The Fractal Pattern of the French Gothic Cathedrals
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Abstract
The classic patterns of Euclidean Geometry were used in the construction of the Gothic cathedrals to provide them with proportion and beauty. Still, there is also another complex concept related to them: the unevenness of their structures, which determines their spacefilling ability, that is, their level of roughness. In this paper we use the techniques of Fractal Geometry to generate parameters which provide a measure of roughness. In this way we show that the French Gothic cathedrals do not only follow Euclidean geometric patterns, but also have a general nonrandom fractal pattern.
Keywords
French Gothic Fractal parameter Fractal dimension Gothic architectureIntroduction
Benoit Mandelbrot was the main developer of Fractal Geometry in the late 1970s. His theories have evolved and have been used in several fields including architecture. Inspired by Mandelbrot’s work, Bechhoefer and Bovill used the concept of fractal dimension in architectural drawings (Bechhoefer and Bovill 1994; Bovill 1996). As a result of this work, many authors (see Ostwald et al. 2008; Vaughan and Ostwald 2009, 2010, 2011; Ostwald and Vaughan 2009, 2010) used similar techniques to analyze the design of certain architects such as Le Corbusier, Frank Lloyd Wright, Peter Eisenman and Eileen Gray. Specifically, those authors looked for a relation between the structure of the architects’ constructions and the natural or artificial environment where those constructions were projected. These techniques have also been used in other kinds of architectural studies (see Bovill 1996; Eglash 1999; Batty and Longley 2001; BurkleElizondo 2001; Ostwald 2001, 2010; Crompton 2002; Brown and Witschey 2003; Cooper 2003; Sala 2006; Hammer 2006; Joye 2007; Rian et al. 2007; Bovill 2008; Ostwald and Vaughan 2013). These works are based on a geometrical concept which is called fractal dimension. In this paper we will use this concept in order to generate what we will call fractal parameter; and by means of this parameter we will study the existence of a yet unknown pattern in French Gothic cathedrals. This generated fractal parameter will give a measure of the unevenness of their structures, which in turn determines their spacefilling ability, which is also a measure of their level of roughness.
Historical Setting
Gothic cathedrals are widely accepted as being among the most important art creations of mankind. They are the result of religious interests, aesthetic patterns and social influences, and their making brought together great scientific knowhow (Wilson 1990; Baldellou 1995; Toman 1998; Prache 2000; Schütz 2008). The style of architecture we now call Gothic first emerged between the 12th and 15th centuries of the medieval period. It emphasized structural lightness and illumination of the inside naves. It arises in contrast to the massiveness and the inadequate interior illumination of Romanic churches. It evolved mainly within ecclesiastical architecture, especially cathedrals.
Gothic art was born in northern France, in a region called “Île de France” to be precise. Historically, this style was marked by the alliance between the French monarchy and the Catholic Church. The first trial of Gothic architecture took place in Saint Denis under the patronage of abbot Suger, friend and confident of Louis VI. Following the example of Saint Denis, several primeval Gothic buildings were built in the second half of the 12th century. In Laon cathedral (1156–1160) and Notre Dame cathedral (1163), the central nave was raised and the light became the dominant element. It was also then that construction of Chartres cathedral began. There, the architect abandoned entirely the use of the tribune gallery and introduced the use of simple ribbed vaults. From the 13th century onward, after these first trials, the Gothic style entered its classical stage. The best examples are the Reims cathedral (1211) and the Amiens cathedral (1220). Both have a crossshaped floor plan and their elements were combined in pursuit of illumination and structural lightness and regularity. The classical Gothic style was adapted in France into numerous regional ramifications.
Significant Sample and Geometrical Patterns
This sample of cathedrals is considered to be significant in stylistic, chronological and geographical terms. We intend to analyse the constructions geometrically using new parameters which were unknown until now. Since each construction had its very own authorship and circumstances, we have not gone on to compare cathedrals with each other. Instead, we have examined if the design traits and structures of these constructions have a deeper geometrical sense than is known to us yet.
The aim of our investigation is to analyse the geometry of the French Gothic cathedrals in order to show the existence of a general fractal pattern. We are not comparing the constructive processes of the cathedrals nor the will of the architects to implement their designs. Instead, we will demonstrate that the geometry of their compositions shows a common formal pattern.
Fractal Parameter and Method
The first step of our investigation was to collect as many graphic documents as possible from all the cathedrals being studied. The main information sources were: the respective archdioceses of the cathedrals (we thank them for providing us with specially clear and precise graphic papers); JeanCharles Forgeret, in charge of the Médiathèque de l’Architecture et du Patrimoine, who provided us with original historical drawings for the different parts of each cathedral; Martine Mauvieux, in charge of the Département des Estampes et de la Photographie de la Bibliothèque Nationale de France, who offered a large number of photographs and drawings which we used for redrawing; the Archivo del Colegio de Arquitectos de Catalunya; and the Biblioteca de la Escuela Técnica Superior de Arquitectura de Barcelona, where we found some very illustrative monographs (see Murray 1996; Kunst and Schekluhn 1996) which let us complete the study.
