)2166( 68-6 )2( 66 ةيسدنهلا ثوحبلل تاراملإا ةلجم ) ةيماظن ... ·...
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(2166) 68-6( 2) 66مجلة اإلمارات للبحوث الهندسية
(مقالة نظامية )
6
21192161
The fractal geometry has emerged from the Chaos theory within the new scientific group
including complexity theory, which has given a new vision to the natural global system, which is
considered far from regularity, order, linearity, and the power of prediction. The Fractal geometry
shows ability to bridging in gaps between architecture and mathematics, nature and fractal
characteristics, and to describe complex forms.
Many architectural studies focused on western architectural products and did not give an
importance to the products of Islamic architecture and its extensions which represented by
traditional architecture. So, the research problem concentrated on the need for exploring the
nature of the fractal iteration systems in traditional architecture. The research hypothesized that
there is no complete fractal characteristic in traditional architectural structure at all hierarchical
levels, but there is a kind of fractal characteristic in traditional architecture, that expressing fractal
rhythm locating somewhere between order and surprise .
To tackle research problem, analytical descriptive method, and hypothetical conceptual
framework that describing analytical indicators connecting with the characteristics of fractal
shapes like self-similarity, self-affine, hierarchical scaling, infinite iteration, coherence,
raggedness and richness in detail, and methods of the fractal dimension like Self similarity
dimension, measured dimension, and box- Counting dimension depended. 20 heritage house
selected as a research sample. The survey process included two phases. The computer was
depended as survival, analytical, and statistical tool. The software CAD 2008, Excel 2007 used
for analyzing, accounting, and drawing.
The results revealed that the traditional architecture behaves within certain boundaries of fractal
characteristic. There is no infinite iteration which can be occurring on its elements or components.
Hence, it is no complete fractal characteristic at all levels and it can be identified as traditional
architecture with fractal nature.
The conclusion demonstrated that the traditional architecture is completely free and progressive.
It combines between order and surprise. Also, it is depending upon the characteristic of fractal
geometry. It has self-similarity with varying fractal rhythms, hierarchical cooperation, and
cascading of details at all levels.
Key words: Fractal architecture, Iteration systems, Traditional architecture
20
CAD 2008
Excel 2007
2 62166
1
[1]
Type[2]
[3]
Self - Similarity
[4]
golden section[5]
self - exactness
likeness
Kind ship6
self-affine transformation
7
Fractal dimension
8
Roughness
9
Scaling
[10]
[11]Alteration
[12]Feedback and iteration
[13]
, 2003(Salingaros
.[14]
Bangura, 2000 Eglash
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3
[15]
(Lorenz, 2003)
[16](Capo, 2004
Doric CompositeCorinthian
[17](Magdy & Krawczyk,
2004)
[18]
CECA (Centre for Environment &
Computing in Architecture)
fractal
decomposition
lindenmayer systems
marching cubes algorithm
Dust
PlatesBlobs[19](Bovill, 1996)
[20]Schmitt
.[21]
2
1.2
4 62166
Self-similarity
self
- exactnessself-alikeness
[22]
strictly self-similar
statistically self-similar
[23]
Fractal Tiling
[24]
[25]
[26]
XYZ
localized
self-similarity
Golden section
Phi 1 : 1.618
2
[27]
[28]
[29]
[30]
White noise
Brownian noise1/f noise
[31]
[32]
Dilation Symmetry
[33]
[34]
Hierarchical Scaling
62166
5
(2, 3)
e = 2.7
[35]
[36]
[37]
2.2Fractal Dimension
Dsd
Db
DsSelf similarity dimension
1[38] 2
Ds = log (N)/log (1/r) (1)
Ds =[log (n2)-log (n1)]/[Log (r2)- Log (r1)] (2)
n1n2
r1r2
Ds
[39]
d Measured dimension
[40]
u = (constant) (1/r) d (3)
ur
u
(zero).[41]
(Db) box- Counting dimension
Ds
[42]
[43][44]
N(r) = 1/r.
