Fractal Geometry and Architecture Design Case Review Study

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Chaotic Modeling and Simulation (CMSIM) 2: 311-322, 2012

_________________

Received: 29 June 2011 / Accepted 30 March 2012

© 2012 CMSIM ISSN 2241-003

Fractal Geometry and Architecture Design: Case

Study Review

Xiaoshu Lu1,

, Dere! Clements"Croome#, $artti %il&anen

1

1!epart"ent #$ Civi% and Structura% &n'ineerin'( Sch##% #$ &n'ineerin'( Aa%t#

)niver*it+( , .# 12100( IN-0210( &*p##( in%and

&-"ai%: ia#*hucchut$i 2inni*h In*titute #$ ccupati#na% ea%th( in%and

3Sch##% #$ C#n*tructi#n Mana'e"ent and &n'ineerin'( hite5ni'ht*(

)niver*it+ #$ Readin'( , .# 219( Readin' R67 7A( )8

A'stract: he idea #$ ui%din'* in har"#n+ ;ith nature can e traced ac5 t# ancientti"e* he increa*in' c#ncern* #n *u*tainai%it+ #riented ui%din'* have added ne;cha%%en'e* in ui%din' architecturera% de*i'n and ca%%ed $#r ne; de*i'n re*p#n*e*

Su*taina%e de*i'n inte'rate* and a%ance* the hu"an 'e#"etrie* and the natura% #ne*

A* the %an'ua'e #$ nature( it i*( there$#re( natura% t# a**u"e that $racta% 'e#"etr+ c#u%d p%a+ a r#%e in deve%#pin' ne; $#r"* #$ ae*thetic* and *u*taina%e architecturera% de*i'n

hi* paper 'ive* a rie$ de*cripti#n #$ $racta% 'e#"etr+ the#r+ and pre*ent* it* current*tatu* and recent deve%#p"ent* thr#u'h i%%u*trative revie; #$ *#"e $racta% ca*e *tudie* inarchitecture de*i'n( ;hich pr#vide* a rid'e et;een $racta% 'e#"etr+ and architecture

de*i'n

(eywords: racta% 'e#"etr+( Architecture de*i'n( Su*tainai%it+

1) *ntroductionhe idea #$ ui%din'* in har"#n+ ;ith nature can e traced ac5 t# ancient

&'+ptian*( China( 6ree5* and R#"an* At the e'innin' #$ 21*t centur+( the

increa*in' c#ncern* #n *u*tainai%it+ #riented #n ui%din'* have added ne;

cha%%en'e* in ui%din' architecture de*i'n and ca%%ed $#r ne; de*i'n re*p#n*e*A* the %an'ua'e #$ nature <1(2=( it i*( there$#re( natura% t# a**u"e that $racta%

'e#"etr+ c#u%d p%a+ a r#%e in deve%#pin' ne; $#r"* #$ de*i'n #$ *u*taina%e

architecture and ui%din'*

racta%* are *e%$-*i"i%ar *et* ;h#*e pattern* are c#"p#*ed #$ *"a%%er-*ca%e*

c#pied #$ the"*e%ve*( p#**e**in' *e%$-*i"i%arit+ acr#** *ca%e* hi* "ean* thatthe+ repeat the pattern* t# an in$inite%+ *"a%% *ca%e A pattern ;ith a hi'her



>u et a%312

$racta% di"en*i#n i* "#re c#"p%icated #r irre'u%ar than the #ne ;ith a %#;er

di"en*i#n( and $i%%* "#re *pace In "an+ practica% app%icati#n*( te"p#ra% and

*patia% ana%+*i* i* needed t# characteri*e and ?uanti$+ the hidden #rder in

c#"p%e pattern*( $racta% 'e#"etr+ i* an appr#priate t##% $#r inve*ti'atin' *uch

c#"p%eit+ #ver "an+ *ca%e* $#r natura% phen#"ena <2(3= rder in irre'u%ar

pattern i* i"p#rtant in ae*thetic* a* it e"race* the c#ncept #$ d+na"ic $#rce(

;hich *h#;* a natura% phen#"en#n rather than "echanica% pr#ce** In

architecture de*i'n ter"*( it repre*ent* de*i'n princip%e here$#re( $racta%

'e#"etr+ ha* p%a+ed a *i'ni$icant r#%e in architecturera% de*i'n

In *pite #$ it* 'r#;in' app%icati#n*( *uch ;#r5* in %iterature are rather narr#;(

