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Introduction
For students of medieval proportional systems who lack
such documentary evidence as James Ackerman had, in
relative abundance, for his groundbreaking 1949 study of
the Cathedral of Milan, the only choice has been to turn
to the buildings. Yet Ackerman warned that such an exer-
cise was destined to fail: ‘the analysis of remaining monu-
ments,’ he wrote, ‘provides insufficient evidence for [the]
task’ (Ackerman 1949: 85).
Ackerman’s pessimism probably had much to do with
the quality of survey data readily available for medieval
monuments. One could not assume, without knowing
the specific conditions of acquisition, that existing plans
were accurate enough to sustain the scrutiny neces-
sary to resolve differences among potential proportional
schemes. Had the buildings in question been completely
and carefully surveyed? Were the resultant plans and sec-
tions plotted using the drafting techniques necessary to
minimize error? To what extent were measurements recti-
fied — i.e., arbitrarily corrected — out of a desire to repre-
sent the building in a supposed perfected state, or from a
lack of sufficient survey data to represent the building as
it actually stood?
The scholar of proportions had two solutions when
faced with the difficulty of establishing the represen-
tational reliability of drawings created by others. He
might dismiss the problem by arguing that precision
in plan and section was in any case not essential: since
proportional systems were more often than not imper-
fectly executed in the buildings themselves for a host of
reasons having to do with the reality of the construction
site, a certain inaccuracy in an existing plan might be
tolerated as a result. Though this might be satisfactory
in a general sense — in order, for example, to locate a
double square in the nave of a given building — more
complex situations could not be resolved with any clar-
ity. It was simply impossible to know precisely by how
much a proposed proportional scheme was distant from
the constructional reality of the building.1
In this case the only choice was to undertake a new
survey of the building in question using the most precise
means available. Yet this could take weeks, if not months,
and only in the best of situations would it be possible to
acquire data in the upper reaches of the building: scaffold-
ing is expensive and encumbering.
Today, over a half a century later, Ackerman might have
had a different response. Though we are faced now, as much
as we were then, with a dearth of documentary evidence for
proportional planning, our ability to look to the buildings
with confidence — and rapidity — has changed radically, in
large part thanks to laser scanning technology.2
Acquisition, assembly, and sectioning are the three pri-
mary components of laser surveying.
Acquisition
There are two primary means of measuring distance with
a laser: a) by calculating the time of flight for a laser pulse
to be sent, reflected from a surface and returned, and b)
by calculating the phase shift induced in a sinusoidally
encoded beam after travel, reflection, and return. Each
technique presents certain advantages and is chosen gen-
erally as a function of the type of scanning work to be
accomplished. Time of flight measurement, for example,
has been the preferred approach for distances that extend
beyond 150 meters. The technology is advancing with
impressive rapidity, however, driven by a greater demand
for speed, accuracy, and portability, and limitations are
regularly overturned as scanner hardware evolves.3
In a typical scanner the laser beam is distributed over a
range of 360 degrees horizontally and 270 degrees verti-
cally by a rotating mirror, and acquires the distance between
itself and every surface that it can ‘see’ at a rate that can
approach one million measurements per second (Fig. 1).4
The result is what is called a ‘cloud’ of points (Fig. 2). The
* Vassar College, United States
antallon@vassar.edu
RESEARCH ARTICLE
Divining Proportions in the Information Age
Andrew Tallon*
Tallon, A 2014 Divining Proportions in the Information Age.
Architectural
Histories,
2(1): 15, pp. 1-14, DOI: http://dx.doi.org/10.5334/ah.bo
$UFKLWHFWXUDO
+LVWRULHV
The process of reverse engineering proportional systems of historic buildings has long been fraught with
problems. One cannot assume, without knowing the specic conditions of acquisition, that existing plans
are accurate enough to sustain the scrutiny necessary to resolve dierences among potential proportional
schemes. Yet producing a new survey with conventional measurement instruments could take weeks, if not
months, and only in the best of situations would it be possible to acquire data in the upper reaches of the
building—information required to avoid arbitrary dimensional rectication. With the advent of high-speed
and high-precision laser scanning, however, the situation has changed dramatically.