So, in order to establish the fractal parameter of the French Gothic architecture, we leave ornaments aside and we take three basic projections of the cathedral’s structure: floor plan, main elevation and crosssection. Working on these projections, we make the corresponding mathematical calculations. Figure 4 shows the detail level of the redrawing. Figures 7, 8 and 9 show the redrawing made for these basic projections of the 20 cathedrals being studied (redrawing is in a small scale to keep the paper reasonably short).
We have strictly followed precise drawing lines, highlighting the lines which best represent the geometry of the floor plan, main elevation and crosssection. As we have stated before, this redrawing is absolutely necessary, since the documents collected consist of drawings or photographs with shadows, stains, colors, defects, freehand lines, etc.; i.e. all graphic documents show “noise”, so we had to do again each and every one of the drawings which appear in this paper.
Summary of the Fractal Parameter’s Generation Process
Architectural structures, from now on called M, are not fractal objects. Despite that, we can consider M’s unevenness and its spacefilling ability, that is, its level of roughness. Also, we can generate a parameter for these nonfractal objects, the fractal parameter P _{ s }(M). The value provided by this fractal parameter P _{ s }(M) is a measure of the level of roughness of the architectural structure being studied M.
 1.
Given the architectural structure M, first we generate its design in AutoCad vector format, using black color and line width ~0.00 mm. From this AutoCad format we obtain the pdf vector format.
 2.
From the pdf vector format we generate a blackandwhite digital bitmap file, sized 1,024 × υ pixels, showing the architectural structure with its size adjusted to full width and height.
 3.
P _{ s }(M)—using a selfcreated software, we calculate the fractal parameter P _{ s }(M) based on the slope on the last point of a continuous graph ln–ln. In the following sections we explain which continuous graph we are talking about and which calculations are made.
The value P _{ s }(M) is a fractal parameter of the structure M, which gives a measure of its level of roughness.
The reason to create for ourselves a special software is twofold: firstly, we will have total control of the calculations and so we will ensure these calculations are correct. Secondly, commercial software like Benoit does not use the slope on the last point of the continuous graph ln–ln. Benoit uses the slope of the regression line corresponding to the discrete set of points in the graph ln–ln.
However, the slope of the regression line is only similar to the fractal dimension when M is a fractal selfsimilar object, because then its continuous graph ln–ln is a straight line. But an architectural structure M is not a fractal object nor a selfsimilar fractal object, so its continuous graph ln–ln is not a straight line (later on we will define a fractal object and a fractal selfsimilar object). M does not possess fractal geometry but it does have a fractal parameter P _{ s }(M); we will generate this parameter P _{ s }(M) with a process which has been extrapolated from the theoretical limit \( \bar{F}(M) \), called upper fractal dimension of M, M being a non selfsimilar fractal object (this process will be explained later).

3. P _{ r }(M)—using either a selfcreated software or the commercial software Benoit, we calculate the fractal parameter P _{ r }(M) based on the slope of the regression line corresponding to the discrete set of points in the graph ln–ln; and for this calculation we use square meshes, the finest mesh having 4 × 4 pixel squares, and the coarsest mesh having 32 × 32 pixel squares.
The reason for using square meshes with these particular limits of fineness and coarseness will be explained later.

3. P _{ b }(M)—using the commercial software Benoit we calculate the fractal parameter P _{ b }(M) based on the regression line corresponding to the discrete set of points in the graph ln–ln, and for this calculation we use square meshes, the finest mesh having 1 × 1 pixel squares and the coarsest mesh having 256 × 256 pixel squares.
To sum up, we have created software to obtain the value P _{ s }(M) because the structures M being considered are not selfsimilar objects, but we also calculate the values P _{ r }(M) and P _{ b }(M). These three measures are not equal because the calculation methods are different. However, all three give a measure of the roughness of the structure.
In this paper we calculate and use the three parameters P _{ s }(M), P _{ r }(M) and P _{ b }(M) in order to show, beyond any doubt, the existence of a fractal pattern in Gothic structures. At the end of this paper we will see the results and relationships between the three measures.
Hausdorff–Besicovitch Dimension
Readers may skip this very technical subsection and still fully understand the work which is presented in this paper. We provide book references to interest readers. In this section we summarize the dimensions \( T(M), {\mathcal{H}}(M),\bar{F}(M), F(M), H(M), S(M) \) of the fractal objects M; and some of their properties. This is the theoretical basis from which the ideas for generating the fractal parameter P _{ s }(M) arise.