ie the+ "ain%+ $#cu* #n app%icati#n* $#r $racta% de*i'n pattern* #n ae*thetic

c#n*iderati#n* e; ;#r5* have re%ated t# a c#"prehen*ive and uni$ied vie; #$$racta% 'e#"etr+ in *tructura% de*i'n( $#r ea"p%e( a* it i* intended in thi* *tud+

e ai" t# $i%% in thi* 'ap + intr#ducin' $racta%* a* ne; c#ncept* and pre*entin'

it* current *tatu* and recent deve%#p"ent* in architecture thr#u'h an i%%u*trative

revie; #$ *#"e $racta% ca*e *tudie* in de*i'n he paper *h#;* that

inc#rp#ratin' the $racta% ;a+ #$ thin5in' int# the architecture de*i'n pr#vide* a

%an'ua'e $#r an in-depth under*tandin' #$ c#"p%e nature #$ architectura%

de*i'n in 'enera% hi* *tud+ di*ti%* the $unda"enta% pr#pertie* and the "#*t

re%evant characteri*tic* #$ $racta% 'e#"etr+ e**entia% t# architect* and ui%din'

*cienti*t*( initiate* a dia%#'ue and ui%d* rid'e* et;een *cienti*t* and

architect*

) +asic heory o- Fractal Geometry

)1) +asic heoryhe "athe"atica% hi*t#r+ #$ $racta%* e'an ;ith "athe"atician 8ar% eier*tra**

in 1@2 ;h# intr#duced a eier*tra** $uncti#n ;hich i* c#ntinu#u* ever+;here

ut di$$erentia%e n#;here <4= In 1904 e%'e v#n 8#ch re$ined the de$initi#n #$

the eier*tra** $uncti#n and 'ave a "#re 'e#"etric de$initi#n #$ a *i"i%ar

$uncti#n( ;hich i* n#; ca%%ed the 8#ch *n#;$%a5e <=( *ee i'ure 1 In 191(

ac%a; Sie%pin*5i c#n*tructed *e%$-*i"i%ar pattern* and the $uncti#n* that

'enerate the" 6e#r' Cant#r a%*# 'ave an ea"p%e #$ a *e%$-*i"i%ar $racta% <7=

In the %ate 19th and ear%+ 20th( $racta%* ;ere put $urther + enri ,#incare( e%i

8%ein( ,ierre at#u and 6a*t#n Ju%ia In 19( Mande%r#t r#u'ht the*e ;#r5

t#'ether and na"ed it B$racta%B



Cha#tic M#de%in' and Si"u%ati#n CMSIMD 2: 311-322( 2012 313

i' 1 I%%u*trati#n #$ 8#ch Curve

racta%* can e c#n*tructed thr#u'h %i"it* #$ iterative *che"e* inv#%vin'

'enerat#r* #$ iterative $uncti#n* #n "etric *pace* <2= Iterated uncti#n S+*te"

ISD i* the "#*t c#""#n( 'enera% and p#;er$u% "athe"atica% t##% that can e

u*ed t# 'enerate $racta%* M#re#ver( IS pr#vide* a c#nnecti#n et;een $racta%*

and natura% i"a'e* <(@= It i* a%*# an i"p#rtant t##% $#r inve*ti'atin' $racta%

*et* In the $#%%#;in'( an intr#ducti#n t# *#"e a*ic 'e#"etr+ #$ $racta% *et* ;i%%

