Fractal components in the Gothic and

in the Baroque Architecture

Professor Nicoletta Sala, Ph-D

Accademia di Architettura (Academy of Architecture) di Mendrisio.

Università della Svizzera Italiana (University of Lugano).

Largo Bernasconi, 6850 Mendrisio. Svizzera. E-mail:

Abstract

Fractal geometry describes the irregular shapes and it can occur in many different places, for example in Mathematics and in Nature. The aim of this paper is to present an overview which involves the fractal geometry, the property of self-similarity and the Iterated Function System (IFS) applied in two different architectural styles: the Gothic and the Baroque.

1. Introduction

In the different centuries the architecture followed the Euclidean geometry and the Euclidean shapes (for example,  to realize the boards and the bricks). Thus, it  is not a  surprise that the buildings had Euclidean aspects. The geometric properties of the symmetry applied to the buildings and to the temples helped to realize the engineering calculus, and to obtain the structural stability. Some architectural styles, for example the Baroque, found inspiration by the Nature, and the Nature is manifestly irregular and fractal-like. So perhaps there is not difficult to find fractal components in architecture [1, 2, 3, 4, 5]. As this paper will describe, fractals appear in architecture to reproduce some shapes and some patterns present in the Nature.

This  fractal analysis has been divided in two parts:

·        a small scale analysis, for example to determine the  single building shape [4].

·        a large scale analysis, for example to study the urban growth and the urban development using a fractal point of view [6, 7, 8].

The small scale analysis observed:

·        the building's self-similarity, for example to find some  self-similar components in the Baroque churches which repeat  themselves in different scales,

·        the Iterative Function System (IFS), for example to determine some iterative fractal processes present in the Gothic style.

The paper is organized as follow: section 2 presents a short description on fractal geometry, in particular the self-similarity and the IFS. Section 3 and section 4 show the fractal components in the Gothic and in the Baroque architecture respectively. Section 5 contains the conclusions, and the section 6 is dedicated to the references.

2. Fractal Geometry: the self-similarity and the IFS

A fractal object is self – similar if it has undergone a transformation whereby the dimensions of the structure were all modified by the same scaling factor. The new shape may be smaller, larger, translated, and/or rotated, but its shape remains similar [9]. The self-similarity is a property by which an object contains smaller copies of itself at arbitrary scales. “Similar” means that the relative proportions of the shapes’ sides and internal angles remain the same. As described by Mandelbrot (1988), this property is ubiquitous in the natural world [9]. Oppenheimer (1986) used the term “fractal” exchanging it with self-similarity, and affirmed: “The geometric notion of self-similarity became a paradigm for structure in the natural world. Nowhere is this principle more evident than in the world of botany.” [10]

Self-similarity appears in objects as diverse as leaves, mountain ranges, clouds, and galaxies. Figure 1 shows a broccoli (Brassica oleracea) which is an example of self-similar vegetable in the Nature. In figure 2 there is a Sierpinski triangle that is a fractal object which presents the self-similarity. It has been created using simple geometric rules.

Figure 1. The broccoli is an example of                       Figure 2. The Sierpinski triangle is an

self-similarity in the Nature                                          object derived by a simple iterative process

Iterated Function System (IFS) is another fractal that can be applied in the architecture. Barnsley [11, p. 80] defined the Iterated Function System as follow: “A (hyperbolic) iterated function system consists of a complete metric space (X, d) together with a finite set of contraction mappings wn: X® X with respective contractivity factor sn, for n = 1, 2, …, N.  The abbreviation “IFS” is used for “iterated function system”. The notation for the IFS just announced is { X, wn, n = 1, 2, …, N} and its contractivity factor is s = max  {sn :  n = 1, 2, …, N}.”

Barnsley put the word “hyperbolic “ in parentheses because it is sometimes dropped in practice. He also defined the following theorem [11, p. 81]: “Let {X, wn, n = 1, 2, …, N} be a hyperbolic iterated function system with contractivity factor s. Then the transformation W : H(X) ® H(X) defined by:

(1)

For all BÎ H(X), is a contraction mapping on the complete metric space (H(X), h(d)) with contractivity factor s. That is:

H(W(B), W(C)) £ s×h(B,C)                                                                                      (2)

for all B, C Î H(X). Its unique fixed point, A Î H(X), obeys

(3)

and is given by A = lim n®¥ Won (B) for any B Î H(X).”