Let M be a bounded nonempty subset of the ndimensional Euclidean affine space, \( {\mathbb{A}}^{n} \). The definition of the sdimensional Hausdorff measure of M, \( {\mathcal{H}}^{s} (M) \), can be found in (Falconer 1990, 1997; Edgar 1998) or in other books about fractals. There is also a theorem which states as follows: there is a critical value \( {\mathcal{H}}(M) \) such that: \( {\mathcal{H}}^{s} (M) = \infty \) if 0 ≤ s < \( {\mathcal{H}}(M) \), and \( {\mathcal{H}}^{s} (M) = 0 \) if s > \( {\mathcal{H}}(M) \). If s = \( {\mathcal{H}}(M) \), then \( {\mathcal{H}}^{s} (M) \in [0,\infty ] \). This critical value \( {\mathcal{H}}(M) \) is called HausdoffBesicovitch dimension of M. Moreover, M has its topological dimension, T (M). We have the following geometric definition: M is a fractal object if \( {\mathcal{H}}(M) \) > T (M).
The Hausdorff–Besicovitch dimension has an upper bound \( \bar{F}(M) > H(M) \). This upper bound \( \bar{F}(M) \) is called MinkowskiBouligand dimension of M \( \subset {\mathbb{A}}^{n} \), and we will call it upper fractal dimension of M. The object M can be fractal or not, but in any case the upper fractal dimension \( \bar{F}(M) \) is a measure of the level of roughness of M; i.e. it is a measure of the unevenness of the structures, which determines their spacefilling ability.
In the abovementioned books, the reader can see that the definition of \( \bar{F}(M) \) is as follows: let N _{ e }(M) be the smallest number of sets of diameter at most ɛ > 0 which can cover M. The upper fractal dimension of M is defined as \( \bar{F}\left( M \right) = \mathop {\lim \sup }\nolimits_{\varepsilon \to 0} \frac{{In\left( {N_{e} \left( M \right)} \right)}}{  In\left( \varepsilon \right)} \)
If \( \bar{F}\left( M \right) = \mathop {\lim \sup }\nolimits_{\varepsilon \to 0} \frac{{In\left( {N_{e} \left( M \right)} \right)}}{{  In\left( {1/\varepsilon } \right)}} \) is equal to \( \mathop {\lim \inf }\nolimits_{\varepsilon \to 0} \frac{{In\left( {N_{e} \left( M \right)} \right)}}{{In\left( {1/\varepsilon } \right)}} \), then the limit \( \mathop {\lim }\nolimits_{\varepsilon \to 0} \frac{{In\left( {N_{e} \left( M \right)} \right)}}{{In\left( {1/\varepsilon } \right)}} = F(M) \) exists and \( \bar{F}(M) \) = F(M). This limit F(M), if it exists, is called the fractal dimension of M, (box counting fractal dimension of M).
Let (M, N) be a pair of bounded nonempty subsets of \( {\mathbb{A}}^{n} \), such that: \( M = h_{r} (N) \), where h _{ r } is a homothety of ratio r, and \( M = \coprod\nolimits_{i = 1}^{i  m} {g_{i} } (N) \), a disjoint union, where g _{ i } is a displacement. Then M is called a homothetic object, and its homothetic dimension is \( H(M) = \log_{r} (n) \), i.e. \( r^{H(M)} = n \). If M is a homothetic object and H(M) ≠ T(M), then M is a fractal object. A homothetic object is a particular case of the following objects called selfsimilar objects.
Let M be a bounded nonempty subset of \( {\mathbb{A}}^{n} \), such that: \( M = \coprod\nolimits_{i = 1}^{i  m} {S_{i} } (N) \) where S _{ i } is a contractive similarity; i.e. \( S_{i} : {\mathbb{A}}^{n} \to {\mathbb{A}}^{n} \) such that \( \forall (x,y) \in {\mathbb{A}}^{n} \times {\mathbb{A}}^{n} \Rightarrow d(S_{i} (x),S_{i} (y)) = k_{i} d(x,y) \) with 0 < k _{ i } < 1. Then M is called a selfsimilar object, and its selfsimilarity dimension is the value S(M) such that \( \sum\nolimits_{i = 1}^{i = m} {k_{i}^{S(M)} } = 1 \). If M is a homothetic object, then S(M) = H(M). If M is a selfsimilar object and S(M) ≠ T(M), then M is a fractal object.
 1.
\( T(M) \le {\mathcal{H}}(M) \le \bar{F}\left( M \right) \le n \)
 2.
If M is a selfsimilar object, then \( {\mathcal{H}}(M) = F(M) < S(M) \).
 3.
If \( M = \coprod\nolimits_{i = 1}^{i  m} {S_{i} } (M) \) is a selfsimilar object and there exists a nonempty bounded open set \( V \subset {\mathbb{A}}^{n} \) such that V ⊃ ∐ _{ i=1} ^{ i−m } S _{ i }(V), then \( {\mathcal{H}}(M) = F(M) = S(M) \).