e appr#ached $r#" an IS per*pective In a *i"p%e ca*e( IS act* #n a *e'"ent

t# 'enerate c#ntracted c#pie* #$ the *e'"ent ;hich can e arran'ed in a p%ane

a*ed #n certain ru%e* he iterati#n pr#cedure "u*t c#nver'e t# 'et the $racta%

*et here$#re( the iterated $uncti#n* are %i"ited t# *trict c#ntracti#n* ;ith the

.anach $ied-p#int pr#pert+

>et E( dD den#te* a c#"p%ete "etric *pace and H X D the c#"pact *u*et* #$ X (

the au*d#r$$ di*tance i* de$ined a*

h A,BD F "aGd A,BD( d B,ADH ∀ A, B∈ H X D 1D

It i* ea*+ t# pr#ve that h i* a "etric #n H X D M#re#ver( it can e pr#ved that

H X D( hD i* a%*# a c#"p%ete "etric *pace <= ;hich i* ca%%ed the *pace #$ $racta%*

$#r X

A c#ntracti#n "appin'( #r c#ntracti#n w: X X α ha* the pr#pert+ that there i*

*#"e n#nne'ative rea% nu"er k ∈<0(1D( c#ntracti#n $act#r k ( *uch that

d w xD( w yDD ≤ k d x( yD 2D

In n#n-technica% ter"*( a c#ntracti#n "appin' rin'* ever+ t;# p#int* c%#*er in

the "etric *pace it "ap* he .anach $ied p#int the#re" 'uarantee* the

ei*tence and uni?uene** #$ $ied p#int* #$ c#ntract "ap* #n "etric *pace*: I$

w: X X α i* a c#ntracti#n( then there ei*t* #ne and #n%+ #ne x ∈ X *uch that

w xD F x



>u et a%314

he .anach $ied p#int the#re" ha* ver+ i"p#rtant app%icati#n* in "an+

ranche* #$ "athe"atic* here$#re( 'enera%i*ati#n #$ the a#ve the#re" ha*

een eten*ive%+ inve*ti'ated( $#r ea"p%e( in pr#ai%i*tic "etric *pace* he

the#re" a%*# pr#vide* a c#n*tructive "eth#d t# $ind $ied-p#int

An IS <9= i* a *et #$ c#ntracti#n "appin'* wi de$ined #n X ( d D ;ith c#ntracti#n

$act#r* k i $#r i F 1(2(( N e den#te it a*G X wi( i F 1(2(( N H and c#ntracti#n

$act#r k F "aGk i( i F 1( 2(( N H utchin*#n <9= pr#ved an i"p#rtant the#re"

#n a *et #$ c#ntracti#n "appin'* in ;hich IS i* a*ed: >et G X wi( i F 1(2((

N H e an IS ;ith c#ntracti#n $act#r k hen W : DD X H X H α de$ined a*

W BD . Υ N

i

i Bw1

D=

∀ D X H B ∈ 3D

i* a c#ntracti#n "appin' #n H X D( hD r#" .anachB* the#re"( there ei*t* a

uni?ue *et D X H A ∈ ( the attract#r #$ IS( *uch that

A F W AD F Υ N

i

i Aw1

D=

4D

It can e *een that A i* *e%$-*i"i%ar *ince it i* epre**ed a* a uni#n #$

tran*$#r"ati#n* c#pie*D #$ it*e%$ he attract#r A can e ta5en a* a de$initi#n #$

deter"ini*tic $racta%*

)) Fractal dimensions Mande%r#t <2= pr#p#*ed a *i"p%e ut radica% ;a+ t# ?ua%i$+ $racta% 'e#"etr+

thr#u'h a $racta% di"en*i#n a*ed #n a di*cu**i#n #$ the %en'th #$ the c#a*t #$

&n'%and he di"en*i#n i* a *tati*tica% ?uantit+ that 'ive* an indicati#n #$ h#;

c#"p%ete%+ a $racta% appear* t# $i%% *pace( a* #ne K##"* d#;n t# $iner *ca%e*

hi* de$initi#n i* a *i"p%i$icati#n #$ the au*d#r$$ di"en*i#n that Mande%r#tu*ed a* a a*i* e $#cu* #n thi* #ne and rie$%+ "enti#n* #-c#untin'

