “everywhere” is admittedly an overstatement. Geometrically perfect octagons appear in the elevation of the Liebfrauenkirche, rather than in its groundplan, even though the groundplan seems at first glance to include more octagonal forms. 18 It is possible to derive the span of the Liebfrauenkirche nave from the dimensions of the Quadratbau—the old core of Trier Cathedral--using only basic geometrical operations seen in the two buildings, but it is not clear that this procedure was actually followed in the Middle Ages. Starting with the 41.43m octagon shown in Figure 1b, one can create a second octagon, larger by an octature factor of 1.082. From its corners, lines can be struck parallel to the main axes of the building, until they meet the interior faces of the exterior walls. Diagonals connecting these intersection points will coincide with the diagonal lines describing the groins in the four square corner bays of the Quadratbau. An octagon framed by these diagonals will have a face-to-face diameter of 40.27, so that its lateral facets coincide with the window plane in the building’s outer walls. Stepping out by another ocature factor of 1.082, one can arrive at an octagon with face-to-face diameter of 43.59m, which is almost exactly 4 times the 10.87m span between the axes of the Liebfrauenkirche. These results can be called interesting and even plausible, but they cannot be called authoritative until they are checked against a detailed survey of the Quadratbau. 19 The wall thickness introduced irregularities into the layout of diagonal chapels in other buildings, as well. This appears to happen more in the Belgian and German buildings of the Braine-related group than at the French ones, where the walls are generally thinner, as noted previously. It is thus interesting to compare the plans of St-Victor in Xanten, St-Martin at Ypres, and the Onze Lieve Vrouwe Bezoeking in Lissewege, with those of St-Yved at Braine, St-Pierre et St-Paul in Mons-en-Laonnois, and St-Michel-en-Thiérache. For discussion of these buildings, see Klein’s essay in this volume, and his book St-Yved in Braine.

The elevation of the Liebfrauenkirche is simpler and more lucid than the groundplan, in part because the wall thickness has no bearing in the establishment of the elevation. The elevation, in other words, could be conceived in purely geometrical terms, in closer accord with French Gothic norms. More specifically, it is clear that the elevation of the Liebfrauenkirche was organized around a system of interlocking octagons. As Figure 5d shows, a single large octagon whose facet length equals the 10.87m span of the crossing sets the overall height of the church, measured from its floor level up to the prominent molding that ends its Obergadenmauer. The Liebfrauenkirche was by no means the only major Gothic church to have an elevation based on the establishment of a large octagon whose bottom facet corresponds to the interaxial span of the main vessel. As Figure 6b shows, such a system sets the overall proportions of the Cistercian church of Altenberg.20 At Altenberg, the equator of the octagon coincides with the base of the triforium, while the lower lateral corners of the octagon lie at the same height as the tops of the main arcade capitals. Since construction began at Altenberg only in 1259, three decades after work began in Trier, it can hardly be taken as a source for the design of the Liebfrauenkirche. The cathedral of Reims, begun around 1211, was probably a common source for both German buildings (Fig. 6a). At Reims, the equator of the octagon once again corresponds to the baseline of the triforium, while the lower lateral corners of the octagon locate the bottoms of the capitals in the engaged piers. The high vaults now rise beyond the top of the octagon to match the height of the circle circumscribed around it, but this was not the original intention.21 As Henri Deneux recognized, the curve of the vaults was modified during construction, adding approximately 1.70m to their height.22 The original design for Reims, therefore, appears to have been based on strictly octagon-based system like those seen at Trier and Altenberg, and many later buildings.23 And, while the design of the Liebfrauenkirche cannot be taken as a terminus post quem for the redesign of the Reims vaults, the chronologies of the two buildings suggest that the high vaults of the Liebfrauenkirche were completed before those of Reims. It seems, therefore, that the designer of the Liebfrauenkirches was fully conversant with the latest in French Gothic elevation design, as it had developed by the 1220s. In the Liebfrauenkirche, as in Reims and Altenberg, the details of the elevation develop naturally within the main octagonal framework. As Figure 5e shows, for example, the equator of the octagon 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 the alignment was surely deliberate. It is clear, moreover, that the arcade capitals were deliberately placed at a height equal to the span of the main vessel. The base of the clerestory falls an equivalent distance below the top of the octagon, so that the midpoint of the octagon falls halfway between the arcade capitals and the base of the                                                                                                                20 Robert Bork, “Neue Überlegungen zur Geometrie des Chores,” in Norbert Nussbaum and Sabine Lepsky, Gotische Konstruktion und Baupraxis an der Zisterzienserkirche Altenberg, Band 2, Bergisch Gladbach, 2012. 21 The current proportions of the Reims elevation are thus based on octature. 22 On Deneux’s observation and the likely chronology of the vault redesign, see Alain Villes, Notre-Dame de Reims, Reims, 2009. 23 Octagon-based proportions like those of the Liebfrauenkirche would later be seen in the cathedrals of Clermont-Ferrand and Prague, closely related buildings begun in 1248 and 1344, respectively. The slightly steeper proportions seen in Cologne Cathedral and in the older of the Reims Palimpsest façade designs can be found by simply extending the upper diagonal facets of the octagon until they converge on the building axis. See Bork, The Geometry of Creation, esp. 50-54, 208-214, and 397-400.