Fractal Parameter and Calculation Method
After the theoretical base of the upper fractal dimension \( \bar{F}\left( M \right) \) and the properties mentioned in the former subsection, now we will explain how to generate the fractal parameter P _{ s }(M).
Architectural structures M are not fractal, however we can consider their unevenness, which determines their spacefilling ability (i.e. their level of roughness), and we can generate a parameter for those nonfractal objects. This parameter, which we will call fractal parameter P _{ s }(M), provides a measure of M’s roughness.
Given the architectural structure M, first we generate its design in AutoCad vector format using black color and line width ~0.00 mm, with homogeneous graphic display criteria. From this AutoCad format we obtain the pdf vector format. Figure 4 shows the detail level of the structure.
From the pdf vector format, by means of an image processing software, we generate a blackandwhite digital bitmap file N, sized 1,024 × υ pixels (standard resolution 1,024 horizontal, υ vertical). The size of the image in N is adjusted to full width and height.
We have developed our own software in order to have total control of the calculation and guarantee correctness. Besides, as we said in the first subsection, using this selfcreated software we generate the fractal parameter P _{ s }(M) based on the definition of the upper fractal dimension \( \bar{F}\left( M \right) \). The process used is not the slope of a regression line, but the slope of the last point of a continuous graph ln–ln, as we shall now see.
Since N is a pixelated digital image file, the calculation process to generate P _{ s }(M) will have a finite number of steps. The finest mesh used to generate P _{ s }(M) is a 4 × 4 pixel square mesh, because \( 4 = 2^{n} \) gives the finest mesh which is similar to the theoretical meshes of the theorem mentioned in the previous subsection for the theoretical calculation of \( \bar{F}\left( M \right) \). This is true because the 4 × 4 pixel squares have inside points and border points. Therefore, in order to calculate P _{ s }(M) we will use four meshes the squares of which are 4, 8, 16 and 32 pixels in length, respectively. The reason to use four meshes is that, as we will see later, P _{ s }(M) is generated with the slope of a function of a continuous graph ln–ln. Using the classical interpolation methods, four points of a function are enough to find a good approximation of that slope. And we should not use more than four interpolation points because of the wellknown Runge Phenomenon in numerical calculation.
Data corresponding to Fig. 5
g _{ n }  a _{ n }  h _{ n }  s _{ n }  ln (2^{ n })  ln(s _{ n }) 

g _{5}  32  32  799  ln (2^{ 5 })  ln(799) 
g _{6}  16  64  2,689  ln (2^{ 6 })  ln(2689) 
g _{7}  8  128  8,236  ln (2^{7})  ln(8236) 
g _{8}  4  256  22,294  ln (2^{8})  ln(22294) 
As a result of the above mentioned process we obtain the coordinates of four points \( ({ \ln }(2^{5} ), { \ln }(s_{5} )), ({ \ln }(2^{6} ), { \ln }(s_{6} )), ({ \ln }(2^{7} ), ({ \ln }(s_{7} )), ({ \ln }(2^{8} ),{ \ln }(s_{8} )) \) in a graph ln–ln.
Variants of the Fractal Parameter
We have explained that the fractal parameter P _{ s }(M) is generated by means of the slope in the fourth point of the continuous graph ln–ln. However, in the theoretical cases of fractal selfsimilar objects such a graph is a straight line. Therefore, if we generate a fractal parameter under the hypothesis of selfsimilarity, then we can use the slope of the regression line corresponding to the discrete set of the four points belonging the graph \( ln  ln \). So, the calculation is the quotient of the covariances \( \frac{{\sigma_{xy} }}{\sigma xx} \) where \( \sigma_{xy} = \frac{1}{4}\sum\nolimits_{i = 0}^{i = 4} {(y_{i}  \bar{y})((5 + i)\ln 2  \bar{x}), \sigma xx = \frac{1}{4}} \sum\nolimits_{i = 0}^{i = 4} {((5 + i)\ln 2  \bar{x})^{2} ,\bar{x} = \frac{5 + 6 + 7 + 8}{4}\ln 2, \bar{y} = \frac{{y_{0} + y_{1} + y_{2} y_{3} }}{4}} \). This fractal parameter will be called P _{ r }(M). In the case of the rose window displayed in Fig. 5, we have P _{ r }(M) ≃ 1.60.
Some commercial software like Benoit 1.31, by TruSoft Int’l Inc, allow generation of P _{ r }(M) from the pixel file N. In this situation, of course, the value \( P_{r} (M) \) obtained with our software is the same as the value obtained with Benoit.
When many calculation steps are used, with more meshes, commercial software always uses the slope of the regression line. For instance, Benoit 1.31 uses 22 meshes as default variables: the edge of the widest square mesh is 256 pixels long; each of the following meshes has an edge length equal to the preceding edge length divided by 1.3. The new parameter thus generated will be called P _{ b }(M). Taking the example of Fig. 5, we have P _{ b }(M) ≃ 1.41.