di"en*i#n ecau*e #$ it* ;ide practica% app%icati#n* #;ever( it *h#u%d e

n#ted that there are "an+ *peci$ic de$initi#n* #$ $racta% di"en*i#n*( *uch a*

au*d#r$$ di"en*i#n( RLn+i di"en*i#n*( #-c#untin' di"en*i#n and

c#rre%ati#n di"en*i#n( etc( and n#ne #$ the" *h#u%d e treated a* the univer*a%

#ne

#r D X H A ∈ ( %et n A(ε D( ε0( den#te the *"a%%e*t nu"er #$ c%#*ed a%%* #$

radiu* ε needed t# c#ver A I$




ε

ε

ε 1%#'

D(%#'%i"

0

An D

→= D

ei*t*( then D i* ca%%ed the $racta% di"en*i#n #$ A

S# n A(ε D i* pr#p#rti#na% t# ε -D

a* ε → 0 #r the ep#nent D i* in n A(ε D F ε -D

;hich i* the p#;er %a; re%ati#n*hip A p#;er %a; de*crie* a d+na"icre%ati#n*hip et;een t;# #ect* ;hich p#rtra+* a ;ide variet+ #$ natura% and

"an-"ade phen#"ena A 5e+ $eature #$ the p#;er %a; i* that the p#;er %a;

re%ati#n*hip i* independent #$ *ca%e* A '##d ea"p%e #$ intuiti#n #$ $racta%

di"en*i#n i* a %ine ;ith the %en'th #$ε

( ;hereε

i* the "ea*urin' %en'thA**u"e the %ine i* divided in 3 e?ua% part* and ε F

3

1 then the *i"p%i$ied

n A(ε DF ε -D

'ive* 3 F 1/3D- D

;ith D F 1 Si"i%ar%+( the 8#ch curveB* $racta%

di"en*i#n i* D F3%#'

4%#'F 127

,ractica%%+( the $racta% di"en*i#n can #n%+ e u*ed in the ca*e ;here

irre'u%aritie* t# e "ea*ured are in the c#ntinu#u* $#r" Natura% #ect* #$$er a

%#t #$ variati#n ;hich "a+ n#t e *e%$-*i"i%ar he .#-c#untin' di"en*i#n i*"uch "#re r#u*t "ea*ure ;hich i* ;ide%+ u*ed even t# "ea*ure i"a'e* #

ca%cu%ate the #-c#untin' di"en*i#n( ;e need t# p%ace the i"a'e #n a 'rid he

nu"er #$ #e*( ;ith *iKe s1( that c#ver the i"a'e i* c#unted n1D hen thenu"er #$ a *"a%%er 'rid #$ #e*( ;ith *iKe s2( i* c#unted n2D he $racta%

di"en*i#n et;een t;# *ca%e* i* then ca%cu%ated + the re%ati#n*hip et;een the

di$$erence #$ the nu"er #$ #ed #ccupied and the di$$erence #$ inver*e 'rid*iKe* <10= In "#re cha#tic and c#"p%e #ect* *uch a* architecture and de*i'n(

"#re $%ei%e and r#u*t "ea*ure*( *uch a* ran'e ana%+*i*( "idp#int

di*p%ace"ent( etc( can e e"p%#+ed #r "#re detai%ed in$#r"ati#n( reader* "a+

re$er t# .#vi%%B* ##5 <10=

)#) /0amles o- *FS alications racta% 'e#"etr+ i* at the c#nceptua% c#re #$ under*tandin' natureB* c#"p%eit+and IS pr#vide* an i"p#rtant c#ncept $#r under*tandin' the c#re de*i'n #$ the

natura% #ect* a* ;e%% a* appr#i"atin' the natura% de*i'n In thi* *u*ecti#n

;e #ut%ine the ev#%uti#n #$ the idea #$ IS ;ith #ur ca%cu%ati#n ea"p%e*

e 5n#; that the .anachB* $ied-p#int the#re" $#r"* the a*i* #$ the IS

app%icati#n* #;ever( app%+in' the the#re" in practiced rai*e* t;# centra%?ue*ti#n* ne i* t# $ind the attract#r $#r a 'iven IS he #ther i* t# $ind IS