Tallon: Divining Proportions in the Information AgeArt. 15, page 2 of 14
Fig. 1: Leica Geosystems ScanStation C10 at work on the south flank of the Cathedral of Saint-Pierre in Beauvais, June
2013. Photograph: Andrew Tallon.
Fig. 2: Cathedral of Saint-Pierre in Beauvais, laser scan, June 2013: point cloud. Targets are labeled in yellow. Each pixel
in the image represents a measurement; the points have been color mapped using textures acquired photographi-
cally. Image: Andrew Tallon.
Tallon: Divining Proportions in the Information Age Art. 15 , page 3 of 14
maximum positional error for each of these measure-
ments, which in the latest long-range scanners is often
fewer than five millimeters, is a function of the laser acqui-
sition technology employed, the error-correction abilities
of the scanner, and atmospheric conditions (lasers can
produce erroneous data when passing through raindrops,
for example).
Scan resolution is a key consideration. Scanning with
lower point density will take less time, but with too few
points the details of the building will be impossible to
reconstruct from the data. A resolution of a point every
five centimeters, for example, may be sufficient to locate
the plane of a wall and even the curvature of a vault, but
it is largely insufficient to represent the detailed forms
of a capital or base, which require point densities of mil-
limetric order. For this reason multiple resolutions are
often most practical: the general details of walls, floor, and
vaults might be acquired with relatively low density (and
thus relative rapidly), with denser and slower windowed
scans (i.e., a subset of the scanner acquisition sphere)
reserved for key details.
While an individual cloud will supply a great deal of
information, it is but a single viewpoint; to produce a sur-
vey of sufficient density and to minimize occlusions it is
necessary to displace the scanner. Each scanner position,
or station, must be fixed to a network of control points
(Figs. 2 and 3).5 Control points are most often supplied by
reflective targets that can be accurately scanned and then
recognized by the onboard scanner software; typical error
in target recognition is on the order of two millimeters.6 It
is also possible to use architectural elements in the build-
ing as control points, which presents certain advantages
in terms of rapidity (no targets need be placed), but at the
potential cost of precision. Scanning a target that is recog-
nized by the scanner as such is straightforward; assuring
that the corner of a particular abacus is scanned at suf-
ficient density to properly resolve its vertex, for example,
is less so.
The amount of time required to scan an entire building
is a function of the technology used, the scan resolution,
conditions of building access, the size and complexity
of the building, and the efficiency of the operator. In
2008, for example, a survey of 52 stations undertaken
by the author at Bourges Cathedral took fully nine days
to acquire (Tallon forthcoming). In 2013 the author pro-
duced 74 stations at the Cathedral of Beauvais — with a
twenty-fold increase in measurement density — in just
three days.7
Fig. 3: Cathedral of Notre-Dame in Amiens, partial laser scan, June 2009: individual scan positions with targets (1–8)
and final registration (9). Image: Andrew Tallon.
Tallon: Divining Proportions in the Information AgeArt. 15, page 4 of 14
Assembly
Once the desired stations and requisite control points are
acquired the data must be registered — that is, assembled
by computer.8 The software undertakes a series of inter-
polations to create a match among the various control
points, or constraints, with the least error.9 It is often nec-
essary to suppress certain constraints.
Scan-processing software offers two ways to under-
stand and minimize error accrued during the process of
assembly. First, it indicates the extent to which individual
registration constraints are distant from their positions
as indicated by the final interpolation (Fig. 4). Second,
it is possible to inspect visually the resultant registration
for errors. Because laser scanners are often operated in
full-dome mode, scanning through their entire spherical
range, they tend to produce measurements that overlap
with those of many other stations. When the scan data for
a wall plane, for example, are examined at close range, it
will be possible to see if the measurements of this wall as
acquired in these various stations are coincident within
Fig. 4: Cathedral of Saint-Pierre in Beauvais, laser scan, June 2013: list of constraints for the interior registration, with
corresponding error. Note that one of the constraints has been disabled because its error was greater than three mil-
limeters. Image: Andrew Tallon.
Fig. 5: Cathedral of Saint-Etienne in Bourges, laser scan, May 2008: alignment problem due to misnamed target in the
westernmost bays of the nave. Image: Andrew Tallon.