As we have commented in “Summary of the Fractal Parameter’s Generation Process”, if M is not a fractal object nor a selfsimilar object, then the parameter P _{ s }(M) is more approximated to the theoretical calculation of \( \bar{F}\left( M \right) \) than P _{ r }(M) and P _{ b }(M). Therefore P _{ s }(M) is lower than P _{ r }(M) and P _{ b }(M) because \( \bar{F}\left( M \right) \) goes to the minimum value 1.
Redrawing and Calculation
Vectors obtained from the calculations
(s_{5}, s_{6}, s_{7}, s_{8})  Plan  Elevation  Section 

Strasbourg  (955, 2957, 8256, 23092)  (1532, 5414, 17073, 44325)  (1323, 4386, 12034, 29699) 
Troyes  (1530, 4555, 13043, 35786)  (1054, 3465, 10033, 26254)  (1374, 4405, 11715, 28619) 
Sens  (1476, 4398, 12486, 35766)  (1166, 3634, 9919, 24561)  (758, 2524, 7675, 21131) 
Noyon  (1116, 3430, 9792, 26615)  (1315, 3953, 11315, 29582)  (1391, 4261, 12076, 28501) 
Senlis  (1535, 4673, 13513, 37928)  (1500, 4534, 11833, 28228)  (1159, 3760, 11092, 27027) 
Laon  (1121, 3726, 10948, 30914)  (1384, 4716, 13904, 35030)  (1033, 3253, 8551, 18751) 
Paris  (2083, 6427, 17992, 50208)  (1151, 3805, 12038, 32454)  (1241, 3673, 9682, 23367) 
Lisieux  (1217, 3458, 9095, 23534)  (1310, 3844, 10530, 24592)  (1214, 3524, 8307, 18041) 
Tours  (1394, 4280, 11882, 31996)  (1848, 6237, 18417, 49724)  (1684, 5272, 14664, 38914) 
Soissons  (1144, 3312, 8795, 23510)  (1094, 3613, 10772, 27100)  (1489, 4172, 10338, 23298) 
Chartres  (1128, 3654, 10658, 28952)  (1237, 391, 10939, 28075)  (1571, 4888, 12944, 30644) 
Bourges  (1084, 3410, 9666, 26764)  (1256, 4165, 11905, 29997)  (1097, 3544, 11158, 31973) 
Rouen  (1335, 4044, 10705, 28928)  (964, 3324, 10391, 26682)  (2126, 6453, 17031, 42164) 
Reims  (1720, 5023, 14585, 41123)  (1374, 4592, 14584, 40203)  (1579, 4911, 13876, 37304) 
Auxerre  (1551, 4582, 12858, 37109)  (1080, 3508, 10260, 25428)  (1746, 5736, 15140, 37738) 
Amiens  (1449, 4292, 11624, 30418)  (1128, 3866, 11656, 29874)  (1788, 6127, 18364, 50930) 
Metz  (1493, 4509, 11640, 30187)  (1520, 4752, 13326, 33006)  (1405, 4767, 13897, 37464) 
Orléans  (1561, 4891, 13181, 35273)  (1199, 3868, 11726, 29260)  (1778, 5168, 12624, 30641) 
Toul  (1075, 3065, 8553, 23887)  (1524, 5256, 15170, 39088)  (1211, 3510, 9973, 25873) 
Sées  (1162, 3544, 9875, 26931)  (1575, 5060, 15021, 41740)  (1256, 3837, 10523, 25676) 
Parameters P _{ s } obtained from the calculations
P _{ s }  Plan  Elevation  Section 

Strasbourg  1.54  1.20  1.27 
Troyes  1.42  1.33  1.28 
Sens  1.55  1.25  1.39 
Noyon  1.42  1.30  1.06 
Senlis  1.48  1.22  1.10 
Laon  1.51  1.21  1.00 
Paris  1.52  1.26  1.22 
Lisieux  1.39  1.06  1.12 
Tours  1.44  1.39  1.41 
Soissons  1.47  1.18  1.12 
Chartres  1.41  1.32  1.19 
Bourges  1.49  1.25  1.42 
Rouen  1.52  1.17  1.30 
Reims  1.46  1.32  1.41 
Auxerre  1.59  1.16  1.35 
Amiens  1.39  1.22  1.44 
Metz  1.46  1.21  1.41 
Orléans  1.48  1.12  1.36 
Toul  1.49  1.31  1.28 
Sées  1.47  1.43  1.20 
Parameters P _{ r } obtained from the calculations
P _{ r }  Plan  Elevation  Section 

Strasbourg  1.53  1.62  1.49 
Troyes  1.52  1.54  1.46 
Sens  1.53  1.46  1.60 
Noyon  1.52  1.50  1.46 
Senlis  1.54  1.41  1.52 
Laon  1.59  1.55  1.39 
Paris  1.53  1.61  1.41 
Lisieux  1.42  1.41  1.29 
Tours  1.50  1.58  1.51 
Soissons  1.45  1.55  1.32 
Chartres  1.56  1.50  1.43 
Bourges  1.54  1.52  1.63 
Rouen  1.47  1.60  1.43 
Reims  1.53  1.63  1.52 
Auxerre  1.52  1.52  1.47 
Amiens  1.46  1.58  1.61 
Metz  1.44  1.48  1.58 
Orléans  1.49  1.54  1.36 
Toul  1.49  1.56  1.48 
