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CRITICISM INSTRUMENTS ANTÓNIO PEDRO A. N. L. LIMA F.C.T. - Fundação para a Ciência e a Tecnologia C.I.A.U.D. - Centro de Investigação em Arquitectura, Urbanismo e Design Universidade Técnica de Lisboa - Faculdade de Arquitectura António Pedro A. N. L. Lima, Ph.D. E-mail: arq.lima@gmail.com Summary Examples of objects with scale properties and non integre dimension have long been known to mathematicians but it was Benoit Mandelbrot [1] who, introducing the term fractal, first classified such kind of objects, providing a scientific basis for the study of irregular sets. One of the characteristics of these objects is their fractal dimension and one of the methods to determine this dimension is the “box-counting dimension” technique that, for real objects, provides a very good approximation to the determination of this non-topological dimension. This technique is applied covering any two-dimensional representation with a squared mesh and counting the number of boxes containing lines from it knowing that, the tighter the grid, the greater the number of boxes containing lines from the given representation. Being A a one-dimensional object, covered by a squared mesh size є, N(є) represents the number of boxes that contain lines belonging to A. Thus, if D = lim log N (ε ) exists, then D is the box-counting dimension of A, fig. 1. ε→0 log(1/ε) Introduction
Figure 1: First four stages of the box-countig dimension calculation on a von Koch curve. 41
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In 1996, Carl Bovill [2] pointed to the application of some techniques associated with fractal geometry as tools of analysis and criticism of architectural objects through the approximate calculation of the fractal dimension of a single drawing belonging to a particular architectural object, considering different scales and relating them to the visual perception, and incidence angle, of the look of an observer. In 1999, Richard P. Taylor, and others, also presented an application of the box-counting technique to Jackson Pollock´s drip paintings in a very short article published in the Nature Journal [3]. In our opinion, the application of this technique can be extended by the constitution of an instrument defining the characteristics of the work of an architect considering the different projects and their respective drawings. After the encouraging results obtained in a previous short joint paper published in the proceedings of I.S.A.M.A. 2002 [4], organized by The International Society of the Arts, Mathematics and Architecture, the University at Albany, State University of New York, and the Pädagogische Hochschule Freiburg, Germany, where four projects coming out from Frank Gehry´s office, also included in the present work, were submited to the box-countind dimension calculation technique after the respective drawings were scaned, from previously selected publications, and cleaned from all unnecessary information, such as reference lines and alphanumeric information, according to a pre-established criteria of programmatic complexity equivalence, in the present work, as a result of the work developed for our PhD thesys [5], we intend to go even further and apply the same technique to the work of other architects. For that, we will use the same technique on four different sets of drawings, each set belonging to one of a set of four different buildings designed by one of three different architects, Álvaro Siza Vieira (b. 1933), Frank Owen Gehry (b. 1929) and Peter Eisenman (b. 1932), and, at the end, we will compare the obtained results in each project to determine if these values may, or may not, be a determinant of, at least, a specific period of their personal work. Here, we intend to verify the hypothesis, then raised, that it is possible to obtain a specific value, as a determinant feature, for a specific architect´s work, and that, using the same project selection criteria carried out in our first test, the application of this technique is sustainable for other architects. This time, we performed a new scan, with a better resolution, 400 dpi instead of 200 dpi, of all the drawings, including those used in the first test, in order to give them a higher level of detail, which allows a safer use of a greater number of mesh divisions used in this technique and, finally, compared the different results obtained among the representations of the same architectural object and, later on, between architectural objects, to establish a link between them, using the set of all the results to obtain a determinant characteristic of the work of the chosen architect. The dimension of architectural objects Determining the box-counting dimension of all the drawings belonging to a same project was based on the assumption that each project may have a specific dimension, considering the average of the dimension of all the drawings, and the respective dispersion values. To automatically calculate the dimension of all these architectural objects, we 42