N(r) = (1/r)2.
N(r) = (1/r)3.
grid square Boxes
r
N(r)
(r)
N(r)
[N(r)](1/r)
Log-Log
(Db)
(Db)
6 62166
Db =
[Log (N (r2)) – Log (N (r1))]
[Log (1/r2) – Log (1/r1)] (4)
1/r
(Ds)
(d , Db)
(Ds)
(H)
[45]
D = 2- H (5)
3
4
1.4
20
[46][47]
300
(ppi.)
X1
X2
X3
X4
X1
X5
X6
X7
X8
1/f
scannerCanon
CAD 2008
Excel 2007
Pilot study
62166
7
CAD
8*84*42*21*1
.5*.5
scanner
JPEG
CAD2008
.5
8*84*42*2
1*1
hatch
Adobe Acrobat 7.0 ProfessionalAdobe
Photoshop 7.0.1 ME
X1
12
12
2
Mean
Standard
Deviation
8*84*4
2*21*1CAD
1*1
X2
X3
correlation
1/r
N(0, 1)
10
X4
21
X1
Correlation
X5
STD
X6
(1-0)
X7
8 62166
(0,1)1
0
X8
Mode
Van Der
Laan
Major wholeMinor whole
Major partMinor part
Major pieceMinor piece
Major elementMinor element
1/f
1/f
5
X1X7X8
(2)
(5)
(1)1(5)
(1):
Scanner
7212
414 (Warren, p.48,1982)
CAD
8*8
62166
9
4*4
2*2
1*1
.5*.5
1/f
61 62166
(1:)(5)
(2):
2.4
X1
1 2
1.8-1.4
1.5 12%60
1.49
%201.8
% 101.6 %5
1.7%5
X2
4*48*8
%85
%15
X3
1/5
(5)
1/f
62166
66
X4
X1
1.561.951.645
1.825
1.715
1.510
1.95
X1
Correlation0.7
X5
0.250
0.130 0.0215
%5
3
8*81*1
X6
2*2 1*1
1.6 -1.2
1.7
1.3
2
30%
70%
55
10
%35
X7
8*8
62 62166
20,19,18,84,6,20
X8
1/f
5-15
55
1/f
%25
15
1.6
4
1.29.1
0.6
1.2
Van Der Laan
8.4
9.11.5
1.6
5
1-4
62166
63
1/f
REFERENCES
1. Musgrave, F. Kenton, 1999. Fractal Forgeries of
Nature in: Fractal Geometry and Applications:
AJubilee of Benoit Mandelbrot by Michael, L.
Lapidus , Managing Editor, American
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2. Weinstein, Eric W.Concise Encyclopedia of
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3. Batty and Longley, 1994. Fractal Cities,
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4. Mandelbrot, Benoit B., 1983. The Fractal
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24.
5. Bovill, Carl, 1996. Fractal geometry in
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6. Maletz, Andrew Scott, 1999. Developing a
Fractal Architecture , A Thesis Submitted to the
Faculty of Miami University , In partial
fulfillment of The requirements for the degree of
Master of Architecture , Department of Fine Arts
, Miami University , Oxford, Ohio , p.10.
7. Bovill, Carl, 1996. Fractal geometry in
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8. Bovill, Carl, 1996. Fractal geometry in
architecture and design, Ibid, p.2.
9. Ibid, p.43.
10. Schmitt, Gerhard N., 1988. Microcomputer
Aided Design: For Architects and Designers.
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11. Mandelbrot, Benoit B., The Fractal Geometry of
Nature, Ibid, p.60.
12. Bovill, Carl, 1996. Fractal geometry in
architecture and design, Ibid, p.47.
13. Meggs, Philip B., 1989. Type and Image: The
Language of Graphic Design , Van Nostrand
Reinhold, New York, p.97.
