## Introduction

Discussion of proportion has a curiously vexed status in the literature on Gothic
architecture. On the one hand, it is obvious to even the most casual observer that
the proportions of Gothic buildings and their constituent parts, which are often
very tall and slender, contribute significantly to their visual impact by suggesting
upward movement and transcendence. On the other hand, though, it has proven
difficult to explain exactly how these proportions arose in the design process.
Indeed, the shockingly non-classical proportions of Gothic buildings famously led
Renaissance writers like Vasari to conclude that this *maniera
tedesca* was inherently wayward and disorderly.^{1} In the five subsequent centuries, many more sympathetic
authors have attempted to analyze and describe the logic of Gothic architectural
proportions. However, while some valuable work has been done in this direction, the
overall state of the field remains strikingly primitive even today. All too often
such work has been flawed by imprecision, ambiguity, and wishful thinking. Many
scrupulous scholars, therefore, have become skeptical about all such research,
concluding that it reveals more about the pet theories and preoccupations of the
researchers than it does about medieval design practice.

Fortunately, recent developments in the study of drawings, the surveying of
buildings, and the use of computer-aided design (CAD) systems now allow the
proportions of historic monuments to be studied with new rigor. It is finally
becoming possible, therefore, to speak with reasonable certainty about the working
methods of Gothic designers. To show this, the present essay presents two groups of
CAD-based case studies: the first considers medieval drawings related to the design
of the great spired towers at Ulm and Freiburg-im-Breisgau; the second considers the
cross sections of the cathedrals of Reims, Clermont-Ferrand, and Prague, and of the
Cistercian church at Altenberg. These case studies will demonstrate that Gothic
design methods involved the dynamic unfolding of geometrical constructions.^{2} This approach to design produced proportional
relationships qualitatively different than those seen in the more static and
module-based formal order of classicism. In a sense, therefore, Vasari was right to
say that Gothic buildings lacked ‘every familiar idea of order’,
although this comment says more about his own limitations than it does about the
Gothic builders he sought to criticize.

The complex and procedurally based formal order of Gothic architecture, in fact,
offers a highly sophisticated alternative to the classical tradition, one with real
relevance for present-day architectural practice. Gothic buildings often exhibit
patterns of self-similarity, in which details such as pinnacles echo the forms of
larger elements such as spires, creating a rich resonance between microcosm and
macrocosm. Analogous patterns are now seen in the mathematical objects known as
fractals, and in the work of contemporary designers who use computer algorithms to
develop complex and innovative formal systems of their own.^{3} Geometrical analysis of Gothic design thus has the potential
to enrich architectural practice in the twenty-first century, in much the same way
that formal and archaeological analysis of Gothic buildings enriched architecture in
the nineteenth and early twentieth centuries. To see why the present analyses offer
something new and valuable to the discussion of medieval design practice, it will be
helpful to consider briefly some of the historiographical developments that have
shaped scholarly understanding — and misunderstanding — of Gothic
proportioning systems.

## The problem of Gothic proportions, from Villard to Hecht

Gothic builders themselves left behind no very satisfying records of their methods.
This fact is hardly surprising, since their training emphasized visual rather than
verbal communication. The so-called portfolio of the thirteenth-century draftsman
Villard de Honnecourt admittedly provides some scattered commentaries among its many
drawings, but it certainly provides no sustained discussion of Gothic design
practice.^{4} Most surviving medieval documents
of the building process are simply unillustrated construction ledgers, while most
surviving design drawings have no textual glosses. Gothic builders were certainly
able to convey their techniques effectively from master to apprentice, as the
continuity of their traditions over more than four centuries demontrates. However,
Gothic design conventions governed the rules of the process more than the shape of
the final product, which meant that the spatial relationships between building
components varied far more widely in Gothic than in classical architecture.^{5} This, in turn, meant that precision could only
be achieved by explicit demonstration and description, rather than by allusion to
venerated prototypes. The procedural dynamics of Gothic creativity, therefore, could
not easily and concisely be translated into words that would be understandable or
satisfying to an educated layman. In this important sense, the geometrical logic of
Gothic design was ‘unspeakable’ (**Bork
2011a**).

In the decades around 1500, nevertheless, several German late-Gothic authors
attempted to explain their design methods in short pamphlets.^{6} Matthäus Roriczer’s **1486*** Buchlein von der Fialen Gerechtigkeit*, for example, illustrates the
successive steps in the design of a pinnacle (Fig. 1).

Roriczer makes clear that the process began with the geometrical construction of the
pinnacle’s ground plan within a square base. Next, a series of progressively
smaller rotated squares should be inscribed within the original square. Further
permutations of these figures, easily accomplished with the compass and
straightedge, suffice to determine the complete ground plan of the pinnacle,
including the collapsed ‘footprint’ of its vertical shaft. The elevation
of the pinnacle was then determined by stacking up a series of modules based on the
ground plan. This process of extrusion from the ground plan into the third
dimension, which German authors call *Auszug*, or ‘pulling
out’, was fundamental to the Gothic design method as a whole (see **Shelby 1977: 76–79**). Roriczer himself
hints that something more general than pinnacle construction is at stake in his
booklet. On its first page, he explains to his learned patron Wilhelm von Reichenau,
the Bishop of Eichstätt, that his writings will ‘explain the beginning of
drawn-out stonework — how and in what measure it arises out of the
fundamentals of geometry through manipulation of the dividers’ (**Shelby 1977: 82–83**).^{7} Significantly, too, he makes clear that he is describing
traditional design methods rather than his own innovations, since he invokes the
authority of the ‘Junkers of Prague’, the members of the Parler family
who dominated architectural practice in central Europe from the middle of the
fourteenth century. Roriczer may well have chosen to focus on pinnacles for
basically pedagogical reasons, thinking that this simple example could clarify
design principles of wide applicability, but the seeming narrowness of this topic
surely diminished the impact of his writings. The tediously detailed quality of his
text, which combines the worst qualities of a math textbook and a cookbook, also
must have limited the appeal of his work.