The fixed point A Î H(X),  described in the theorem by Barnsley is called the “attractor of the IFS” or “invariant set”.

Bogomolny (1998) affirms that two problems arise [12]. One is to determine the fixed point of a given IFS, and it is solved by what is known as the “deterministic algorithm”.

The second problem is the inverse of the first: for a given set AÎH(X), find an iterated function system that has A as its fixed point [12]. This is solved approximately by the Collage Theorem [11,  p. 94].

The Collage Theorem states: “Let  (X, d), be a complete metric space. Let LÎH(X) be given, and let e ³ o be given. Choose an IFS (or IFS with condensation) {X, (wn), w1, w2,…, wn} with contractivity factor 0 £ s £ 1, so that

(4)

Where h(d) is the Hausdorff metric. Then

(5)

Where A is the attractor of the IFS. Equivalently,

(6)

for all LÎH(X).”

The Collage Theorem describes how  to find an Iterated Function System whose attractor is "close to" a given set, one must endeavour to find a set of transformations such that the union, or collage, of the images of the given set under transformations is near to the given set. Next figures 3 and 4 show respectively, the first two steps to create an image of a fern using the IFS, and  the Collage Theorem applied to a region bounded by a polygonalized leaf boundary [11, p. 96].

Figure 3. The first two steps to create an image  of a      Figure 4. The Collage Theorem applied

fern [13, p. 116]                                                                    to a region bounded [11, p. 96]

The IFS are produced by polygons that are put in one another and show a high degree of similarity to nature, such as the fern presented in figure 5 [13, p.117]. The IFS form the connection between the true mathematical fractals and the Nature.

The next sections describe some applications of the self-similarity and of the IFS in the Gothic and in the Baroque architecture.

Figure 5. The first five steps to generate a fern using IFS [13, p. 117]

3. Fractal components in the Gothic Architecture

The Gothic is a style developed in northern France that spread throughout Europe between the 12th and 16th centuries. The term “Gothic” was first used during the later Renaissance by the Italian artist Giorgio Vasari (1511-1574) as a term of contempt. He wrote: "Then arose new architects who after the manner of their barbarous nations erected buildings in that style which we call Gothic".

Fulcanelli, the 20th century most enigmatic alchemist, gave another explication of the term Gothic, which is connected to the language of the alchemy[1].

Some fractal components are present in the Gothic churches; an example is shown in figure 6 which reproduces the facade of the Reims’ Cathedral (1210-1241, Reims, France). The white arrows point out the fractal components [13, p. 86]. The self-similarity  is also present inside the Gothic Cathedrals, as shown in figure 7.

Figure 6. Reims’ Cathedral (1210-1241)          Figure 7. Notre Dame (c. 1163-1250, Paris)

shows fractal components                                 shows a kind of the self-similarity

Gothic architecture can be observed using the iterative function system. The method is similar to the Wright’s approach [15]. He dissected a fern in to similar part, and he  marked some  triangles on these parts which are similar to the whole, as shown in figure 8a). An affine maps was determined by how they map a single triangle to another triangle. This allowed Wright  to convert out dissection of the fern into four affine maps. Figure 8b shows the original four parts together with a triangle corresponding to the whole fern, it is drawn in bold lines.

a)                                   b)

Figure 8. Dissection of the fern into similar parts a),

mapping triangles for the fern b) [15]

Figure 9a illustrates an attempt to find a IFS which could generate the ideal Gothic Church conceived by Eugène-Emmanuel Viollet-le-Duc (1814-1879). The figure 9b is dedicated to applied the same approach to a flower (Celosia plumosa) with is manifestly fractal-like.

a)                                                                 b)

Figure 9. An attempt to dissect a Gothic church in self-similar parts a), the same approach applied to a flower (Celosia plumosa)

In the Italian Gothic style there are many examples which show the presence of the fractal components. In Venice there are many palaces (Ca' Foscari, shown in figure 10,  Ca' d'Oro, Duke Palace, and Giustinian Palace) that  have  a rising fractal structure; for this reason  Fivaz  (1988)  named this town: "fractal Venice" [16].