Tallon: Divining Proportions in the Information Age Art. 15 , page 5 of 14
a range of error that is consistent with that of the laser
itself (typically around five millimeters). If they are not,
the flawed data must be traced back to the station to
which they belong, and the constraints carefully verified
to determine the source of the problem. Figure 5 illus-
trates a dramatic error of several meters in the west end of
the nave at Bourges Cathedral, the result of a target mis-
takenly given the name of another: the data in blue are
meant to align with those in green.
Sectioning
Once the data are registered, sections, plans, and views
can be created by limiting visible points. The rectangular
excerpt of the plan of the abbey church of Saint-Denis
in Figure 6, for example, is limited in vertical terms to
the crypt and an adjoining meter of the chevet; it is gra-
dated in color according to elevation. Because the scanner
generates points, not planes, all surfaces are transparent;
it becomes possible to observe that the chevet piers (in
orange) are not aligned with those of the crypt (in blue).
If we assume that the builders in fact attempted to posi-
tion one directly over the other, the disjunction could be
attributed to a faulty positional translation necessitated
by the presence of the crypt vaults and complicated, per-
haps, by portions of the previous church still present on
the site. An apparent miscalculation such as this would be
of great interest if it could be taken as a benchmark for
planning precision — for the builders’ ability to control
constructional error elsewhere in the church.10
In a similar way, a section through the nave of the abbey
church of Saint-Leu-d’Esserent (Fig. 7) reveals informa-
tion about the building with far greater clarity than could
be had with the conventional tools of steel tape, plumb
bob, or total station, for which multiple measurements of
this density would be laborious at best. Such a section — a
representation as visually explicit as it is precise — makes
it possible to quantify, with a level of detail on the order of
five millimeters, the vault-induced outward deformation
of the building (Tallon 2012: 173–93).11
Analysis
Any such subset of the cloud data can be imported into
computer-aided design software such as AutoCAD, for
the iteration of potential proportional schemes, using the
robust shape generation and mensuration tools proper
to such programs. The chevet plan of the Cathedral of
Notre-Dame in Paris (Fig. 8), from a scan undertaken by
the author in 2010, will supply an example (Sandron and
Tallon 2013: 30–31, 183). The data reveal that the columns
in the hemicycle and ambulatory of Notre-Dame were
located using a series of concentric circles with propor-
tional radii — as would be done subsequently at the geo-
metrically proximate cathedrals of Bourges (Fig. 9) and
Coutances.12 Further, the scan data indicate that certain
plinth faces among the intermediate ambulatory piers are
curved according to the radius of the circle used to place
them (A and B in Fig. 10).
This is a detail that is difficult to see in the building,
and is easily overlooked when measuring by traditional
means: had one assumed, for example, that the plinths
were square like their adjacent counterparts, one might
well have fixed them in plan using their corner points
alone. Finally, not only does the outermost concentric cir-
cle in the choir of the Cathedral of Paris probably locate
the outer extremity of the original choir buttresses, but
there is a direct correspondence between the penultimate
circle and an equilateral triangle that appears to have
determined the sectional envelope of the choir (Fig. 11).13
The entire spatial and structural system might thus be
accounted for in plan.
Given the resemblance in plan and section of the
cathedrals of Paris and Bourges, it could be expected that
the transverse matrix at Bourges would also have been
determined using an equilateral triangle. Yet this is a
difficult hypothesis to sustain because of the perpetual
uncertainty, in the absence of documentary evidence,
concerning the key points of reference for an imposed
geometrical shape. Should the base of the triangle spring
from the inner faces, the outer faces, or the centers of
the outer walls? To which vertical points must the trian-
gle altitude correspond to be considered legitimate? The
transverse vault rib, the severies just above, or the extra-
dos of the vault?
That no accurate section of Bourges existed until
recently (Fig. 13) has not made the quest any easier.
Eugène-Emmanuel Viollet-le-Duc, for example, derived
the sectional proportions of the building using a draw-
ing he had copied, apparently without having visited the
building, from a highly confected section made by dioc-
esan architect Hippolyte Roger in the early nineteenth
century.14 Viollet-le-Duc’s image was widely disseminated,
as was a rectified version (Fig. 12) published by Georg
Dehio and Gustav von Bezold (Dehio and von Bezold
1894: plate 376).15 In the late 1950s Robert Branner and
Pierre Capron took a series of measurements of the build-
ing for Branner’s doctoral dissertation (Branner 1953).