Three decades after the publication of Roriczer’s booklet, the noted Heidelberg
court architect Lorenz Lechler tried to explain Gothic design more quickly and
economically in his *Unterweisungen*, a more comprehensive compendium
of architectural advice for his son Moritz (**Coenen
1990: 15–25 and 146–152**). Although Lechler’s known
architectural works, such as the sacrament house of S. Dionys in Esslingen, are
formidable in their geometrical complexity, his writings present mostly short rules
of thumb based on simple numerical ratios. He recommends, for example, that side
aisle spans should be one half as great as the free span of the main central vessel,
which he takes as his fundamental module. The thicknesses of the walls and piers, he
suggests, should equal one tenth of this module. The capitals of the main vessel
should sit either one module, or alternatively one and a half modules, above the
floor. Lechler’s short modular recipes are less tiresome to read than
Roriczer’s detailed geometrical instructions, but they ultimately prove
frustrating to the modern researcher, since they fail to explain the origins of the
complex dynamic forms that make German late-Gothic design so interesting. These
examples, moreover, are unillustrated, at least in the three surviving manuscripts
of the *Unterweisungen*. Lechler’s manuscripts do, however,
include several illustrations showing how combinations of geometrical and
arithmetical subdivision could be used to generate the cross sections of window
mullions. The small and large mullions are shown to have lengths of 5 and 7 units
respectively, which at first sounds like a simple modular relationship. However, the
smaller mullion is shown within a square circumscribed by a circle framed by a large
square, which demonstrates that the 5:7 ratio is really just an approximation to the
1:√2 proportion that emerges geometrically from the operation of square
rotation, which is often called quadrature.^{8}
Gothic architects used a wide variety of similar constructions, often inscribing
other regular polygons within circles to set the proportions of their building
components.^{9} They could also easily unfold
the diagonals of a half-square to create the so-called Golden Ratio ϕ, which
relates a whole harmonically to the sum of its parts.^{10} Convenient numerical approximations to the resulting lengths might
then be chosen to facilitate construction using fixed foot units or blocks of
standardized sizes. Franklin Toker (**1985**) has
aptly called this design method ‘pseudo-modular’, but Lechler provided
no very clear exposition of the relationship between geometry and modularity in his
work. The late medieval design handbooks, in fact, are fairly cavalier about
theoretical niceties. Like many other medieval technical texts, in fact, they are
essentially just compilations of recipes, rather than polished treatises with clear
organization and argument structure.

In terms of scope and rhetorical sophistication, late medieval design booklets like
Roriczer’s and Lechler’s were no match for the comprehensive
architectural treatises that began to emerge contemporaneously in Renaissance
Italy.^{11} Vitruvius’s impressively
comprehensive and genuinely Roman *De architectura* had been known
throughout the Middle Ages, but it received greater attention after its
popularization by Poggio Bracciolini early in the fifteenth century (**Kruft 1994: 38–39**). Leon Battista Alberti
wrote its most direct Renaissance successor, *De re aedificatoria*,
in eloquent Ciceronian Latin that would appeal to well-educated humanist courtiers,
in a way that the Gothic design booklets never could. Alberti’s discussion of
proportion, meanwhile, emphasized fixed whole-number ratios, rather than the more
flexible relationships that could emerge in the geometrical dynamics of the Gothic
tradition.^{12} The proportions of Renaissance
buildings, therefore, could be captured much more readily in simple graphics than
those of Gothic buildings. This fact contributed to the success of illustrated
Renaissance treatises, including most notably those produced by Serlio, Palladio,
and Vignola, whose publications helped to spread Italianate architecture throughout
Europe in the sixteenth century. From this perspective, the eclipse of the Gothic
tradition can be understood in part as a consquence of its practitioners’
inability to provide verbal and visual explanations for their methods as compelling
and accessible as those provided by their Renaissance rivals.

Over the past five hundred years, therefore, the logic of the Gothic design process
has been less well understood, and less celebrated, than that of classical
architecture. While the module-based systems of classical design were actively
taught to generations of students, most classically inclined writers followed
Vasari’s lead in dismissing Gothic architecture as lawless and
disproportionate. Romantic writers who were more sympathetic to the Middle Ages,
meanwhile, often saw the seeming freedom of the Gothic tradition as a virtue; thus
they rarely devoted sustained attention to figuring out the logic of the Gothic
design system. Although a fairly substantial literature on the topic had begun to
emerge by the middle of the twentieth century, two complementary problems kept this
work from enhancing the relative prestige of Gothic builders. First, rigorous
historians like James Ackerman demonstrated that Gothic planning methods could be
strikingly unsystematic and ad hoc. To add insult to injury, Ackerman’s famous
article on the chaotic progress of Milan Cathedral appeared in 1949, the same year
that Rudolph Wittkower’s *Architectural Principles in the Age of
Humanism* argued for a close connection between the modularity of
Renaissance design and the elegant harmonies of musical theory (**Ackerman 1949**; **Wittkower 1949**). The second and larger problem with research on Gothic
proportion is that much of it appeared fanciful and unreliable, revealing more about
the preconceptions of its authors than about the working methods of the Gothic
designers themselves. In his magisterial 1960 review of writings on the Gothic
period, therefore, Paul Frankl wrote in apparent frustration that ‘the
question of what is actually gained by such research becomes urgent. There can be no
doubt that Gothic architects made use of triangulation and the like, but the
excogitated networks made up of hundreds of lines to determine all points has not
been proved and is probably undemonstrable and unlikely’ (**Frankl 1960: 722**). The most devastating
critique of this research tradition came from Konrad Hecht, whose work occupies a
singular place in the historiography of Gothic proportion.