Sées  1.51  1.58  1.45 
Parameters P _{ b } obtained from the calculations
P _{ b }  Plan  Elevation  Section 

Strasbourg  1.60624  1.57518  1.49741 
Troyes  1.60898  1.51084  1.44870 
Sens  1.61128  1.48047  1.52771 
Noyon  1.59964  1.49272  1.74688 
Senlis  1.62527  1.44926  1.48583 
Laon  1.63119  1.52501  1.41866 
Paris  1.63493  1.56197  1.40060 
Lisieux  1.53833  1.41831  1.36375 
Tours  1.59779  1.56567  1.49519 
Soissons  1.55476  1.46949  1.34814 
Chartres  1.59455  1.48041  1.45659 
Bourges  1.62432  1.52556  1.56154 
Rouen  1.57932  1.55053  1.51279 
Reims  1.61244  1.59039  1.50352 
Auxerre  1.61642  1.49271  1.47486 
Amiens  1.56189  1.54350  1.56477 
Metz  1.54586  1.49182  1.51680 
Orléans  1.58828  1.50319  1.42449 
Toul  1.58842  1.54721  1.44217 
Sées  1.59304  1.53743  1.44756 
Results and Discussion
Fractal geometry has often been used to study certain elements of architecture, but this is the first time that it has been used to find a new geometric pattern in the French Gothic architecture.
Means obtained from the calculations
m _{ i }  P _{ s }  P _{ r }  P _{ b } 

Strasbourg  1.33  1.55  1.56 
Troyes  1.34  1.51  1.52 
Sens  1.40  1.53  1.54 
Noyon  1.26  1.49  1.61 
Senlis  1.27  1.49  1.52 
Laon  1.24  1.51  1.52 
Paris  1.33  1.52  1.53 
Lisieux  1.19  1.38  1.44 
Tours  1.41  1.53  1.55 
Soissons  1.25  1.44  1.46 
Chartres  1.30  1.49  1.51 
Bourges  1.39  1.56  1.57 
Rouen  1.33  1.50  1.55 
Reims  1.40  1.56  1.57 
Auxerre  1.37  1.51  1.53 
Amiens  1.35  1.55  1.56 
Metz  1.36  1.50  1.52 
Orléans  1.32  1.47  1.51 
Toul  1.36  1.51  1.53 
Sées  1.37  1.51  1.53 
We will show that the fractal parameters P _{ r } and P _{ b } in Table 6 have a very strong linear correlation among them, so we can consider the fractal parameters P _{ s } and P _{ r } only. From Table 6 we can see that the total mean m _{ r } of the data from P _{ r } is m _{ r } ≃ 1.50 and the total mean m _{ b } of the data from P _{ b } is \( m_{b} \simeq 1.53 \). The standard deviation of the 20 results from Table 6 is σ _{ r } ≃ 0.041 for P _{ r }, and σ _{ b } ≃ 0.037 for P _{ b }. The covariance between the results for P _{ r } and the results for P _{ b } is σ _{ rb } ≃ 0.0012. Therefore, the Pearson correlation coefficient between the two sets of results is \( R_{rb} = \frac{{\sigma_{rb} }}{{\sigma_{r} \sigma_{b} }} \simeq 0.80 \), and the Pearson determination coefficient between the two sets of results is R _{ rb } ^{2} ≃ 0.63. Using Student’s t test with 20–2 degrees of freedom and eight decimal digits (we use eight decimal digits because we want to show the real magnitude of the probability) we reject the null hypothesis of “no correlation between the two sets of results” with a probability P _{ rb } ≃ 0.99998620. Therefore, in order to study the fractal pattern we will only use the data sets P _{ r } and P _{ s } in Table 6.
Now we will prove that the 20 means m _{ si } from Table 6, corresponding to P _{ s }, are very concentrated around their total mean m _{ s } ≃ 1.33. To that effect, we calculate the standard deviation of these 20 results, which is σ _{ s } ≃ 0.058, and the variance σ _{ s } ^{2} 0.0034. Therefore, Pearson’s coefficient of variation \( \frac{\sigma }{m} \) is 4 %. In general, when the Pearson coefficient of variation is under 25 % it is considered that there is little scattering around the mean. Since the coefficient in this case is 4 %, we conclude that the total mean m _{ s } is highly representative and shows very little scattering.
