Daniele CapoVia Mazzetta, 10 01100 Viterbo ITALIA
As a guide, we can take the definition of fractal sets F
laid out by Flaconer [1990: xx-xxi]:
In the field of architecture, Carl Bovill [1996] performed a fractal analysis by measuring, by means of the method of box-counting, the fractal dimension of some works of Wright and Corbusier. In the present paper I would like to make some observations on the implications of an architecture with a fractal nature, and in particular, to demonstrate how a discussion of this kind is well suited to the architectural orders (it appears that thus far the architectural orders have never been subject to a fractal interpretation). The architectural orders can be taken as meaningful case studies for several reasons. In the first place, the object of study is easily defined; secondly, the analysis can be limited only to vertical successions of elements that are clearly disparate; through focussing on only one dimension, an analysis can be perfomed in a systematic and precise fashion. In spite of its simplicity, or perhaps because of it, the example of the architectural orders furnishes a very clear image of how fractal analysis can be applied to architecture in general and contributes to the resolution of the ambivalence concerning the meaning of the term "fractal" in the field of architecture.
With the first method (Fig.
2), we count the number of small squares that are "occupied"
by some point of the set being investigated, and at each successive
passage, we divide the side of the square by two. Rather than
using the classic method of box-counting, we will use a slightly
modified version, "the information dimension" [1], which
takes into consideration the number of points that fall in each
square. The values obtained are placed on a graph so that the
number of squares occupied are laid out on the x-axis, and the
logarithm of the inverse of the side of the square are laid out
on the y-axis.[2]
By means of a statistical analysis the straight line is obtained
that best approximates the distribution of points and the coefficient
of correlation, which tells how valid it is. The slant of the
line represents the fractal dimension of the set of points for
the architectural order. In order to obtain the dimension of
the architectural order (that is, the dimension of the original
drawing, taking into account only the horizontal rows), it is
sufficient to add 1 to the result obtained.[3]
The "information dimension" method shows that all three of Palladio's orders maintain a certain consistency of the data up to the eighth level, indicating that the value of the dimension is demolished only when the count is based on squares with a small side that is equal to 1/256 of the height of the entire order. If we consider a total height of 10 meters, we can conclude that the fractal coherence is maintained down to a detail of 4 centimeters, which is not surprising considering that, in the architectural orders, that are mouldings that are exceedingly small. The second method validates the results of the first, showing how the number of elements continues to increase as their height gradually diminishes, a characteristic that is essential to fractal objects. In this case, as above, the most important result is the large interval on which the fractal interpretation has been effected. The trend not being perfectly linear would seem to deny that this is true, even though the coefficient of correlation is still very high, but if the form of the graph is carefully considered, it can be discerned that the most important element is the tendency of the details to grow as their height decreases. The "control" order, explicitly constructed with a fractal recursion, furnishes results that are very similar to those obtained from the analysis of Palladio's orders, reinforcing still more this interpretation. This helps us to understand that the jumps present in the graph relative to the second method are inevitable.[7] The fractal dimension measured is not fixed, as can be easily verified in the graphs, but oscillates. The important thing is to notice that it also oscillates for the control order, and is greater than than the theoretical one that is known in this case. The problem therefore reduces down to two facts: the first is that the method itself has limits, as was noted above; the second is that we must not consider the orders as "simple" fractals, but rather as examples of multi-fractals in which diverse dimensions coexist.[8] Even taking into account these limits, it can be brought out in any case that the dimension runs between 0.6 and 0.7.[9] Knowing that these values are approximate by excess, we can in any case affirm that the dimension can be collocated in a position that tends to mediate between 0 and 1, where 0 represents the null set (a total absence of any element of interest) and 1 the completely full set (visual chaos, where every part is filled) (Fig. 5). The geometry of these architectural objects strike a balance between the two extremes, a fact that is held to be extremely important by both Mandelbrot [1981: 45-47] and Eglash [1999: 171]. Understanding the orders, which for centuries have provided
the basis for Western architecture, in light of the analysis
presented above, brings us to certain considerations. The first
is that it allows us to observe, through the analysis of numerical
data, how small elements are inserted in a continuous and coherent
whole. If we interpret this structure fractally we do not distinguish
between the essential and the inessential; everything is essential
and so creates in this way a greater (fractal) coherence. It
could be said, in this light, that the general form is not what
counts the most, but rather, what is really important is the
way in which parts hold together. For example, by means of this
analysis it could be said that the "abstractions" that
reduced the architectural orders to their principal elements
(as in certain architectures of totalitarian regimes of the first
half of the last century) did not grasp this fact, while an architect
like Wright [10],
even while not replicating the form of the orders, realized an
architecture which, from the point of view of fractals, came
very close to them.
[2] For our purposes we consider the height of the entire
architectural order, from the ground to the uppermost moulding,
to be one unit. [3] This problem can be taken back to that of the multiplication
of two sets with different fractal dimensions, the dimension
of which is equal to the sum of the dimensions. Cf. [Falconer
1990]. [4] For a discussion of Salingaros's position on "fractal"
architecture, see [Salingaros and West 1999]. The method that
we suggest can also be extracted from the analysis of Mandelbrot
[1987] of the Cantor set. The links between the architectural
orders and the Cantor set will become evident later in the text.
[5] A fractal object such as the Cantor set will never
tend to zero but, in the case of real objects, we cannot ever
arrive at this level of abstraction. For this reason we have
introduced the definition of architecture as having a "fractal
nature," to indicate that architecture which, within certain
limits, behaves in a way that is similar to a fractal. [6] It is possible to propose a different means of representation
which would still present the same advantages given above. [7] In essence, the form of the graph is effected by
the fact that the length u has been halved each time. The jumps
represent the fact that in those points the height of the elements
"jumped" in a more rapid manner. [8] We are unable to find actual examples of applications
of methods of multi-fractal analysis to architecture. Within
the limits of this brief paper we can only advance the hypothesis
that a similar approach can furnish new information about the
geometry of architecture. [9] 0.6 and 0.7 are the dimensions of the set of points
that has been found with our analysis. The dimension of the succession
of mouldings varies therefore between 1.6 and 1.7. [10] For studies of Wright's architecture with regards
to fractals, see [Bovill 1996] and [Eaton 1998].
Bovill, Carl. 1996. Eaton, Leonard K. 1998. Fractal Geometry in
the Late Work of Frank Lloyd Wright: The Palmer House. Pp. 24-38
in Eglash, Ron. 1999. Eisenman, Peter. 1986. L'inizio, la fine e
ancora l'inizio. Falconer, K. 1990. Jencks, Charles. 1997a. ______. 1997b. New Science=New Architecture.
Mandelbrot, Benoit B. 1981. Scalebound or
Scaling Shapes: A Useful Distinction in the Visual Arts and in
the Natural Sciences. ______1987. Ostwald, Michael. 2001. "Fractal
Architecture": Late Twentieth-Century Connections Between
Architecture and Fractal Geometry. Palladio, Andrea. 1992. Peitgen, Heinz Otto, et. al. 1998. Salingaros, Nikos and Bruce West. 1999. A
universal rule for the distribution of sizes.
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