Hecht, writing in the years around 1970, aggressively challenged the authors who had
tried to explain Gothic design in geometrical terms. Hecht argued that Gothic
builders used a modular and numerical approach, rather than geometry, to define the
proportions of their buildings. Hecht paid particular attention to the tower and
spire in Freiburg im Breisgau, which had figured prominently in many earlier studies
of Gothic proportion. Taking advantage of a recent survey, Hecht effectively
demonstrated that most previously proposed geometrical theories about the
spire’s proportions were untenable. This fact, of course, does not mean that
the Gothic builders of the tower did not use geometrical methods, but Hecht argued
vociferously in this direction. To provide an alternative framework, Hecht attempted
to show that module use could explain the proportions in the Freiburg tower, and in
the elevation drawings for the tower of Ulm Minster.^{13} Hecht’s critique of poorly done geometrical scholarship was
well motivated, but his modular schemes explain very little, since he gave no reason
why their proportions should involve the modules he proposed. He was, in essence,
just presenting numerical approximations to sets of proportions that could easily
have been determined by geometrical means. His analyses thus amount to little more
than quantified descriptions, which give no insight into the form-giving strategies
used by medieval designers. When Hecht tried to achieve precision, moreover, he
generally did so by transforming his subject buildings and drawings into nearly
indigestible tables of numbers, thus obscuring the visual relationships that would
have been paramount for a medieval builder or draftsman. Because Hecht’s
densely argued critical writings outwardly appear so rigorous, though, they continue
to discourage research on Gothic architectural geometry even today. The impact of
his writings has been particularly pronounced in the German-speaking world, where
such work had formerly flourished.^{14}

## Towards a new understanding of Gothic geometry

Despite the widespread skepticism that Hecht’s work radically exemplified,
research into the geometrical bases of Gothic architectural design has a great deal
to offer. And, while the field has not thrived in the past half century, enough good
work has been done in recent decades to demonstrate the potential of such research
(see, for example, **Wu 2002a**). Most
importantly, perhaps, scholars including Stephen Murray have demonstrated
convincingly that the overall proportions of Gothic buildings can often be explained
by fairly simple sequences of dynamically unfolding geometrical operations. As
Murray, Toker, and Peter Kidson have begun to show, this geometrical approach to
design was compatible with modular approaches to construction and building layout
(see **Murray and Addiss 1990**; **Kidson 1993**; and **Toker 1985**). As noted previously, in fact, modular dimensions
were often chosen to approximate geometrically determined proportions, as
Toker’s term ‘pseudo-modular’ effectively suggests.

A variety of new technical and methodological approaches are now beginning to
converge productively, in ways that are allowing decisive steps forward in the study
of Gothic architectural geometry. First, truly accurate building surveys are
beginning to become more widely available. Some nineteenth- and twentieth-century
surveys were already quite precise, and current scholars still have good reason to
conduct careful manual surveys; the Regensburg Cathedral survey project and the work
of Matthew Cohen described in this volume provide good recent examples of such
work.^{15} But the field of building
measurement is being rapidly transformed by the spread of photogrammetric and
especially laser-based survey methods. As Andrew Tallon’s essay in this volume
demonstrates, such methods can dramatically increase the precision of building
surveys, putting the field of geometrical research onto a new and strongly
reinforced empirical foundation. A second and closely related development has been
the spread of computer-aided design (CAD) systems, which permit scholars to draw
exact geometrical figures, and to compare them to the forms seen in medieval
buildings.^{16} Together, these trends are
rendering obsolete the concerns about imprecision and sloppiness that had formerly
engendered much well-justified skepticism of geometrical research; this can be seen
in the supplement to this article (see http://dx.doi.org/10.5334/ah.bq.s1),
concerning the plan of Notre-Dame in Paris. Even with the world’s most precise
building surveys, though, some ambiguity about the intentions of the designers
remains, because errors and changes may have been introduced in the course of the
construction process, and because it is not always easy to tell from the fabric of
even a well-constructed building which elements had conceptual priority for the
designers. For these reasons, the study of surviving medieval architectural drawings
can be a helpful adjunct to the study of the monuments.^{17} Drawings are the documents, after all, that were produced by the
designers themselves. Their proportions thus tend to reflect the designer’s
intentions more directly than the buildings do.^{18} Drawings, moreover, include blind lines, compass prick marks, and
other traces of the draftsman’s labor, which can help to reveal the logic of
the design’s conception. The scribed lines often marking pier and buttress
centerlines, for example, clearly attest to the importance of these axes in the
layout of the drawings. The vast majority of the visual information in a drawing,
though, appears in the inked lines describing the architectural forms themselves,
which have rarely received the careful geometrical scrutiny that they deserve.

It is crucial to recognize, in this context, that dynamic geometry was not simply a means that Gothic designers used to establish the overall proportions of their buildings; rather, it was a comprehensive form-giving strategy that determined the shapes of individual building components as well as the relationships between large- and small-scale forms. The geometrical steps of the Gothic design process, of course, did not take place in a vacuum. Tradition, functional requirements, and educated guesses about structural stability, all would have informed the design process, establishing the basic outlines of the architectural scheme in ways that geometry by itself never could. Most Gothic designers, therefore, probably had at least a rough idea in mind even before sitting down at the drafting table. Geometrical experimentation with the compass and rule then served to sharpen the focus, by generating specific trial lines that could be accepted or rejected depending on their usefulness in the overall scheme. In a sense, therefore, a Gothic design can be seen as an architectural topiary, in which geometry provides the quasi-random growth factor, while artistic judgment guides the pruning process. This dialog between growth and pruning helps to explain the organic quality characteristic of Gothic design.