Twoway table of m _{ s }
I _{1}  I _{2}  I _{3}  I _{4}  I _{5}  I _{6}  I _{7}  I _{8}  I _{9}  I _{10}  \( { \ldots } \)  I _{20}  Total  

m _{ si } ∊ I _{ k }  0  0  0  1  2  3  6  7  1  0  \( { \ldots } \)  0  20 
m _{ si } ∉ I _{ k }  20  20  20  19  18  17  14  13  19  20  \( { \ldots } \)  20  380 
Total  20  20  20  20  20  20  20  20  20  20  \( { \ldots } \)  20  400 
Now we repeat the study with the 20 means \( m_{ri} \) from Table 6.
We will prove that the means \( m_{ri} \) from Table 6 corresponding to P _{ r } are very concentrated around their total mean \( m_{r} \simeq 1.50 \). To that effect, we calculate the typical deviation of these 20 results, which is \( \sigma_{r} = 0.041 \), and the variance σ ^{2} ≃ 0.0017. Therefore Pearson’s coefficient of variation \( \frac{\sigma }{m} \) is 3 %. According to this, we conclude that the total mean is highly representative and shows very little scattering.
Twoway table of m _{ r }
I _{1}  \( { \ldots } \)  I _{7}  I _{8}  I _{9}  I _{10}  I _{11}  I _{12}  I _{13}  \( { \ldots } \)  I _{20}  Total  

m _{ ri } ∊ I _{ k }  0  \( { \ldots } \)  0  1  1  6  10  2  0  \( { \ldots } \)  0  20 
m _{ ri } ∉ I _{ k }  20  \( { \ldots } \)  20  19  19  14  10  18  20  \( { \ldots } \)  20  380 
Total  20  \( { \ldots } \)  20  20  20  20  20  20  20  \( { \ldots } \)  20  400 
Consequently, applying Pearson’s Chi squared test with 19 degrees of freedom we conclude that Table 8 is a nonrandom table, because the probability of being nonrandom is P ≃ 0.99999999.
So, all things considered, the fractal parameters given by the means m _{ ri }, m _{ si } and m _{ bi } have been obtained using a mathematical mechanism and objective results. In addition, these means are highly concentrated around their total means, m _{ s }, m _{ r } and m _{ b }, respectively; and the probability of them being random results is negligible.
Therefore, we claim that the existence of a fractal pattern (which can be measured by any of the three means m _{ s }, m _{ r } or m _{ b }) is a general, nonrandom property of the French Gothic cathedrals.
Conclusion
The classic patterns of the Euclidean Geometry were used in the construction of the Gothic cathedrals to provide them with proportion and beauty. Still, as well as the Euclidean elements, there is another complex concept in them: the unevenness of their structures, which determines their spacefilling ability, i.e. their level of roughness. The best tool to describe this concept is given by Fractal Geometry, by means of a ratio called fractal dimension. In this paper we use this concept to generate fractal parameters which give a measure of roughness. These parameters are not only attributable to Euclidean elements, but also given by the final architectural result of the constructions.
The 20 cathedrals being studied in this paper are a highly representative sample of the French Gothic style, one which prevents arbitrariness. By means of this sample, we analyze the geometry of the resulting construction using new parameters which were unknown until now. Since each construction had its very own authorship and circumstances, we have not gone on to compare cathedrals with each other. Instead, we have examined if the design traits and structures of these constructions have a deeper geometrical sense than was known to us yet.
In this paper we prove that the French Gothic cathedrals do not only follow the Euclidean geometric patterns, but also have another characteristic pattern which is determined by fractal parameters.
We conclude the existence of a general nonrandom pattern within the fractal dimension of the French Gothic cathedrals.