$#r a 'iven attract#r( an inver*e pr#%e" #$ the $ir*t



>u et a%317

#r the $ir*t pr#%e"( the attract#r can e #tained + *ucce**ive

appr#i"ati#n* $r#" an+ *tartin' p#int the#retica%%+ r#" a c#"putati#na%

p#int #$ vie;( t;# techni?ue*( deter"ini*tic and *t#cha*tic( can e app%ied he

deter"ini*tic a%'#rith" *tart* ;ith an aritrar+ initia% *et t# reach the attract#r

he *t#cha*tic a%'#rith" i* #$ten "#re c#"p%e ut "#re e$$icient A *t#cha*tic

a%'#rith" attriute* t# the IS *+*te" ;ith a *et #$ pr#ai%itie* + a**i'nin' a

pr#ai%it+ t# each "appin'( ;hich i* u*ed t# 'enerate a rand#" ;a%5 I$ ;e

*tart ;ith an+ p#int and app%+ tran*$#r"ati#n* iterative%+( ch#*en acc#rdin' the

pr#ai%itie* attached( ;e ;i%% c#"e aritrari%+ c%#*e t# the attract#r he

a**#ciated pr#ai%itie* deter"ine the den*it+ #$ *patia%%+ c#ntracted c#pie* #$

the attract#r here$#re( the pr#ai%itie* have n# e$$ect #n the attract#r ut

in$%uence *i'ni$icant%+ the renderin' #$ it* appr#i"ati#n*

he *ec#nd pr#%e"( the inver*e pr#%e"( can e *#%ved + .arn*%e+B* C#%%a'e

he#re"( a *i"p%e c#n*e?uence #$ .anachO* $ied p#int the#re" Such

pr#cedure ;a* i%%u*trated nice%+ thr#u'h the B.arn*%e+ $ernB in <9= and <11= u*in'

$#ur-tran*$#r"ati#n IS ;ith a**#ciated pr#ai%itie* i'ure 2 *h#;* #ur

ca%cu%ati#n ea"p%e* #$ $racta%* u*in' $#ur-tran*$#r"ati#n IS ;ith variati#n*

and their a**#ciated pr#ai%itie* pr#duced + Mat%a( ;here 20000 iterati#n*

;ere *et he*e $racta%* actua%%+ have "#re than #ne attract#r In i'ure 2( the

$#ur-tran*$#r"ati#n "atrice* are

=

100

01@00

000

A

−

=

100

1230220

020190

B

−

=

100

44020240

02@010

C

−=

100

19000

004070

D 7D

! ha* the pr#ai%it+ 0 and #ther* 00@3

i' 2 Ca%cu%ati#n ea"p%e* #$ $racta%* u*in' IS ;ith variati#n*




#) Alications o- Fractal Geometry to Architecture Design

#)1) Alications o- *FSIS pr#vide* ;ide ran'e #$ architecture de*i'n app%icati#n* in pattern* and

*tructure* Per+ #$ten( IS c#de* are u*ed t# 'enerate $racta%* #r ea"p%e(t#p#%#'+ %a+#utD #pti"iKati#n ha* een pr#p#*ed and i* a*ed #n IS

repre*entati#n* ;ith vari#u* app%icati#n* <12= Chan' <13= pr#p#*ed a

hierarchica% $ied p#int-*earchin' a%'#rith" $#r deter"inin' the #ri'ina%

c##rdinate* #$ a 2-! $racta% *et direct%+ $r#" it* IS c#de he IS c#de cane e

"#di$ied acc#rdin' t# the de*ired tran*$#r"ati#n i'ure 3 *h#;* the Ca*t%e

;ith di$$erent re$%ecti#n directi#n* 'enerated + the "#di$ied IS c#de*

(

i' 3 Ca*t%e ea"p%e 'enerated + the "#di$ied IS c#de* <13=

#)) Alications o- -ractal geometryracta% 'e#"etr+ ha* een app%ied in architecture de*i'n ;ide%+ t# inve*ti'ate