With this perspective in mind, it becomes possible to achieve a geometrically informed understanding of Gothic proportion far more plausible, and far more satisfying, than the strictly modular accounts provided by anti-geometrical skeptics like Hecht. The investigative method employed in the following case studies, therefore, closely emulates the Gothic design process just described. Here, once again, basic geometrical operations have been used to generate trial lines. In this context, though, the importance of a line can be judged by how well it matches lines already determined by the medieval designers, rather than by how well it matches a vague phantom in the mind’s eye. This distinction, of course, makes the investigative process less open-ended than the original design process, but the resonance between the two has great methodological importance. In order to generate plausible hypotheses for testing, the researcher has to empathize with the original designer, imagining how a given design can be brought forth step by step on an initially blank sheet.

The following case studies show how the use of CAD systems permits both the fruitful
harnessing of this creative empathy, and the rigorous testing of geometrical
hypotheses. All of the associated graphics were created using the Vectorworks CAD
system, in a three-stage process. First, source images of the drawings or buildings
in question were scanned and imported into the system. Second, their relative
proportions were carefully checked against published dimensions and measurements
made in the field, and corrected where necessary; these adjustments were generally
quite small, thanks to the quality of the source images.^{19} Finally, the CAD system was used to draw trial lines and
polygons on top of the source images. The geometries of these added lines are
perfect, in the sense that the squares are square, the circles circular, the
verticals vertical, and so forth. These figures, in other words, have never been
adjusted or ‘fudged’ to match the source images. The computer, moreover,
treats these figures as assemblages of perfectly thin lines, so that the user never
has to worry about finite line width introducing imprecision into the
geometries.^{20} The goal in creating all of
these figures was to find coherent sequences of geometrical operations that would
cumulatively built up the outlines of the medieval forms. In cases where original
design drawings survive, the presence of compass prick marks and blind lines
provided valuable evidence about the constructions actually used by the medieval
draftsmen, as noted previously. The combination of CAD use and careful on-site
examination of drawings, therefore, minimizes the problems of imprecision and
ambiguity that had troubled critics of earlier geometrical research. This method, in
fact, allows modern researchers to test geometrical hypotheses with unprecedented
rigor.

The graphics in the rest of this essay, and in the larger study from which they have
been drawn, are meant to illustrate the geometrical logic of the designs in
question.^{21} They thus explicitly show
geometrical figures to make visible operations that the original draftsmen likely
used in creating their design drawings. The draftsmen themselves, however, would not
have had to draw complete figures like these in order to establish the layout of
their compositions. A designer wishing to establish points outside an already
constructed square, for example, might have used his compasses to unfold the
diagonals of the square to its baseline, but he would have had no need to actually
draw in the arcs describing the path of the compass. Indeed, he would have had good
reason not to, since such visible arcs would have appeared intrusive and distracting
in the final drawing.^{22} So, while Gothic
drawings and buildings have a strongly geometrical character, the logic of their
designs becomes apparent only when extra lines and figures are superimposed over
them, as the following case studies will demonstrate.

## Confronting the Hechtian legacy at Ulm and Freiburg

It makes sense to begin this geometrical discussion with consideration of the Ulm and Freiburg tower projects, both because of the prominent roles they played in Hecht’s discussion, and because the pinnacle-like format of these towers facilitates comparison with Roricizer’s pinnacle design booklet. Since Hecht recognized that the analysis of original drawings can provide an even more intimate perspective on the medieval design process than the analysis of buildings, he dedicated the culminating chapter of his book on Gothic proportions to the great elevation drawings associated with the Ulm Minster workshop. This decision made good sense, not only because these drawings are among the most spectacular of medieval ‘blueprints’ but also because they can be fruitfully compared with the structure of the present tower, whose construction they guided. Hecht’s analysis of the Ulm elevation drawings must be criticized as perverse and unhistorical, however, not only because he chose to atomize these masterpieces of Gothic draftsmanship into tables of numbers, but also because he ignored much of what medieval sources reveal about Gothic design. Since Roriczer’s first step in designing a pinnacle was to establish its ground plan, and since ground plans also have priority over elevations in the booklets written by his near-contemporary Lorenz Lechler, it is odd that Hecht chose not to analyze the ground plans associated with the Ulm tower project.

Two closely related plan drawings survive to document early planning on the Ulm tower. One drawing, now preserved in London, shows the tower mostly at ground level, while another, which remains in Ulm, shows mostly the transition to the octagonal story (figs. 2b and 2a).

Until recently, both were generally dated to the 1390s and associated with the career
of the first designer involved with the project, Ulrich von Ensingen.^{23} In their recent catalog of drawings from the
Ulm region, Hans Böker and his team have plausibly proposed later datings,
attributing the plans to two of Ulrich’s followers, Hans Kun and Matthäus
Ensinger, but both drawings clearly reflect the geometrical givens established by
Ulrich von Ensingen in his design for the tower base and its buttresses.^{24} Both drawings fit neatly into the same
geometrical framework, which is shown in Figures 2c and 2d. Within the basic square
footprint of the tower, the walls and buttresses are one fourth as wide as the open
space between them, so that the salience of the buttresses beyond their centerlines
equals one tenth of the interval between those centerlines. This simple modular
relationship, shown by the small dotted arcs at the top of Figure 2c, echoes the recommendations for wall thickness
published in Lechler’s booklet.^{25}

Within this simple modular armature, though, Ulrich von Ensingen soon constructed complex geometrical figures whose subtleties would go on to influence all later contributors to the tower project. Most obviously, he constructed octagons within the square framework of the tower base, establishing the basic symmetry pattern for the tower and spire superstructure. The smallest octagon visible in Figure 2a stands slightly but measurably inboard of the buttress edges, corresponding to the dotted octagon shown below in Figure 2c, rather than to the solid lines framing the buttresses. As the labels at left indicate, their distances from the tower center are 0.765 and 0.800 times as great, respectively, as the distance to the buttress axes, which can be called one unit for convenience. The large dotted circle in Figure 2c illustrates the relationship between these geometries. The radius of the circle is established by the point where the rays aiming for the octagon corners intersect the centerlines of the main buttresses; these points are indicated by the larger black dots in the figure. The large circle thus defined then sweeps through the principal diagonals of the tower plan, creating the intersection points shown by the smaller black dots in the figure. These points define the corners of the dotted square in Figure 2c, which frames the dotted octagon corresponding to the inner octagon shown in Figure 2a. This octagon stands inset from the buttresses, since the 0.765 unit span determined by this unfolding geometrical construction differs from the 0.800 unit span given by the simple modular frame of the buttress outlines. The proportional relationship between the tower octagon and the buttresses, in other words, can only be understood by considering the interaction of modular and geometrical design strategies.