Capron then created a new section drawing (Fig. 13) that
was printed on a large fold-out piece of paper placed at
the back of the 1962 edition of Branner’s monograph
(Branner 1962: plate 1).16
These were the drawings that Peter Kidson probably
had at his disposal — though he did not identify which
he used — when he attempted to address the questions
posed above in an article published in 2000 (Kidson 2000:
147–56). Kidson proposed that the distance between the
exterior faces of the outer choir walls at Bourges was cal-
culated using 30 perches of 1.42 meters, each equal to five
28.5 centimeter-long feet — an unusual unit whose origin
he did not discuss.17 The altitude of an equilateral trian-
gle with a side of 30 is 26 — two numbers, Kidson argued,
that were chosen for a specific reason (Kidson 2000: 155).
The ratio of 26 to 15 (half of 30) was the most precise
arithmetic approximation in common use for √3, a num-
ber necessary to calculate the triangle altitude (altitude =
½ · √3 · side length). Kidson believed that Bourges had
been dimensioned literally according to the formula — an
idea that he found particularly compelling because of an
apparent conceptual link with the cathedral in Paris. An
Tallon: Divining Proportions in the Information AgeArt. 15, page 6 of 14
Fig. 7: Abbey church of Saint-Leu-d’Esserent, laser scan, June 2011: section through the nave. Image: Andrew Tallon.
Fig. 6: Abbey church of Saint-Denis, laser scan, June 2011: color-coded plan and section of the sanctuary and crypt. The
crypt piers and bases are in blue, the crypt vaults are in green, and the sanctuary piers and bases are in orange. Image:
Andrew Tallon.
Tallon: Divining Proportions in the Information Age Art. 15 , page 7 of 14
equilateral triangle with sides of 26 perches corresponds,
in Kidson’s words, ‘as nearly as no matter’ with the sec-
tional matrix of Notre-Dame (Kidson 2000: 155).
Kidson’s triangle does indeed align well with Branner’s
section (Fig. 13) and ‘as nearly as no matter’ with that
of Dehio (Fig. 12); the same must have been true for
whatever section of Notre-Dame he tested. Yet when
imposed on a laser-generated section of the choir of
Bourges (in white in Fig. 14), the correspondence is less
compelling. The exterior faces of the outer choir walls are
ten centimeters further apart than dictated by the 1.42
meter perch — admittedly a fairly minimal difference.
Fig. 8: Cathedral of Notre-Dame in Paris, laser scan, January 2010: plan of the choir with concentric circles of propor-
tional radii. The circles have radii of 6.65 m, 12.42 m, 18.19 m, and 23.96 m. Image: Andrew Tallon.
Tallon: Divining Proportions in the Information AgeArt. 15, page 8 of 14
Yet what to make of the triangle peak, which extends 40
centimeters above the vault extrados?18 As for the corre-
spondence with Notre-Dame in Paris, is the actual width
of the choir, expressed in Kidson’s enigmatic perch units,
25.6, sufficiently close to 26 to forge the link?
Another triangle lurks in the wings, that generated from
the outermost concentric circle of the plan of Figure 9,
with diameter of 40.14 meters (in red in Fig. 14). Yet its
peak falls 98 centimeters below the vault keystone — not
much of an improvement in correspondence with respect
to the triangle proposed by Kidson. There are two poten-
tial morals of this story: first, it may be time to aban-
don the supposed equilateral triangularity of Bourges
in favor of a scheme that better reflects constructional
Fig. 9: Cathedral of Saint-Etienne in Bourges, laser scan, May 2008: plan of the choir with concentric circles of propor-
tional radii. The circles have radii of 7.53 m, 13.8 m, and 20.07 m. Image: Andrew Tallon.
Tallon: Divining Proportions in the Information Age Art. 15 , page 9 of 14
Fig. 10: Cathedral of Notre-Dame in Paris, laser scan, January 2010: the inner faces of the intermediary aisle columns
that frame the axial bay (A and B) are curved. Image: Andrew Tallon.