Santa Croce, the church of the Franciscans in Florence, is one of the finest examples of Italian Gothic architecture. It was begun in 1294, in the period that served as the transition from Medieval times to the Renaissance.  It has been designed by Arnolfo di Cambio (1240-1302), and it was finished in 1442, with the exception of the 19th century Gothic Revival facade and campanile. The church is simple basilica style with a nave and two isles. Figure 11 illustrates the west facade of Santa Croce, and an attempt to dissect it in triangles to find the IFS connected to the church.

Figure 10. Ca' d'Oro (Venice, Italy) (1421-1440) shows a fractal structure

Figure 11. Santa Croce (Florence,  Italy ) an attempt to find the IFS

4. Fractal components in the Baroque architecture

The Baroque (1600-1750) was born in Italy, and adopted in France, Netherlands, Germany, and Spain. The term “Baroque” was probably derived from the Italian word “barocco”, which was a word used by the philosophers during the Middle Ages to describe a hindrance in a schematic logic. After, this have been used to describe  any contorted process of thought or complex idea. Another possible meaning derives by the Spanish “barrueco”, Portuguese form “barroco”, used to describe an imperfect or irregular shaped pearl. This word has survived in the jeweller’s term “Baroque pearl”.

This style suggested movement in static works of art, and it influenced important  challenges in architecture [17]. Baroque architecture was based on the mathematics [18]. The Baroque architecture could be analysed using a fractal point of view [19].

Figure 12 shows a kind of self-similar components present in the which illustrates the plan of church of Saint Karl (1715-1737, Vienna) where the oval is repeated in three different scales.

Figure 12.  The plan of the church of  Saint Karl (Vienna)

shows some self-similar shapes

Another example of self-similar component is present in  the church of San Carlo alle Quattro Fontane, conceived by Francesco Borromini (1599-1667). The Swiss architect used the octagons, the Greek crosses and other shapes for the coffering of the dome of San Carlo alle Quattro Fontane. The figure 13a illustrates the valve lattice of the shell (Cakadia) which provided the brunched coordinates that map out the Greek crosses and the octagons, shown in figure 13b,  that  Borromini used to cover the  dome of  San Carlo alle Fontane. The figure 14 illustrates the dome interior where the ends of each lozenge and of each rhombus are unequal, the upper half of each octagon is smaller than the lower half, and the top of the upright in each Greek cross is shorter than the bottom of the lower part of the cross’ upright [18]. Observing figure 14, it is possible to see the presence of two directional compressions, horizontal and vertical at the same time, over a (much shallower) dished plan. These compressions introduces a kind of self-similarity in the dome [19].

a)                                                        b)

Figure 13. The valve lattice of Cakadia  a), the lattice used to map a detail in the

Borromini’s dome b) [20, p. 52]

Figure 14. Dome of  San Carlo alle Quattro Fontane, Rome.

The arrows connect the self-similar shapes

An other example of self-similarity in the Baroque architecture is in the dome of Church of San Lorenzo (Turin, 1666-1680), designed by the Italian architect Guarino Guarini (1624-1683). Norwich (1975) wrote: "The Church of San Lorenzo, Turin, was begun by Guarino Guarini in 1668 for the Theatine Order, of which he was a member. The plan is remarkable for its curved bays pressing into the central domed space—an idea developed from Borromini—but the dome is even more remarkable. It is a masterpiece of ingenious construction—the ribs actually carry the lantern above them—which is also used to produce dramatic contrasts of light and shade" [21, .p 176]. Guarini used the octagonal star to define the bearing structure of the dome. The self-similar components are an octagon and an octagonal star which are repeated in different scales, as shown in figure 15 [22, p. 85].

Figure 15. The dome of San Lorenzo (Turin, Italy) shows some self-similar components

5. Conclusions

Fractal geometry and its connection between the complexity can help to introduce the new paradigm in architecture [2, 5, 7, 8, 9, 10, 18]. This paper introduces only an approach to observe the Gothic and the Baroque architecture using a fractal point of view. The property of the self-similarity present in these two different styles has been chosen for an aesthetic sense; in fact  the Gothic and the Baroque architects did not know the fractal geometry, because it is a recent discovery. Thus,  it is possible to  refer as an “unintentional” use of the fractal geometry.