Figure 2d shows how a similar construction explains one crucial subtlety of Ulrich von Ensingen’s tower design; namely, the way the tower buttress axes pinch inward above ground level. The white dots in the figure indicate the points where a large circle inscribed within the overall tower footprint intersects the principal diagonals and the rays to the octagon corners. Lines projected forward from these intersection points define the edges of the buttresses in the second tower story. The inner and outer edges are 0.849 and 1.109 units from the building centerline, respectively, as the labels along the bottom of the figure show. The centerlines, shown in bold in the figure, thus stand 0.979 units from the tower centerline. So, as the heavy lines within the salient buttresses indicate, their centerlines are indeed slightly inboard of the original dotted buttress axes defined at ground level. The inward stepping of the buttresses that results can be seen not only in the Ulm ground plans, and in the tower itself, but also in the elevation drawings that helped to guide its construction.

The medieval elevation drawing most closely related to the final form of the Ulm tower is the so-called Ulm Riss C, created by Matthäus Böblinger around 1477 (Fig. 3).

Böblinger necessarily took as his point of departure the proportions established
by Ulrich von Ensingen at ground level, and the structure of the tower base erected
in the first three quarters of the fifteenth century, but he modified the design by
introducing a taller belfry story and a simpler overall silhouette than his
predecessors had foreseen. Böblinger himself was unable to finish the tower
because of structural problems that arose in the 1490s, but his Riss C eventually
went on to inform the nineteenth-century campaigns that made the spire the
world’s tallest masonry structure upon its completion in 1890. Discussion of
all the drawing’s intricacies would take more space than this short essay
permits, but several basic points deserve emphasis.^{26} First, the geometrical armature shown in Figure 3 explains the forms of the drawing with great precision. This
is evident, for example, in the upper zone, where the spire is drawn within a stack
of three equally sized squares, with the successive horizontals in this stack
locating the crockets flanking the spire cone. More subtly, the overall geometrical
armature appears to have also governed details such as the location of stringcourses
and tracery panels in the buttress articulation. These relationships, and similar
ones seen in many other drawings, demonstrate that the placement of Gothic
architectural ornament often reflected the underlying geometrical logic of the whole
design. So, while Gothic ornament can appear strangely flexible and capricious,
especially when seen from a classical perspective, its deployment was anything but
random. Further evidence for the importance of the geometrical armature in
Böblinger’s Riss C comes from the arrangement of its constituent
parchment pieces. Thus, for example, the parchments in the top part of the drawing
were left wide enough to include the two major vertical axes framing the cage of
diagonal lines in the upper spire zone. These lines correspond precisely to the
pinched buttress axes introduced by Ulrich von Ensingen in his designs for the tower
plan, already described in Figure 2d. These are
also the lines used to define the square modules stacked in the spire zone.^{27}

It thus becomes clear that Ulm Riss C incorporates the same mixture of geometrical and modular design principles described in Roriczer’s booklet on pinnacle composition, but in more complex and convoluted form. In both cases, the forms established in the ground plan go on to influence the elevation through processes of extrapolation and stacking. These findings thus help to demonstrate how the simple exercises described in late medieval texts related to actual building projects, including even the most ambitious tower-building projects of the era.

Since the Ulm tower was in many ways just an updated and enlarged version of the
tower at Freiburg im Breisgau, it would be natural to suspect that some of the same
design strategies were at work there. Geometrical analysis demonstrates that this
was indeed the case, even though the Freiburg project began already in the late
thirteenth century. The base of the Freiburg tower, which is quite spartan in
appearance, was probably designed around 1270. The lacy upper tower and openwork
spire, though, clearly reflects a different vision, suggesting strongly that it was
designed only around 1300, with construction of the spire lasting through the first
quarter of the fourteenth century.^{28} The
Freiburg tower is not as well documented in original design drawings as the later
Ulm tower. Seven medieval drawings depict variants of the Freiburg design, but since
none of them relates very precisely to either component of the structure, Hecht left
them entirely out of his account. In recent years there has been a growing
recognition that these drawings, even if they postdate the spire, may record
valuable information about the logic of its conception. The drawing that holds
inventory number 16.869 in the spectacular collection of the Viennese Academy of
Fine Arts, in particular, records a scheme likely connected with an intermediate
phase of the Freiburg, project, conceived between the completion of the tower base
and the design of the far more complex tower superstructure.^{29} Geometrical analysis of this drawing and of the two main
components of the tower supports this conclusion, demonstrating both continuity and
development in the Gothic design tradition.

The lowest section of the Freiburg tower is not only the oldest part of the
structure, but also the simplest, which makes analysis of its proportions
comparatively straightforward. The tower base is a plainly articulated masonry box,
with two buttresses emerging from each face. The corners of the box are just visible
between adjacent buttresses, forming a salient masonry flange in the space between
them. Some of the proportional relationships between these components are quite
obvious. The span across the outer faces of the lateral buttresses, 12.39 meters, is
almost exactly twice the 6.19-meter span between the axes of the forward-facing
buttress. Hecht believed that these dimensions were to be understood at 80 feet and
40 feet respectively, with the size of his postulated foot units being based on a
convoluted and ultimately implausible statistical argument. The span between the box
corners, 15.71 meters, he described as 50 feet 6 inches, without suggesting any
rationale for why the tower designer would have chosen this dimension (**Hecht 1979: 344**).