14. Salingaros, Nikos A., 2003. Connection the
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towns and town planners in Europe Barcelona,
http://www.math.utsa.edu/sphere/salingar/Lifean
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Brunswick, New Jersey: Rutgers University
Press.
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planning and architecture , Vienna University of
Technology ,Vienna, www.fractalarchitect.com
p.128
17. Capo, Daniele “Generating Fractal Architecture:
The Fractal Nature of the Architectural Orders"
64 62166
Nexus Network Journal, Vol. 6, spring 2004,
http:// www.nexusjournal .com/ capo. htm/.
18. Ibrahim, Magdy & Krawczyk, Robert,2004. http
: // homepage . vel . ac. uk / 1953r / watfrac.htm
19. CECA (Centre for Environment & Computing in
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& Blobs", Paul S. Coates with: Tom Appels, and,
Corinna Simon, University of East London,
Stratford London UK,p. p.1-9.
20. Bovill, Carl, 1996. Fractal geometry in
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21. Oliver, Dixon, 1992. Fractal Vision: Put Fractals
to Work for You. Carmel, IN: Sams, p.29.
22. Bovill, Carl, 1996. Fractal geometry in
architecture and design, Ibid, p.2.
23. Garcia, E., 2005. Fractal Motifs and Iterated
Function Systems, Article 2 of the series The
Fractal Nature of Semantics,
www.miislita.com/fractals/fractal.html.
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Nature, Ibid, p.539.
25. Gordon, Nigel Lesmir & Rood, Will & Edney,
Ralph, 2006. Introduction, Fractal geometry,
Edited by Richard Appignanesi , Tooem Books
USA and Icon Books UK, pp.6-48.
26. Salingaros, Nikos, 2001. Fractals in the new
Architecture, Katarxis Nº 3, Department of
Applied Mathematics, University of Texas at San
Antonio, San Antonio, Texas 78249, USA.
27. Oswald, Michael J., 2001. Fractal Architecture:
Late Twentieth Century Connection between
Architecture and Fractal Geometry, Nexus
Network Journal, vol. 3, no. 1 (Winter),
http://www.nexusjournal.com/Ostwald-
Fractal.html .
28. Ching, Francis D.K., 1979. Arch: Form, Space
and Order, Van Nostrrand Reinhold Company,
p.62.
29. Scott, Robert Gilliam, 1951. Design
Fundamentals, McGraw Hill Book Company
Inc., p.55.
30. Bovill, Carl, 1996. Fractal geometry in
architecture and design, Ibid, p.2.
31. www.classes.yale.edu/fractals/IntroToFrac/welco
me.htm
32. Washburn, D. K. and Crowe, D. W., 1988.
Symmetries of Culture, Seattle: University of
Washington Press.
33. Musgrave, F. Kenton, 1999. Fractal Forgeries of
Nature, Ibid, p.539.
34. Bovill, Carl, 1996. Fractal geometry in
architecture and design, Ibid, p.47.
35. Halliwell J, 1995.Arcadia, Anarchy, and
Archetypes, New Scientist 12 August, pp.34-38.
36. Licklider H., 1966. Architectural Scale. New
York, The Architectural Press, p.22.
37. Salingaros, Nikos A., 1998. A Scientific Basis
for Creating Architectural Forms Journal of
Architectural and Planning Research, volume 15
1998. © Locke Science Publishing Company,
pp.283-293.
38. Massopust, Peter R., 1994. Fractal Function,
Fractal Surfaces, and Wavelets, Sam Houston
State University Departement of Mathematics
Huntsville,Texas, Academic Press , p.105.
39. Gordon, Nigel Lesmir & Rood, Will & Edney,
Ralph, Introduction, Fractal geometry, Ibid,
pp.60 – 63.
40. School of Wisdom, 1999. The Five Dimension,
http://www.fractalwisdom.com/FractalWisdom/d
im.html
41. Ho, M.W., 1993. The Rainbow and the Worm,
the Physics of Organisms, World Scientific,
Singapore.