Fig. 11: Cathedral of Notre-Dame in Paris, laser scan, January 2010: section through the third straight bay east of the
chord of the hemicycle with superimposed equilateral triangle. Image: Andrew Tallon.
Tallon: Divining Proportions in the Information AgeArt. 15, page 10 of 14
reality — one that can account for the disposition of the
inner and outer aisles, for example.19 Second, and perhaps
more importantly, working with existing plans and sec-
tions for which the precise conditions of acquisition and
rendering are unknown is sufficiently fraught with prob-
lems as to render the prospect untenable, as Ackerman
seems to have suggested. Had Kidson seen the imper-
fect consonance of his triangle with the actual building,
as indicated here by the laser scan, he might have been
reluctant to propose what in retrospect appears to be a
foot unit concocted to fit the formula.
In the alignment of proposed proportional schemes
with an actual building, how close is close enough? 40
centimeters? Or 4 centimeters? Laser scanning cannot tell
you this — but it can tell you where you stand. By mak-
ing it possible to represent structures in a highly accurate,
explicit, and time-efficient way, laser scanning supplies
the means to combat imprecision and its correlate, the
numerological wizardry sanctioned thereby — and thus
has potential to revolutionize the venerable but vexed
process of reverse engineering proportional systems
directly from the building fabric.20
Fig. 12: Cathedral of Saint-Etienne in Bourges, section drawing (Dehio and von Bezold 1894: plate 376), with equilateral
triangle superimposed by Andrew Tallon.
Tallon: Divining Proportions in the Information Age Art. 15 , page 11 of 14
Fig. 13: Cathedral of Saint-Etienne in Bourges, section drawing (Branner 1989: plate 1), with equilateral triangle super-
imposed by Andrew Tallon. Note the distortion introduced into the southern outer aisle, chapel, and buttress by a
fold in the paper — another hazard of using existing plans.
Tallon: Divining Proportions in the Information AgeArt. 15, page 12 of 14
Acknowledgments
I am grateful to Matthew Cohen and Michael Davis for
their helpful comments.
Notes
1 Nigel Hiscock’s account of his struggle with these
problems is particularly revealing: Hiscock (2000:
293–98). See also Fernie (1990: 230).
2 For a discussion of the changes brought by photo-
grammetry and laser scanning to the discipline of
architectural surveying and drawing see Sartor (2011:
90–103). See also Davis (2011: 219–33).
3 The four major manufacturers of laser scanners used
for surveying purposes are at present Leica Geosystems
(http://www.leica-geosystems.com), Faro (http://
www.faro.com), Trimble (http://www.trimble.com/3d-
Fig. 14: Cathedral of Saint-Etienne in Bourges, laser scan, May 2008: section through the chord of the hemicycle with
superimposed equilateral triangles. In white, the hypothesis of Peter Kidson; in red, an alternative hypothesis based
on the diameter of the outer circle in Figure 9. Both triangles are placed 30 centimeters below the current pavement
to match its original level. The section includes scan data of the roof acquired by the 3d surveying and imaging com-
pany Art Graphique et Patrimoine; I am grateful to directors Gael Hamon and Didier Happe for permission to use
them. Image: Andrew Tallon.
Tallon: Divining Proportions in the Information Age Art. 15 , page 13 of 14
laser-scanning), and Riegl (http://www.rieglusa.com).
For further discussion of the technical aspects of laser
scanning, see García-Gómez et al. (2011: 25–44).
4 As with Leica Geosystem’s current top-of-the-line scan-
ner, the P20.
5 It is standard practice to use a total station in conjunc-
tion with a laser scanner to supplement this control
point network, although the technology is advancing
in such a way that this method may soon be outmoded.
Leica Geosystems’s Nova series, for example, combines
the functions of total station and laser scanner.
6 It is possible, in some scan processing software, to rec-
ognize a series of targets, distributed liberally through-
out the building, after the scan has been completed.
While this practice increases the speed of operation in
the building, because no time is spent scanning tar-
gets or other control points, it can entail several impor-
tant risks: first, that an insufficient number of targets
might be placed; second, that a given target might not
be scanned with sufficient resolution to be properly
recognized by the scan processing software; and third,
that a target critical for registration might be lost,
unbeknownst to the operator, when someone visiting
or working in the space inadvertently interrupts the
laser beam at the very moment that the scanner passes
over the target in question — something that happens
with unfortunate regularity.