As the lower portion of Figure 4a indicates, a
straightforward geometrical construction involving the proportions of the square and
the equilateral triangle suffices to determine the width to the corner flange of the
box. A line with a 30-degree slope departing from the base of the trumeau intersects
the outer buttress face at the 1.15-unit height, where a unit is defined once again
as the space between the building centerline and the axis of the forward-facing
buttress. A shaded triangle fills the space between this line and another, with a
slope of 45 degrees, that rises from the trumeau base to intersect that buttress
axis at height 1.00, before bouncing down to meet the outer buttress face at its
base. The right-hand corner of the shaded triangle, which is the intersection point
between the falling 45-degree line and the rising 30-degree line, falls 1.268 units
to the right of the building centerline. This simple construction thus defines the
width of the basic box even more precisely than Hecht’s ad hoc numerical
description.^{30} With these fundamental
dimensions in hand, many other elements in the tower base can be located. The
horizontal moldings at height 2.15 and 3.15, for example, are found by stacking 1.00
unit boxes on the already established baseline at height 1.15. The span to the outer
buttress face after its first setback is 1.793 units, which is exactly √2
larger than 1.268; this can be seen in the large arc at the top of the figure, which
also sets the chapel height up to level 4.95.

It is interesting and significant that many of these same elements recur, in somewhat
altered form, in the elevation drawing number 16.869 (Fig. 4b), which may well record the oldest surviving design for
Freiburg’s openwork spire.^{31} As Figure
4b shows, the tower base depicted in the
drawing differs slightly in its proportions from the built structure, and its portal
gable is more sharply pitched, with larger and more florid crockets. These
elaborated details suggest that the drawing postdates the construction of the tower
base. Since the main purpose of the drawing was probably to present a design for the
tower superstructure and spire, its creator does not appear to have been concerned
about creating an absolutely precise depiction of the tower base.

As in the present tower base, though, the intersection of 30- and 45-degree lines
seems to have been used to set the 1.268 span to the corner flange in the drawing.
This dimension was then added twice along the vertical axis to locate the horizontal
moldings at heights 2.42 and 3.68, which thus rise measurably higher than their
analogs on the real tower, where the stacked elements are only one unit high. In the
drawing, moreover, the large arc at the top of the figure sets the height of the
chapel including its terminal balustrade; the same strategy was also used in
Böblinger’s Ulm Riss C.^{32}

Above this first balustrade drawing 16.869 shows a large belfry zone topped by a
second balustrade, which is not seen in the present Freiburg design (Fig. 5b). This discrepancy has raised questions about
whether the scheme in the drawing predates or postdates the actual tower
superstructure. The coherent and almost facile geometry of the 16.869 design
supports the former reading.^{33} In the drawing,
the buttress axes continue uninterrupted past the first balustrade, and the belfry
zone appears to have a simple square plan. The belfry is also a perfect square in
elevation; its height and its width are both twice the 1.268 dimension established
in the tower base. The belfry thus rises between heights 5.48 and 9.07, measured
between the tops of the two balustrades.

Above the second balustrade, the upper tower and spire in the drawing fit precisely into a stack of four square modules, each 1.268 units per side. The corners of an octagon inscribed within the lowest of the four locates the corner flanges of the octagonally symmetrical story just below the spire base. As in the case of Ulm Riss C, the parchment is squared off at the top, so as to encompass the full rectangular armature. And, as in the Ulm drawing, the spire fits into a stack of three boxes. While the crockets of Ulm Plan C counted out this rhythm, though, that role falls in the Freiburg drawing to the tracery roundels of the spire, two of which are centered at the box-bounding heights 12.87, 14.14 and 15.41. The tip of the spire finial is at height 19.21 units; if the drawing were scaled so that the distance between its buttress axes measured the same 6.19 meters seen in the present Freiburg spire base, this would work out to an overall height of 118.91 meters.

The present Freiburg spire is slightly shorter than the one depicted in 16.869, but
its geometry is far more sophisticated, suggesting that the present design was
developed later. And, while the geometry of the drawing is quite consistent from
ground level to spire tip, the complex format of the actual upper tower and spire
differs markedly from the simple boxy format of the tower base. As Figure 5a shows, the buttresses of the tower base were
abruptly terminated in a short transitional zone capped by a single balustrade that
runs between heights 6.05 and 6.26. In plan, this balustrade describes a complex
twelve-pointed star. Its format can best be understood as the result of the
dynamically unfolding process illustrated in Figures 6a–f.^{34}

As Figure 6a shows, the basic frame of the
figure is a square circumscribed about an octagon, a circle, and a smaller square,
whose corners coincide with the corner pinnacles flanking the octagonal tower core;
these pinnacles lie 1.000 units out from the building centerline, so that they align
with the axes of the buttresses in the tower base below. However, while the designer
of the tower base used combinations of square and equilateral triangular geometries
in elevation, the designer of the superstructure combined these figures in the plan.
So, as Figure 6b shows, the basic star shape
within the frame can be found by drawing wedges 30 degrees wide within the 45-degree
wedges created by the octagonal geometry of the overall plan. Then, as Figure 6c shows, equilateral triangles can be inserted
into the four corner wedges, forming the basic twelve-pointed figure. Further
elaborations in Figures 6d and 6e produce the final form shown in Figure 6f, which agrees superbly well the the plan of
the tower as recorded in survey drawings. The basic dimensions established in the
plan, moreover, can be stacked to give the crucial points in the elevation, which
are shown in Figure 5a.^{35} Here once again, the dynamics of geometry provide an
explanation for the proportions of the structure far more compelling, and far more
historically plausible, than Hecht’s ad hoc modular schemes. The Freiburg and
Ulm spire designs both involve design strategies very similar to those seen in
Roriczer’s pinnacle booklet.

