(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

62166

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

Mathematical Society , Providence , Rhode

Island , 2004 . Volume 72, Part 2, p.537.

2. Weinstein, Eric W.Concise Encyclopedia of

Mathematics, CD-ROM edition 1.0, Chapman &

Hall/CRCnetBASE.

3. Batty and Longley, 1994. Fractal Cities,

Academic Press Inc., ISBN 0-12-4555-70-5, p.3.

4. Mandelbrot, Benoit B., 1983. The Fractal

Geometry of Nature. New York: Freeman, pp.23-

24.

5. Bovill, Carl, 1996. Fractal geometry in

architecture and design , school of architecture ,

University of Maryland , Birkhäuser Boston,

ISBN 3-7643-3795-8,p.2.

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

architecture and design, Ibid, p.43.

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.

New York: John Wiley & Sons, pp. 61-85.

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

fractal City" Keynote speech, 5th Biennial of

towns and town planners in Europe Barcelona,

http://www.math.utsa.edu/sphere/salingar/Lifean

dComp.html .

15. Eglash,Ron, 1999. African Fractals: Modern

Computing and Indigenous Design' New

Brunswick, New Jersey: Rutgers University

Press.

16. Lorenz Wolfgang E., 2003. Fractal and Fractal

Architecture, Department of computer aided

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

Architecture), 2001, Three projects: Dust, Plates

& 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

architecture and design, Ibid, pp.103-175.

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.

24. Musgrave, 1999. F. Kenton, Fractal Forgeries of

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