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used 2D Fractal Dimension software, coded by Lukacs Zsolt Sandor at Babes-Bolyai University, Faculty of Mathematics and Computer Science of Cluj-Napoca, Romania. Knowing that the smaller box that any computer program of this nature should use is the same supported by the computer, and also by the scanned image, the size of a pixel, the smaler size of the box can´t be less than the drawing line thickness in orther to avoid the measurement of the line dimension itself, which are onedimensional from the topological point of view. In all the calculations we made, from the set of used parameters, we determined that the divisions of the mesh sides should be from 2 to 10 raised to the power of two, in other words, 2²; 3²;...; 10². Thus, the total number of boxes for each mesh should be, respectively, 16, 81, 256, 625, 1.296, 2.401, 4.096, 6.561 and 10.000. In our first test, if the difference between the box-counting calculation results of all the drawings is less than a small positive number we can say that they have almost the same box-counting dimension. Knowing that we can have a variance in the box-counting dimension calculation of no more than about 1%, if we don’t have a deviation of more than this small value between the several results obtained with the drawing measurements, we can also assume that all of them have almot the same dimension. Because the results are not equal we can say that they are almost the same. Assuming that we would study a certain number of projects conceived after the publication of the concepts related to the issues of complexity and of fractal geometry, we chose two architects that consciously used these concepts to justify their formal and compositional options, Frank O. Gehry and Peter Eisenman. However, we also studied four projects from a third architect, Álvaro Siza, who, aparently, didn´t use such concepts, serving, by the contrary, as a counterpoint, or confirmation, of the results obtained by the application of this technique as an instrument of analisys and criticism. This way, we obtained a set of results by drawing, by project and, finally, by architect, as follows: -Álvaro Siza Vieira Valência University´s Rectory and Library building (1990) Results of the box counting dimension calculation: Elevation – East: Elevation – West: Elevation – North: Elevation – South: Plan – Ground floor: Plan - 1st floor: Plan - 2nd floor: Plan - 3rd floor: Plan - 4th floor: Section – Cross: Section – Longitudinal: Average: Variance: Standard deviation: 43
1,315437058 1,349754246 1,350122622 1,292854589 1,558277805 1,527305123 1,536557667 1,524598183 1,463721091 1,520270221 1,529903739 1,451709304 0,010506249 0,102499996
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Alicante University´s Rectory and Library building (1995-1998) Results of the box counting dimension calculation: Elevation – East: Elevation – West: Elevation – North: Elevation – South: Plan – Ground floor: Plan - 1st floor: Section – Cross: Section – Cross: Section – Cross: Section – Longitudinal: Average: Variance: Standard deviation:
1,501926450 1,571638297 1,499957944 1,568324521 1,691278886 1,749134660 1,596442783 1,654239224 1,613908296 1,665630274 1,611248133 0,006466406 0,080413967
Manzana del Revellin´s Cultural Center buiding (1997) Results of the box counting dimension calculation: Elevation – East Elevation – West Elevation – North Elevation – South Plan – Underground floor Plan – Ground floor Plan - 1st floor Plan - 2nd floor Plan - 3rd floor Section – Cross Section – Cross Section – Cross Section – Cross Section – Longitudinal Section – Longitudinal Section – Longitudinal Section – Longitudinal Average Variance Standard deviation
1,293171080 1,389313071 1,327800406 1,341616426 1,576550977 1,597206404 1,587006663 1,550554438 1,547014085 1,577840953 1,610416566 1,472967046 1,566433710 1,515806210 1,514975303 1,592527998 1,558321124 1,507030733 0,010775314 0,103804211
Aveiro University´s Library building (1988-1994) Results of the box counting dimension calculation: Elevation – Northeast Elevation – Southwest Plan – Ground floor Plan - 1st floor nd Plan - 2 floor
1,459438784 1,447460183 1,561339050 1,535971040 1,566045620
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Plan - 3rd floor Section – Cross Section – Cross Section – Longitudinal Section – Longitudinal Section – Longitudinal Average Variance Standard deviation
1,595266010 1,539526170 1,541310378 1,598353055 1,579034939 1,622018196 1,549614858 0,002989885 0,054679844
Results sumary: After obtaining all the values resulting from the box-counting dimension calculation of the several drawings from the four selected architectural objects, we will proceed with the determinant definition for this architect and the respective measures of dispersion, fig. 2. Average of all of the drawings 1,525440195 ≈ 1,53 Variance 0,01064964 ≈ 1% Standard deviation 0,103197094 With the analysis of all the 49 drawings that we gathered for the four Álvaro Siza´s projects, we obtained an approximate average value of 1.53 that we can consider the determinant value for this architect. Since the variance is less than 1%, according to the pre-established parameters, we can say that these drawings, from the statistical point of view, have almost the same dimension. Figure 2: Graphic of all Álvaro Siza´s drawings box-counting dimension calculation.