$racta% *tructure* #$ citie* <14= and *ucce**$u%%+ in ui%din' 'e#"etr+ <1(17=

and de*i'n pattern* <10=

&ar%+ $racta% ui%din' pattern* can e traced t# ancient Ma+a *ett%e"ent* .r#;n

et a% ana%+*ed $racta% *tructure* #$ Ma+a *ett%e"ent* and $#und that $racta%*

ehiit #th ;ithin c#""unitie* and acr#** re'i#n* in vari#u* ;a+*: at the

intra-*ite( the re'i#na% %eve%* and ;ithin archae#%#'ica% *ite* M#re#ver( *patia%

#r'ani*ati#n in 'e#"etric pattern* and #rder are a%*# $racta%*( ;hich pre*ent* in

the *iKe-$re?uenc+ di*triuti#n( the ran5-*iKe re%ati#n a"#n' *ite* and the'e#'raphica% c%u*terin' #$ *ite* <1=

In &ur#pe( $racta%* ;ere $#und in the ear%+ 12th centur+ ui%din'* he $%##r #$

the cathedra% #$ Ana'ni in Ita%+ ui%t in 1104 i* ad#rned ;ith d#Ken* #$ "#*aic*

in a $#r" #$ a Sierpin*5i 'a*5et $racta% See i'ure 4D



>u et a%31@

i' 4 he $%##r #$ the cathedra% #$ Ana'ni in Ita%+ <1@=

racta%* have een app%ied t# "an+ e%evati#n *tructure* t# ec%u*ive%+ addre**

p#;er and a%ance S#"e ver+ ece%%ent ea"p%e* #$ c%a**ica% architecture can

e *een in "an+ part* #$ the &ur#pe( in the Midd%e &a*t and A*ia ;hich have

e$$ect* #$ $racta% e%evati#n*( $#r ea"p%e( Rei"*O cathedra% and Saint ,au%

church in rance( Ca*te% de% M#nte in Ita%+ and "an+ pa%ace* in Penice caO

#*cari( CaO dOr#( !u5e ,a%ace( 6iu*tinian ,a%aceD in Ita%+ Penice ha* een

#ne #$ the "#*t ta%5ed a#ut $racta%* <1@= *ee i'ure D M#re vita% evidence

*h#;* that $racta%* ei*t in 6#thic cathedra%* in 'enera% he p#inted arch( an

i"pre**i#n #$ e%evati#n( appear in entrance( at ;ind#;* and the c#*ta% arch ;ith

"an+ *ca%e* and detai%* <19= i'ure 7 di*p%a+* the e%evati#n* #$ a $ive-$%##rtene"ent ui%din' in the hi*t#rica% part #$ .arce%#na ;hich *h#;* *e%$-

#r'ani*ati#n *tructure

i' racta% ui%din' in Penice <1@=




i' 7 A tene"ent h#u*e in the hi*t#rica% part #$ .arce%#na( Spain: the

e%evati#nB* ph#t#'raph $r#" the 90-* %e$t part D the 'e#"etric *+nthe*i* *h#;*

the #ri'ina% architecture de*i'n "idd%e partD <20=

In the Midd%e &a*t( $racta% pattern* have een ad#pted ;ide%+ in de*i'nin'