## Polygons and ‘irrational’ proportions in Gothic church elevations

Geometrical design strategies were used throughout the Gothic era to set the
proportions not only of pinnacle-shaped spires, but also of church cross sections.
In the literature on Gothic design, such sections are often described as being
designed either *ad quadratum* or *ad triangulum*,
i.e. to the proportions of a square or to those of a triangle. This simple binary,
of course, hardly suffices to describe the full palette of options employed by
Gothic designers. As Ackerman showed decades ago in the case of Milan, even the term
*ad triangulum* could have a variety of meanings, depending on
whether they involved equilateral triangles or other types, and depending on how
these geometric figures were applied in relation to the elevation. Ackerman’s
article on Milan also placed great emphasis on the efforts of the mathematician
Stornaloco to find a modular approximation for the proportions that result from the
construction of an equilateral triangle, which are called ‘irrational’
in the mathematical sense because they cannot be expressed as a ratio of whole
numbers (**Ackerman 1949: esp.
90–96**).^{36} To gain a
complementary perspective on this issue, the following paragraphs present case
studies of several buildings whose elevations appear to have been governed by great
octagons: the cathedrals of Prague, Clermont-Ferrand, and Reims, and the Cistercian
church of Altenberg. Octagons, of course, can be neatly inscribed within squares,
but it would be too simple to describe any of these buildings as being designed
*ad quadratum*, since their proportions evidently depend on the
‘irrational’ relationships deriving from the geometries of the
octagons.

The planning for Prague Cathedral deserves particularly close attention in this context, for several reasons: first, because much of the building was designed by Peter Parler, the first and most influential of the ‘Junkers of Prague’ cited by Roriczer as the authoritative practitioners of his Gothic tradition; second, because an original drawing survives to document the planning of the cathedral’s section; and third, because analysis of this drawing helps to shed light on the relationship between Peter Parler and his French predecessor Matthias of Arras, who began construction of the cathedral in 1344. Comparison of the Prague section with those of Clermont-Ferrand, Reims, and Altenberg will show that the octagon-based planning strategy seen in the Prague drawing was already being used in France and Germany by the middle of the thirteenth century.

Drawing 16.821 in the Vienna Academy collection shows the cross section of Prague
Cathedral’s choir aisles and the flying buttressses that soar over them to
brace the choir wall (see **Böker 2005:
74–78**; **Bork 2011b:
207–212**). The detailing of the buttresses is somewhat simpler than
in the actual structure, suggesting that the drawing may have been produced under
Peter Parler’s direction fairly early in the design process. In this drawing,
the overall proportions are set by the right half of a great octagon, whose height
equals the span from the floor to the top of the upper flying buttresses (Fig. 7).

The center of the octagon coincides with a small mask that gazes out from the middle of the triforium. The ray from the center of the octagon to its upper right corner passes through the two gargoyles on the flying buttresses, which are thus used as geometrical markers. The height of this upper right corner coincides with the height of the capitals in the main elevation; this height can be called 1.707, where 1.000 is equal to the combined width of the two equally sized aisles. The aisles also rise to height 1.000, so that they fit into a square. When an octagon is inscribed within this square, and a rotated square placed around the octagon, its right-hand tip falls on the outer face of the lateral buttress. The midline of the outer wall aligns with a circle circumscribed about this octagon.

The geometrical principles governing the drawing were adopted quite faithfully in the actual choir structure, as Figure 8 shows. The buttress articulation in the real building is more complicated, as noted above, and the intermediate buttress pinnacle now terminates a bit lower than in the drawing, but the proportions of the main elements are effectively identical. Importantly, too, this graphic shows that the central vessel of the Prague choir has proportions determined quite precisely by a single great governing octagon. These proportions occur for two reasons: first, because the geometry of the drawing uses a great half-octagon to relate the elevation of the main vessel to the width of the aisles; and second, because the central vessel is exactly twice as wide as the aisles, as the ground plan in the bottom of the graphic shows. These results together mean that the half-octagon seen in Figure 7 can slide over into the main vessel, where symmetry about the building axis then produces the full octagonal scheme seen in Figure 8.

Geometrical analysis suggests that Peter Parler owed more than has usually been
imagined to his French predecessor Matthias of Arras. Matthias had designed the
radiating chapels and the beginnings of the lower story in the straight bays of the
choir; these are shown in dark grey in the ground plan, while the later portions
completed under Parler’s direction are shown in light grey. It was Matthias,
therefore, who established the 2:1 relationship between the width of the main vessel
and the aisles. And it was Matthias who began to define the elevation by
establishing the height of the aisle and chapel vaults. But evidence from Prague
cannot, by itself, say what Matthias intended for the upper stories. It is
significant, in this connection, that precisely the same octagon-based geometry seen
in Parler’s drawing and in the present Prague choir also governs the
proportions of the cathedral at Clermont-Ferrand, as Figure 9a shows.^{37} Since Matthias
worked in southern France before coming to Prague, he surely would have known
Clermont Cathedral, which was begun in 1248. Matthias probably had the Clermont
scheme in mind when he began the Prague project, for which he likely produced
elevation drawings.