-Frank Owen Gehry Iowa University´s Laboratories building (1987-1992) Results of the box counting dimension calculation: Elevation – East Elevation – West Elevation – North Elevation – South Plan – Ground floor Plan - 1st floor Plan - 2nd floor 45
1,749788874 1,806175319 1,774912658 1,780405148 1,696719987 1,522828190 1,449880062
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Plan - 3rd floor Section – Cross Section – Cross Average Variance Standard deviation
1,447790109 1,649456845 1,65877605 1,653673324 0,018465762 0,017543863
Frederick R. Weisman Museum (1990-1993). Results of the box counting dimension calculation: Elevation – East Elevation – West Elevation – North Elevation – South Plan – Underground floor Plan – Ground floor Plan - 1st floor Section – Cross Section – Longitudinal Average Variance Standard deviation
1,505442151 1,666893357 1,661868388 1,539272794 1,487153395 1,488422013 1,524594341 1,501104553 1,613672145 1,554269237 0,005368366 0,073269135
Paris American Centre building (1988-1994) Results of the box counting dimension calculation: Elevation – West Elevation – South Plan – 4th floor Plan – 5th floor Plan - 7st floor Section – Cross Section – Longitudinal Average Variance Standard deviation
1,672388018 1,675716152 1,560604321 1,531373194 1,550720261 1,612973140 1,607498582 1,601610524 0,003311581 0,057546338
Bilbao Guggenheim Museum (1991-1997) Results of the box counting dimension calculation: Elevation – East Elevation – West Elevation – North Elevation – South Plan – Underground floor Plan – Ground floor Plan - 1st floor Plan - 2nd floor Plan - 3rd floor Section – Cross
1,617043204 1,586608592 1,58760485 1,608412503 1,433416007 1,623604009 1,542814338 1,526086757 1,519360147 1,487551730
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Section – Cross Section – Longitudinal Section – Longitudinal Average Variance Standard deviation
1,582732474 1,578720488 1,517564737 1,554732296 0,003149147 0,056117262
Results sumary: After obtaining all the values resulting from the box-counting dimension calculation of the several drawings from the four selected architectural objects, we will proceed with the determinant definition for this architect and the respective measures of dispersion, fig. 3. Average of all of the drawings 1,588408972 ≈ 1,59 Variance 0,008838042 < 1% Standard deviation 0,094010862 With the analysis of all the drawings that we gathered for the four projects from Frank O. Gehry, we obtained an approximate average value of 1.59 that we can consider the determinant value for this architect. Since the variance is less than 1%, according to the pre-established parameters, we can say that these drawings, from the statistical point of view, have almost the same dimension. Figure 3: Graphic of all Frank Gehry´s drawings box-counting dimension calculation.