42. Habermas, Jurgen, 1984. The Theory of
Communicative Action, Vol.2; Trans by Thomas
McCarthy, Boston, Beacon, p.22.
43. Bovill, Carl, 1996. Fractal geometry in
architecture and design, Ibid, p.42.
44. www.classes.yale.edu/fractals/IntroToFrac/Box-
Counting Dir.htm
45. Bovill, Carl, 1996. Fractal geometry in
architecture and design, Ibid, pp.42-87.
46. Amanat al-Assima, 1980. Conservation of
Traditional Houses, Iraqi National Library
Registration, Baghdad, No.1330.
47. Warren, J., Fethi, I, 1982. Traditional Houses in
Baghdad, Coach Publishing House Limited,
England.
62166
65
8*8 2*2 4 4 0 0.0000
4*4 4*4 16 15 1 0.0667
2*2 8*8 64 39 25 0.6410
1*1 16*16 256 111 145 1.3063
1/r
Log(1/r)
N Log(N)
D=Log(N)/Log(1/r)
Log(N2)-
Log(N1)
Log(1/r2)-
Log(1/r1) D
2 0.301 4 0.602 2.000 0.574 0.301 1.907
4 0.602 15 1.176 1.953 0.415 0.301 1.379
8 0.903 39 1.591 1.7620.454 0.301 1.509
16 1.204 111 2.045 1.699
log-log 1.598
stv 0.275233592
D 0.99754232
8*8 2*2 4 4 0 0.0000
4*4 4*4 16 15 1 0.0667
2*2 8*8 64 45 19 0.4222
1*1 16*16 256 200 56 0.2800
2 0.301 4 0.602 2.000 0.574 0.301 1.907
4 0.602 15 1.176 1.953 0.477 0.301 1.585
8 0.903 45 1.653 1.831 0.648 0.301 2.152
16 1.204 200 2.301 1.911
log-log 1.881
stv 0.284386143
D 0.998413252
X1
1 2 3 4 5 6 7 8 9 10
1.55 1.53 1.57 1.86 1.59 1.82 1.61 1.49 1.57 1.51
11 12 13 14 15 16 17 18 19 20
1.51 1.50 1.49 1.58 1.59 1.53 1.52 1.48 1.49 1.70
66 62166
X2
1 2 3 4 5 6 7 8 9 10
2.000 2.096 1.631 1.000 2.000 0.00 2.048 2.000 2.000 2.000
1.792 1.951 1.750 1.500 1.953 2.000 1.987 1.792 1.730 1.850
1.762 1.853 1.613 1.619 1.762 1.904 1.869 1.749 1.681 1.696
1.665 1.726 1.595 1.646 1.699 1.820 1.802 1.615 1.675 1.635
11 12 13 14 15 16 17 18 19 20
2.096 2.000 1.631 1.000 1.631 2.000 2.000 2.000 2.000 1.000
1.879 1.725 1.672 1.404 1.672 1.904 1.872 1.839 1.850 1.387
1.788 1.686 1.582 1.486 1.661 1.762 1.830 1.721 1.667 1.431
1.715 1.675 1.536 1.432 1.601 1.646 1.713 1.661 1.615 1.455
X3
1 2 3 4 5 6 7 8 9 10
0.9991 0.9977 0.9963 0.9994 0.9975 0.9990 0.9985 0.9978 0.9996 0.9989
11 12 13 14 15 16 17 18 19 20