The modern architecture uses the self-similarity appears in intentional way [1, 2, 4, 13]. The iterated function system applied in the Gothic cathedrals could help us to understand the generative processes of these complex buildings.

Recent studies reveal that the IFS could help to create a new pseudo urban models  based on fractal algorithms [23]. Thus, it could be possible to encode simplified 2D½ city models using an IFS compression technique.

6. References

[1] Bovill, C. Fractal Geometry in Architecture and Design, Birkhäuser, Boston, 1996.

[2] Salingaros, N. I frattali nella nuova architettura, retrieved October 29, 2005, from:

http://www.archimagazine.com/afrattai.htm

[3] Sala, N. The presence of the self-similarity in architecture : some examples, in M.M.Novak (ed), Emergent Nature, World Scientific, 2002, pp. 273-283.

[4] Eaton, L. K.  Fractal Geometry in the Late Work of Frank Llyod Wright: the Palmer House. Williams, K. (ed.), Nexus II: Architecture  and Mathematics, Edizioni Dell’Erba, Fucecchio, pp. 23 - 38,  1998.

[5] Sala, N., Cappellato, G. Viaggio matematico nell’arte e nell’architettura, Franco Angeli, Milano, 2003.

[6] Frankhauser, P. La Fractalité des Structures Urbaines, Collection Villes, Anthropos, Paris,1994.

[7] Frankhauser, P. L’approche fractale : un nouvel outil de réflexion dans l’analyse spatiale des agglomérations urbaines,  Université de Franche-Comté, Besançon, 1997.

[8] Batty, M., Longley, P.A. Fractal Cities: A Geometry of Form and Function, Academic Press, London and San Diego, 1994.

[9] Mandelbrot, B. The Fractal Geometry of Nature. W.H. Freeman and Company,  1988.

[10] Oppenheimer, P. Real time design and animation of fractal plants and trees, Computer Graphics, 20(4), pp. 55–64, 1986.

[11] Barnsley, M.F. Fractals everywhere. Academic Press, Boston, 2nd edition, 1993.

[12] Bogomolny, A. The Collage Theorem. retrieved September 15, 2005,  from:

http://www.cut-the-knot.org/ctk/ifs.shtml

[13] Sala, N., Cappellato, G.  Architetture della complessità. La geometria frattale tra arte, architettura e territorio, Franco Angeli, Milano, 2004.

[14] Fulcanelli, Il mistero delle cattedrali e l'interpretazione esoterica dei simboli ermetici della Grande Opera, Edizioni Mediterranee, Roma, 2000.

[15] Wright, D.J. (1996). Designing IFS's: the Collage Theorem, retrieved, September 30, 2005, from: http://www.math.okstate.edu/mathdept/dynamics/lecnotes/node47.html

[16] Fivaz, R.  L’ordre et la volupté, Press Polytechniques Romandes, Lausanne,1988.

[17] Harbison, R., Reflections on Baroque, Reaktion Book, London, 2000.

[18] Hersey, G., Architecture and Geometry in the Age of the Baroque, University of Chicago Press, Chicago, 2000.

[19] Sala, N., Cappellato G. The generative approach of Botta’s San Carlino. Proceedings 6th Generative Art Conference, Milano, Italy, pp. 328-337, 2003.

[20] Hersey, G., The monumental impulse, The Mit Press,  London, 1999.

[21] Norwich, J.J. (ed.) Great Architecture of the World. Mitchell Beazley Publishers, London, 1975.

[22] Götze, H. Castel del Monte, Hoepli, Milano, 1988.

[23] Marsault, X. Generation of textures and geometric pseudo-urban models with the aid of IFS, Chaos and Complexity Letters, Sala N. (ed.), Special issue dedicated to the Chaos and Complexity in Arts and Architecture, 2005 (in print).

[1] Fulcanelli wrote: “Alcuni autori perspicaci, e non superficiali, colpiti dalla similitudine che esiste tra gotico e goetico, hanno pensato che ci dovesse essere uno stretto rapporto tra Arte gotica e Arte goetica o magica. Per noi art gotique non è altro che una deformazione ortografica della parola argotique la cui omofonia è perfetta. La cattedrale è quindi un capolavoro d’art goth o d’argot.” [14, p. 46]