*tucc#( a t+pica%%+ ,er*ian art $#r" $#r the dec#rati#n #$ d#"e interi#r* In

i'ure ( the pattern in the d#"e interi#r ha* $#ur attract#r* *urr#undin' the

"ain #ne at the center Sarhan'iD

i' Stucc# d#"e interi#r in a private h#u*e in 8a*han <21=



>u et a%320

In A*ia( architecture* ;ith $racta% *tructure* have a%*# een $#und in u"a+unO*

Mau*#%eu"( Shiva Shrine in India and the Sacred Stupa ,ha hat >uan' in

>a#* racta%* have een u*ed t# *tud+ indu te"p%e* In China( *#"e "#*?ue*

in the ;e*t ;ere "#re %i5e%+ t# inc#rp#rate *uch d#"e* ;hich are $racta%* ne

i"p#rtant $eature in Chine*e architecture i* it* e"pha*i* #n *+""etr+ ;hich

c#nve+* a *en*e #$ 'randeur <22=

.e*ide* 'e#'raphica% %#ca%itie*( in recent ti"e*( the c#ncept #$ $racta%* ha* een

etended t# "an+ ;e%% 5n#;n architecture* inc%udin' ran5 >%#+d ri'htO*

BR#ie #u*eB( Ba%%in';aterB( B,a%"er h#u*eB and BMari#n C#unt+ CivicB( ;hich

de"#n*trate that $racta%* have univer*a% appea% and are vi*ua%%+ *ati*$+in'

ecau*e the+ are a%e t# pr#vide a *en*e #$ *ca%e at di$$erent %eve%* ri'ht i* pr#a%+ the "#*t repre*entative #$ the #r'anic architect* i* de*i'n* 're; #ut

#$ the envir#n"ent ;ith re'ard* t# purp#*e( "ateria% and c#n*tructi#n <10=

racta%* have in*pired "an+ 'reat "#dern de*i'ner* *uch a* Qaha adid( !anie%

>iee*5ind( ran5 6ehr+ and #ther* ;ith "an+ n#ta%e $racta% architecture*

<20= Indeed( acc#rdin' t# Irahi" et a%( architect* and de*i'ner* *tarted t# ad#pt

$racta%* a* a de*i'n $#r" and t##% in the 19@0* <10= e**i#* et a% ;a* a"#n'

the $ir*t uti%i*in' $racta%* and $racta% 'e#"etr+ de*i'n in architecture <23= he+

deve%#ped a c#"puter pr#'ra" t# aid architecture u*in' $racta% 'enerat#r* In

1990th

( !ur"i*evic and Ci$tci#'%u app%ied $racta% tree a* an indicat#r #$ a r#ad

in$ra*tructure in the architecture de*i'n and uran p%annin' <24=

en et a% e*ta%i*hed the $racta% di"en*i#n re%ati#n* "atri ta%e ana%+*i* t#

c%a**i$+ architecture de*i'n *t+%e pattern* $#r the "a*terpiece* #$ three "#dernarchitecture "a*ter*: ran5 >i#+d ri'ht( >e C#ru*ier and Mie* van der R#he

<2= i'ure @ *h#;* the re*u%t* It can e *een that the te"p#ra% trend* #$

individua%* var+ he $racta% di"en*i#n* #$ ran5 >i#+d ri'ht are avera'e

;ith %#; e'innin' in the ear%+ 1900th

and end in the "id 1930th

he trend #$

>e C#ru*ier '#e* d#;n*ide ;ith 'ent%e *%#pe $r#" "id 1900th

t# "id 190th

#r the peri#d *h#;n in the 'rape( the trend #$ Mie* van der R#he ha* the *a"e

trend a* that #$ ran5 >i#+d ri'ht $r#" the ear%+ 1900th

t# the "id 1930th

he

avera'e trend #$ the*e three "a*ter* '#e* d#;n in 'enera% *tartin' $r#" 1930th




i' @ racta% di"en*i#n* $#r the "a*terpiece* #$ three "#dern architect*

2) Conclusionshi* paper ha* i%%u*trative%+ revie;ed the $unda"enta% c#ncept* and pr#pertie*

#$ $racta% 'e#"etr+ the#r+ e**entia% t# architecture de*i'n( a* ;e%% a* the current