clerestory. As Figures 5f and 6c show, the upper capitals were also 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 framed 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 previously described in the context of the Quadratbau. This principle of octature also helps to set the proportions of the tower at the Liebfrauenkirche. As the green boxes in Figures 5f and 6c, show, the first story of the tower fits neatly into a square 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 they were surely meant to coincide perfectly. The upper margin of the masonry in the tower, which appears in the figures as a black horizontal, can be found by striking diagonals inward from the corners of the green square. The capitals in the tower windows, similarly, can be found by striking diagonals in from the corners and midpoint of the square. If one calls 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 to create another octature relationship.24 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 the violet lines in the figures show. As the preceding discussion demonstrates, geometrical analysis can help to clarify the inherent formal logic of the Liebfrauenkirche design, and its relationship to both local and international precedents. The overall scale of the Liebfrauenkirche was likely chosen to match that of the late antique Quadratbau, which has a rich geometrical structure of its own. Modular design strategies helped to determine the proportions of certain key elements in the Liebfrauenkirche design, including most importantly the wall thicknesses. Because many of the ribs and buttress axes converge to margins of the arcade walls rather to their axes, the geometry of the Trier groundplan includes many awkward distortions unseen in closely related French buildings such as St-Yved in Braine. This suggests that the builders of the Liebfrauenkirche were working in a very different tradition of architectural practice than their French colleagues, whose ideas in some cases may have reached them only indirectly through the medium of schematic drawings. The elegant octagon-based geometries of the Liebfrauenkirche, though, show that the building’s designers were conversant with some of the latest ideas coming from the workshops of prestigious French buildings such as the cathedral of Reims. A great deal clearly remains to be said about the geometrical strategies used in the design of the Liebfrauenkirche, but this short preliminary discussion already demonstrates that geometrical analysis can provide a valuable perspective on this “Schlüsselbau” and its special place in the history of medieval architecture.

                                                                                                               24 The same octature-based wall proportions were already seen in the 11th-century crossing tower of Jumièges Abbey, as a forthcoming essay by the author will demonstrate.  

Captions Figure 1: Ground plans of Trier Cathedral and Liebfrauenkirche with geometrical overlays by the author.

A) Trier Cathedral, plan B) Trier Cathedral, plan with geometrical overlay, stage 1. C) Trier Cathedral and Liebfrauenkirche, plans with geometrical overlay, statege 2.

Original source drawing by C.W. Schmidt, 1836, reproduced in Nikolaus Irsch, Der Dom zu Trier, Düsseldorf, 1931. Figure 2: Ground plans of St-Yved in Braine and Liebfrauenkirche with geometrical overlays by the author.

A) St-Yved, plan B) Liebfrauenkirche, plan C) St-Yved, plan with squares based on the crossing square D) Liebfrauenkirche, plan with squares based on the crossing square, and with trial buttress axes extending

diagonally from the pier centers E) St-Yved, plan with Golden rectangles unfolding to create the transept, and with diagonals struck back to

create the chapel baseline F) Liebfrauenkirche, plan with arcade walls added in, and with revised buttress axes extending diagonally

from points on the wall surfaces G) St-Yved, plan with chapel geometry added in H) Liebfrauenkirche, plan with chapel geometry added in

Original source drawing of Braine; original source drawing of Liebfrauenkirche by Das Büro Leonhardt für Architektur & Denkmalpflege, 2012. Figure 3, Ground plans of crossing pier from Liebfrauenkirche with geometrical overlays by the author.