The octagon-based proportioning scheme seen at Prague and Clermont was already being
used early in the thirteenth century to establish the elevation of Reims Cathedral,
as Figure 9b shows. As at Clermont, the aisles
teminate at the equator of a great octagon whose lower facet corresponds to the
floor of the main vessel, measured between the arcade axes. In both cases,
therefore, the proportions of the main vessel are ‘irrational’ in the
mathematical sense, although the designs are geometrically quite lucid. At Reims,
the corners of the great octagon establish the baselines of the capitals in the
aisles, and the midlines of the capitals in the arcades. At Reims, the steeply
pitched main vaults surpass the height of the great octagon’s upper facet,
which might at first seem to represent either a breakdown in architectural order, or
a problem with the geometrical analysis. In fact, though, scrutiny of the vaults
springers early in the twentieth century convinced Henri Deneux that the vaults were
originally planned to be about 1.70m lower than they are today, which would place
their keystones on the top facet of the octagon.^{38} As Figure 9b shows, moreover,
the current transverse arches now rise to meet the circle circumscribed around the
octagon, demonstrating that even the vault revision took account of the
building’s overall geometrical order.

Since Reims Cathedral was greatly admired already in the thirteenth century, as many
drawings by Villard de Honnecourt attest, it is not surprising that ideas from Reims
soon began to influence the design of buildings not only in southern France, as at
Clermont, but also in the German-speaking world. The octagon-based elevation scheme
of Reims was copied, for example, at the Liebfrauenkirche in Trier, begun most
likely around 1227, and at the Cistercian church of Altenberg, begun in 1259.^{39} These projects demonstrate that the
geometrical planning strategies seen at Reims and Clermont had begun to enter the
Germanic world a century before Matthias of Arras began his work at Prague. At
Altenberg, the proportions of the choir section are again set by a single great
octagon, as Figure 10 shows.

As at Reims and Clermont, the midpoint of the octagon aligns with the base of the triforium, instead of with the midpoint of the triforium, as it does at Prague. This variation helps to illustrate the flexibility that Gothic designers enjoyed, even when working within a crisply defined geometrical framework like that of the octagon. At Altenberg, as at Reims, the geometry of the elevation involved not just the main octagon, but also the circles related to it. So, while the tops of the arcade capitals coincide with the lower corners of the octagon, at a height equal to 0.707 of the main vessel span, the smaller capitals of the high vault fall at height 1.669, coinciding with the level were the rays to the octagon corners cut the circle inscribed within it. Many other crucial heights in the Altenberg section can be found by logical extension of this system, but the preceding examples should already suffice to demonstrate the relevance of the basic octagonal framework.

The case of Altenberg has the potential to reveal a great deal about Gothic building
practice, because recent studies of the building are starting to show how members of
the Altenberg workshop used modular dimensions together with dynamic geometry to
develop the design. This can be seen in both plan and elevation. At Altenberg, as at
Cologne Cathedral, the overall groundplan of the chevet was set by a dodecagon.^{40} In each case, one facet of this twelve-sided
figure would correspond to a single radiating chapel. In Cologne, the geometry is
particularly clear, with the array of chapels thus corresponding to exactly 7/12 of
a regular dodecagon. In Altenberg, though, the geometry is less regular, because of
a complex interaction between geometrical and arithmetical design modes. The
relative widths of the choir, aisles, and buttresses were definitely set by the
geometry of a regular dodecagon; the relationships are quite precise. But, while all
of the columns of the chevet sit on the circles defined by these radii, their
positions on these circles were not set by a regular dodecagon. Instead, as Norbert
Nussbaum and Sabine Lepsky have demonstrated, they are separated by intervals of 2.5
column diameters, where each column diameter of 83 centimeters in turn equals 2.5
feet of 33.2 centimeters. These same units seem to have been used throughout the
construction of the choir. In elevation, for example, the height to the top of the
main arcade capitals can be expressed as 9 column diameters, or 9 x 0.830 m = 7.470
m; this almost perfectly matches the geometrically determined height seen with the
octagon corner in Figure 10, which is
1/√2 x the choir span, or 0.707 x 10.56 m = 7.468 m. Interestingly, too, the
same geometrically determined heights seen in Figure 10 continue to govern the elevation of the west façade at
Altenberg, which was built in the fourteenth century using a slightly larger foot
unit of 33.55 centimeters.^{41} The use of these
two distinct modular systems to approximate geometrically determined dimensions
provides an excellent example of Toker’s principle of
‘pseudo-modularity’.

## Conclusion

The preceding case studies illustrate several important points about the use of geometrical proportioning strategies in Gothic architecture. They show, first of all, that centuries of sophisticated tradition informed the work of late Gothic authors like Roriczer and Lechler, even if their writings were not eloquent enough to compete with the work of their Renaissance rivals. They demonstrate, moreover, that Konrad Hecht was wrong to dismiss the importance of geometry in Gothic form generation. Numerical and module-based thinking certainly played a role in Gothic design practice, but not to the exclusion of dynamic geometry. Instead, these were complementary strategies: sometimes geometrical constructions could be unfolded within modularly defined armatures, as in the Ulm ground plans; in other cases, modules could be combined to approximate geometrically determined proportions, as in the Altenberg choir elevation. Most fundamentally, though, these examples begin to hint at the rich variety of geometrical planning strategies employed by Gothic designers, which deserve far more detailed and rigorous exploration than they have received to date. With the increasingly widespread availability of reliable building surveys and CAD systems, and with the rapid progress of research on Gothic drawings, there is good reason to be optimistic that more of this kind of scrutiny will soon be forthcoming. Enough good work has already been done in this field, though, to demonstrate that Gothic architecture embodied a complex procedurally based formal order whose conventions governed the dynamic unfolding of geometry, rather than fixed canons of proportion like those seen in classical architecture. In this sense, Gothic designers anticipated the work of their twenty-first century successors, who are now beginning to use computer algorithms to explore similarly dynamic approaches to form generation. Research on Gothic geometry thus has the potential to enrich not only the scholarly discourse on medieval architecture, but also a larger and broader conversation about the character of architectural order and proportion.

## Supplementary Material

Please visit the following link to view the supplementary material:

Supplementary Material: http://dx.doi.org/10.5334/ah.bq.s1