-Peter Eisenman Centro de convenções de Columbus Convencion Center building (1988-1993) Results of the box counting dimension calculation: Elevation – East 1,638598847 Elevation – West 1,801013152 Elevation – North 1,793582489 Elevation – South 1,763613203 Plan – Ground floor 1,597418735 Plan - 1st floor 1,518384263 Average 1,685435115 Variance 0,013802491 Standard deviation 0,117484005
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Emory University´s Scenic Arts Centre building (1991-1993) Results of the box counting dimension calculation: Elevation – East Elevation – North Elevation – South Plan – Roof Section – Cross Section – Cross Section – Longitudinal Section – Longitudinal Section – Longitudinal Section – Longitudinal Section – Longitudinal Average Variance Standard deviation
1,610927414 1,557013208 1,493042451 1,596717203 1,583185615 1,569999543 1,494755829 1,536720113 1,542535862 1,492664488 1,499630661 1,548452790 0,001831820 0,042799766
J. W. Goethe University´s Biocenter building (1987) Results of the box counting dimension calculation: Elevation Elevation Plan Plan Plan Plan Section – Cross Section – Longitudinal Section – Longitudinal Average Variance Standard deviation
1,638361843 1,660332240 1,598270745 1,631528390 1,631391373 1,613019894 1,642835691 1,620141091 1,588074921 1,624884021 0,000509177 0,022564942
Max Reinhardt Haus (1992) Results of the box counting dimension calculation: Elevation – East Elevation – South Plan – Underground floor Plan – Ground floor Plan - 1st floor Plan - 3rd floor Plan - 13th floor Plan - 26th floor Plan - 29th floor Section – Longitudinal Section – Longitudinal Average Variance
1,828633206 1,788745798 1,570020317 1,553829974 1,579636693 1,654888244 1,622729819 1,536444953 1,532757317 1,665414349 1,641068200 1,634015408 0,009667158
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Standard deviation
0,098321706
Results sumary: After obtaining all the values resulting from the box-counting dimension calculation of the several drawings from the four selected architectural objects, we will proceed with the determinant definition for this architect and the respective measures of dispersion, fig. 4. Average of all the drawings 1,616535118 ≈ 1,62 Variance 0,007573361 < 1% Standard deviation 0,087025058 With the analysis of all the drawings that we gathered for the four Peter Eisenman´s projects, we obtained an approximate average value of 1.62 that we can consider the determinant value for this architect. Since the variance is less than 1%, according to the pre-established parameters, we can say that these drawings, from the statistical point of view, have almost the same dimension. Figure 4: Graphic of all Peter Eisenman´s drawings box-counting dimension calculation.
Conclusion With the set of results obtained by calculating the dimension of a series of architectural objects using the box-counting technique, we were able to verify the validity of this method in obtaining the determinants for, at least, a certain period of the work of three different architects, facing the possibility of extending its application to the work of other architects. In addition, we can foresee the possibility of using this methodology in other situations related to analysis and criticism in an objective and scientific way. References [1] Mandelbrot, B. (1975). Les objects fractals: Forme hasard et dimension. Paris,Flammarion. [2] Bovill, C. (1996). Fractal geometry in architecture and design. Boston: Birkhäuser. [3] Taylor, R. P, Micolich, A. P., Jonas, D. (1999). Fractal analysis of Pollock´s drip paintings. Nature, International Weekly journal of Science, vol. 399, June 3rd. [4] Soós, A, Lima, A. P. (2002). Fractal analysis of architectural objects. Guderian, D. (editor). I.S.A.M.A. 2002 proceedings, University of Freyburg, Germany. Ebringen, Bannstein-Verlag. 49
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[5] Lima, A. P. (2007). Referenciais geométricos na arquitectura e no design. Para além da geometria euclidiana. Unpublished PhD Thesis, U.T.L.-Faculdade de Arquitectura, Lisboa, Portugal. Complementary Bibliography Falconer, K. (1990). Fractal geometry. Mathematical foundations and applications. Chichester: John Willey & Sons Ltd. Falconer, K. J. (1997). Techniques in fractal geometry. Chichester, John Willey & Sons Ltd. Mandelbrot, B. B. (1977). The fractal geometry of nature. New York, W. H. Freeman and Company. Ouellette, J (2001). Pollock´s fractals. Discover magazine, published online November 1st.
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