A) Liebfrauenkirche, crossing pier plan B) Liebfrauenkirche, crossing pier plan with shaft and torus diameters shown C) Liebfrauenkirche, crossing pier plan with core and basic plinth dimensions shown D) Liebfrauenkirche, crossing pier plan with final plinth details added.

Original source drawing of Liebfrauenkirche by Das Büro Leonhardt für Architektur & Denkmalpflege, 2012. Figure 4, Detail view of Liebfrauenkirche groundplan with geometrical overlays by the author. Original source drawing of Liebfrauenkirche by Das Büro Leonhardt für Architektur & Denkmalpflege, 2012. Figure 5, Ground plan and elevation of Liebfrauenkirche with geometrical overlays by the author.

A) Liebfrauenkirche, ground plan B) Liebfrauenkirche, ground plan with squares based on the size of the crossing C) Liebfrauenkirche, elevation D) Liebfrauenkirche, elevation with great octagon based on size of the crossing E) Liebfrauenkirche, elevation with details added into main octagon F) Liebfrauenkirche, elevation with details added into tower zone

Original source drawings of Liebfrauenkirche by Das Büro Leonhardt für Architektur & Denkmalpflege, 2012. Figure 6, elevations of churches from Reims, Altenberg, and Trier, with geometrical overlays by the author.

A) Reims Cathedral, elevation with octature scheme B) Cistercian church from Altenberg, elevation with octagon scheme C) Trier Liebfrauenkirche, complete elevation scheme

Original source drawing of Reims after Dehio and von Bezold, 1901; Source drawing of Altenberg after Steinmetz, 1911; Original source drawing of Liebfrauenkirche by Das Büro Leonhardt für Architektur & Denkmalpflege, 2012.

Figure 1: Ground plans of Trier Cathedral and Liebfrauenkirche with geometrical overlays by the author.

Figure 2: Ground plans of St-Yved in Braine and Liebfrauenkirche with geometrical overlays by the author.

Figure 3, Ground plans of crossing pier from Liebfrauenkirche with geometrical overlays by the author. E) Liebfrauenkirche, crossing pier plan F) Liebfrauenkirche, crossing pier plan with shaft and torus diameters shown G) Liebfrauenkirche, crossing pier plan with core and basic plinth dimensions shown H) Liebfrauenkirche, crossing pier plan with final plinth details added.

Original source drawing of Liebfrauenkirche by Das Büro Leonhardt für Architektur & Denkmalpflege, 2012.

Figure 4, Detail view of Liebfrauenkirche groundplan with geometrical overlays by the author. Original source drawing of Liebfrauenkirche by Das Büro Leonhardt für Architektur & Denkmalpflege, 2012.

Figure 5, Ground plan and elevation of Liebfrauenkirche with geometrical overlays by the author.

G) Liebfrauenkirche, ground plan H) Liebfrauenkirche, ground plan with squares based on the size of the crossing I) Liebfrauenkirche, elevation J) Liebfrauenkirche, elevation with great octagon based on size of the crossing K) Liebfrauenkirche, elevation with details added into main octagon L) Liebfrauenkirche, elevation with details added into tower zone

Original source drawings of Liebfrauenkirche by Das Büro Leonhardt für Architektur & Denkmalpflege, 2012.

Figure 6, elevations of churches from Reims, Altenberg, and Trier, with geometrical overlays by the author. D) Reims Cathedral, elevation with octature scheme E) Cistercian church from Altenberg, elevation with octagon scheme F) Trier Liebfrauenkirche, complete elevation scheme

Original source drawing of Reims after Dehio and von Bezold, 1901; Source drawing of Altenberg after Steinmetz, 1911; Original source drawing of Liebfrauenkirche by Das Büro Leonhardt für Architektur & Denkmalpflege, 2012.