0.9999 0.9985 0.9980 0.9970 0.9987 0.9971 0.9979 0.9996 0.9985 0.9977
X5
1 2 3 4 5 6 7 8 9 10
0.1661 0.2344 0.3425 0.1365 0.2752 0.174 0.2031 0.2413 0.0995 0.1654
11 12 13 14 15 16 17 18 19 20
0.0519 0.1867 0.2171 0.2773 0.1803 0.2576 0.2450 0.0907 0.2018 0.263
X4
1 2 3 4 5 6 7 8 9 10
1.67 1.60 1.77 1.95 1.88 1.82 1.87 1.77 1.67 1.59
11 12 13 14 15 16 17 18 19 20
1.89 1.66 1.61 1.67 1.79 1.67 1.64 1.63 1.56 1.86
62166
67
X6
1 2 3 4 5 6 7 8 9 10
1.585 1.722 1.939 2.000 1.907 2.000 1.845 1.585 1.459 1.700
1.700 1.599 1.258 1.858 1.379 1.807 1.477 1.663 1.585 1.387
1.373 1.269 1.532 1.727 1.509 1.652 1.512 1.212 1.656 1.452
11 12 13 14 15 16 17 18 19 20
1.536 1.290 1.737 1.807 1.737 1.807 1.615 1.585 1.700 2.000
1.551 1.585 1.350 1.652 1.632 1.478 1.707 1.415 1.300 1.544
1.455 1.635 1.372 1.268 1.386 1.300 1.244 1.445 1.459 1.543
X7
1 2 3 4 5 6 7 8 9 10
155 527 225 32 145 20 1150 168 152 163
%1.53 %2.18 %1.41 %0.33 %1.30 %0.45 %1.49 %1.90 %1.46 %1.75
1.55 1.53 1.57 1.86 1.59 1.82 1.61 1.49 1.57 1.51
11 12 13 14 15 16 17 18 19 20
535 371 252 75 222 160 645 380 168 90
%2.29 %1.80 %1.90 %1.41 %1.37 %1.66 %1.70 %1.93 %1.90 %0.88
1.51 1.50 1.49 1.58 1.59 1.53 1.52 1.48 1.49 1.70
(8) Major whole 7 8.4
(7) Minor whole 5 ¼ 6.3
(6) Major part 4 4.8
(5) Minor part 3 3.6
(4) Major piece ¼2 2.7
(3)Minor piece¾ 1 2.1
(2) Major element¼1 1.5
(1) Minor element1 1.2
6 2.3 6.8 3.5 2.4 1.9 3 2.6 1.6 4 6.3 3 3 4 1.6
2 7.8 6.7 7.2 6.8 2.6 3.6 9.6 4.5 6.3 9.6 6.3
3 3.4 3.6 6.4 6.5 4.8 2 3.6 6.9 3.4 3.4 4.8 6.4
4 6.9 3.9 3.6 2.5 7 3.6 3.6 7 6.9
5 2 6.9 6.4 4.3 4 6.6 2.6 6.2 4.3 4.3 4.3 6.2
6 3 1.8 1.7 6 8 2.3 1.7 1.7 3 1.7
7 7.5 4.7 3.7 6.8 2.2 6.8 2.2 6.8 8.2 4.6 4.8 6.8 2.2 6.4 6.8 2.2 7.2 6.8 8.2 6.8
8 2.7 5.4 1.6 4.6 6.3 6.6 6.6 4.5 6.6 5.4 1.6
9 4.7 6.6 3.6 4.7 4.6 4.7 6.6 3.6
61 3 3.8 6.6 4.6 6.6 2.5 6 6 4.6 6.6
66 2.6 6.7 3.9 3.5 2.8 6.3 2.8 2.8 6.7 2.6
62 4.6 5.5 6.2 4.6 6.2 2.6 6.7 4.6 2 3.2 3.3 4.6 5.5 6.2
63 2 2.6 6 6.4 2.6 3.3 2.4 3.5 2 4 6.9 2 4 6
64 6 2.9 6 6.9 3.4 2.2 6 3.4 6
65 2.3 4.5 6.3 4 4.8 5.5 6.4 4 4 6.4 6.3
66 6.6 2 3.3 5.4 3.3 3.3 6.6 2
68 62166
1.6
4
1.2
1.1
4.1
6.1