References
 Baldellou, M. 1995. Catedrales De Europa. Madrid: Espasa Calpe. (in Spanish).Google Scholar
 Batty, M., and P. Longley. 2001. The fractal city. In AD, Urban Environments, ed. E.G. Mapelli. New York: Wiley.Google Scholar
 Bechhoefer, W., and C. Bovill. 1994. Fractal analysis of traditional housing in Amasya, Turkey. Traditional Dwellings and Settlements Working Paper Series 61:1–21.Google Scholar
 Bechmann, R. 1985. Villard de Honnecourt. La Pensée Technique au XIIie Siècle et sa Communication. Paris: Picard Éditeur. (in French).Google Scholar
 Bovill, C. 1996. Fractal geometry in architecture and design. Boston: Birkhäuser.CrossRefzbMATHGoogle Scholar
 Bovill, C. 2008. The doric order as a fractal. Nexus Network Journal 10:283–290.CrossRefGoogle Scholar
 Brown, C.T., and W.R.T. Witschey. 2003. The fractal geometry of ancient Maya settlement. Journal of Archaeological Science 30:1619–1632.CrossRefGoogle Scholar
 BurkleElizondo, G. 2001. Fractal geometry in Mesoamerica. Symmetry. Culture and Science 12:201–214.Google Scholar
 Cooper, J. 2003. Fractal assessment of streetlevel skylines: a possible means of assessing and comparing character. Urban Morphology 7:73–82.Google Scholar
 Crompton, A. 2002. Fractals and picturesque composition. Environment Planning B 29:451–459.CrossRefGoogle Scholar
 Edgar, G. 1998. Integral, probability, and fractal measures. New York: Springer.CrossRefzbMATHGoogle Scholar
 Eglash, R. 1999. African fractals: modern computing and indigenous design. New Brunswick: Rutgers University Press.zbMATHGoogle Scholar
 Falconer, K.J. 1990. Fractal geometry, mathematical foundations and applications. Chichester: Willey.zbMATHGoogle Scholar
 Falconer, K.J. 1997. Techniques in fractal geometry. Chichester: Willey.zbMATHGoogle Scholar
 Hammer, J. 2006. From fractal geometry to fractured architecture: the Federation Square of Melbourne. Mathematical Intelligencer 28:44–48.CrossRefMathSciNetGoogle Scholar
 Joye, Y. 2007. Fractal architecture could be good for you. Nexus Network Journal 9:311–320.CrossRefzbMATHGoogle Scholar
 Kunst, H., and W. Schekluhn. 1996. La Catedral de Reims. Madrid: Siglo Veintiuno. (in Spanish).Google Scholar
 Murray, S. 1996. NotreDame, Cathedral of Amiens, the power of change in Gothic. Cambridge: University of Cambridge.Google Scholar
 Ostwald, M.J. 2001. Fractal architecture: late 20th century connections between architecture and fractal geometry. Nexus Network Journal 3:73–84.CrossRefGoogle Scholar
 Ostwald, M.J. 2010. The politics of fractal geometry in Russian paper architecture: the intelligent market and the cube of infinity. Architecture Theory Review 15:125–137.CrossRefGoogle Scholar
 Ostwald, M.J., and J. Vaughan. 2009. Calculating visual complexity in Peter Eisenman’s architecture: a computational fractal analysis of five houses (1968–1976). CAADRIA 2009: Proceedings of the Fourteenth Conference on Computer Aided Architectural Design Research in Asia, 75–84.Google Scholar
 Ostwald, M.J., and J. Vaughan. 2010. Comparing Eisenman’s House VI and Hejduk’s House 7: a mathematical analysis of formal complexity in plan and elevation. Aesthetics and design: 21st Biennial Congress of IAEA. Dresden: Technische Universitat.Google Scholar
 Ostwald, M.J., and J. Vaughan. 2013. Representing architecture for fractal analysis: a framework for identifying significant lines. Architectural Science Review 56(3):242–251.Google Scholar
 Ostwald, M.J., Vaughan, J., and S. Chalup. 2008. A computational analysis of fractal dimensions in the architecture of Eileen Gray. Biological processes and computation: Proceedings of the 28th Annual Conference of the Association for Computer Aided Design in Architecture (ACADIA), 256–263.Google Scholar
 Prache, A. 2000. Cathedrals of Europe. Cornell: University Press.Google Scholar
 Rian, I.M., J.H. Park, H.U. Ahn, and D. Chang. 2007. Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho. Building and Environment 42: 4093–4107.CrossRefGoogle Scholar
 Sala, N. 2006. Fractal geometry and architecture: some interesting connections. WIT Transactions on The Built Environment 86:163–173.Google Scholar
 Schütz, B. 2008. L’art des Grandes Cathédrales. Paris: Hazan. (in French).Google Scholar
 Simon, O. 1985. La Catedral gótica. Madrid: Alianza Editorial. (in Spanish).Google Scholar
 Toman, R. 1998. El Gótico. Arquitectura, Escultura y Pintura. Paris: Konemann. (in Spanish).Google Scholar
 Vaughan, J., and M.J. Ostwald. 2009. A quantitative comparison between the formal complexity of Le Corbusier’s PreModern (1905–1912) and Early Modern (1922–1928) architecture. Design Principles and Practices: An International Journal 3:359–372.Google Scholar
 Vaughan, J., and M.J. Ostwald. 2010. Using fractal analysis to compare the characteristic complexity of nature and architecture: reexamining the evidence. Architectural Science Review 53:323–332.CrossRefGoogle Scholar
 Vaughan, J., and M.J. Ostwald. 2011. The relationship between the fractal dimension of plans and elevations in the architecture of Frank Lloyd Wright: comparing the Prairie Style, Textile Block and Usonian Periods. Architecture Science ArS 4:21–44.Google Scholar
 Wilson, C. 1990. The Gothic Cathedral, the architecture of the Great Church. London: Thames and Hudson.Google Scholar