7 At Bourges, 123,204,667 measurements were
acquired; at Beauvais, 2,915,108,374. I am grateful to
Leica Geosystems France for having supplied me with
their latest laser scanner, the P20, for the Beauvais
survey; to El Mustapha Mouaddib at the University of
Amiens and Stephen Murray for their collaboration;
and to Columbia University graduate student Nicole
Griggs for her assistance with acquisition. Supplemen-
tal scans were produced using a Leica Geosystems C10
and a Faro Focus Focus3D X 330.
8 Typically, the software used for registration is that pub-
lished by the manufacturer of the survey machine. The
software used by the author is called Cyclone, by Leica
Geosystems.
9 Certain scan processing software packages are able to
recognize common forms and planes in raw scan data
through superimposition and error calculation to cre-
ate constraints, but the technique depends on high
scan density and large planar surfaces and is ultimately
less precise than when targets are used.
10 On the question of planning precision at Saint-Denis
see Crosby (1987: 233–41); Kidson (1987: 11–17); Van
Liefferinge (2011: 147–57); and Bork (2013: 55–68).
See also Cohen (2008: 18–57).
11 For a more general discussion of deformation and its
analysis, see Tallon (2013: 530–54).
12 The circles at Notre-Dame have radii of 6.65 m, 12.42
m, 18.19 m, and 23.96 m; the difference in radius
between each adjacent circle is exactly 5.77 m. The
octagon-based theory of Stefaan van Liefferinge (2010:
496–502) is untenable given its lack of correspond-
ence with the building fabric.
13 The buttresses were later extended outward by roughly
1.5 m, according to Viollet-le-Duc (1856: 293). See Tal-
lon (2007: 150–53).
14 Roger’s section is published in Martin and Cahier
(1841–1844). For Viollet-le-Duc’s sectional analysis,
see (Viollet-le-Duc 1864: 546–49); his transverse sec-
tion of Bourges is found in Viollet-le-Duc (1854: 199,
Fig. 34). The earliest section — though only partial
— appears to be that by François-Narcisse Pagot of
1833 (Charenton-le-Pont: Médiathèque du Patrimoine
82/18/1002 no. 14305).
15 Dehio and von Bezold, though they often redrew sec-
tions, had good reason to rectify this image in par-
ticular: the drawing as printed in the Dictionnaire was
somewhat skewed. Two further transverse sections,
little known outside the world of the Monuments
historiques, were created in 1889 by Paul Boeswill-
wald (Charenton-le-Pont: Médiathèque du Patrimoine
82/18/1002 no. 14302) and in 1943 by G. Desmarest
(Charenton-le-Pont: Médiathèque du Patrimoine
82/18/2004, no. 81194).
16 The section drawing published in 1962 is missing a
scale reference; it was included in the posthumous
English edition (Branner 1989: plate 1).
17 Kidson (2000: 155–56) noted only that he had had
‘occasion to argue that one of the more widely used
masonic yardsticks had a length of ca. 1.42 m’ that
‘could be subdivided in various ways to produce a
series of documented foot measures; and it is a moot
point whether in any given case the conceptual unit
was the foot or the yardstick’. Unfortunately his discus-
sion of this perch, during the Mellon Lectures of 1980,
was never published.
18 The triangle in Figure 14 is set at the level of the orig-
inal choir pavement, 30 cm lower than at present.
19 Viollet-le-Duc (1864: 546), despite having employed
a faulty section for his analysis, wrote that ‘tout le
système des proportions de la cathédrale de Bourges
dérive du triangle isocèle rectangle, et non point du
triangle équilatéral’.
20 Stephen Murray’s forthcoming work on the geom-
etry of the chevet and crypt at Saint-Denis is a case in
point.
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How to cite this article: Tallon, A 2014 Divining Proportions in the Information Age.
Architectural Histories,
2(1): 15, pp. 1-14,
DOI: http://dx.doi.org/10.5334/ah.bo
Published: 20 June 2014
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