*tate #$ it* app%icati#n* racta% 'e#"etr+ ha* i"p#rtant i"p%icati#n* $#r

ui%din'* he repre*entative revie; *h#;* that architecture de*i'n i* n#t "ade

t# e i*#%ated ut t# anticipate chan'e* in the envir#n"ent Accu"u%ati#n #$

techn#%#'ica% "#derni*ati#n*( de*tr#+in'( adaptin' and "an+ chan'e* have

cau*ed the de*i'n te"p#ra% and *patia% diver*it+ and c#"p%eit+ M#re

*peci$ica%%+( *u*taina%e deve%#p"ent in a ui%din' can e %##5ed up#n a*

adaptai%it+ and $%eii%it+ #ver ti"e ;hen it c#"e* t# re*p#ndin' t# chan'in'

envir#n"ent* Cha#* and "an+ #ther n#n%inear the#rie* have ep%ained that

etre"e%+ deter"ini*tic and %inear pr#ce**e* are ver+ $ra'i%e in "aintainin'

*tai%it+ #ver a ;ide ran'e #$ c#nditi#n*( ;herea* cha#tic and $racta% *+*te"*

can $uncti#n e$$ective%+ #ver a ;ide ran'e #$ di$$erent c#nditi#n*( there+

#$$erin' adaptai%it+ and $%eii%it+ In thi* c#ntet( $racta% 'e#"etr+ the#r+

#$$er* an a%ternative $#r *u*taina%e architectura% de*i'n hi* paper pr#vide* a

rid'e et;een ui%din' en'ineerin' and architecture and $racta% 'e#"etr+

the#r+

Re-erences1 . Mande%r#t Fractals, Form, Chance and dimension( ree"an( San ranci*c#(

192 .Mande%r#t The Fractal Geometry o Nat!re ree"an and C#"pan+( 19@2

3 ,A .urr#u'h racta% di"en*i#n* #$ %and*cape* and #ther envir#n"enta% data Nat!re

294: 240242( 19@1



>u et a%322

4 8 eier*tra**( n c#ntinu#u* $uncti#n* #$ a rea% ar'u"ent that d# n#t have a ;e%%-de$ined di$$erentia% ?u#tient( in 6 &d'ar( ed Classics on Fractals( Addi*#n-e*%e+(Readin'( Ma**achu*ett*( 3-9( 1993

v#n 8#ch n a c#ntinu#u* curve ;ith#ut tan'ent* c#n*tructi%e $r#" e%e"entar+'e#"etr+( in 6 &d'ar( ed Classics on Fractals( Addi*#n-e*%e+( Readin'(

Ma**achu*ett*( 2-4( 19937 6 Cant#r n the ,#;er #$ ,er$ect Set* #$ ,#int s in 6 &d'ar( ed Classics on

Fractals( Addi*#n-e*%e+( Readin'( Ma**achu*ett*( 11-23( 1993 M .arn*%e+ Fractals "#erywhere Acade"ic ,re**( 19@@@ ,eit'en19@@ anta*tic !eter"ini*tic racta%*( in: ,eit'en( ! Saupe The

$cience o Fractal %ma&es( 202( Sprin'er( Ne; #r5( 19@@

9 J& utchin*#n racta%* and *e%$ *i"i%arit+ %ndiana 'ni#( )ath( *( 30: 134(19@1

10 C .#vi%% Fractal Geometry in Architect!re and Desi&n( .ir5hau*er( .#*t#n( 199711 M &rahi"i( &R Pr*ca+ Se%$-*i"i%arit+ in i"a'in'( 20 +ear* a$ter Tracta%*

&ver+;hereT( http://tic*pc*tut$i/i"a'e*/3/3$/Cr1023-%au*annepd$

12 Chan' Aritrar+ a$$ine tran*$#r"ati#n and their c#"p#*iti#n e$$ect* $#r t;#-di"en*i#na% $racta% *et* %ma&e and +ision Com!tin& 22: 111-112( 2004

13 a"da( J#uve( & >utt#n( M Sch#enauer( M Sea'( C#"pact un*tructuredrepre*entati#n* $#r ev#%uti#nar+ t#p#%#'ica% #pti"u" de*i'n Al %ntell 17: 139-

1( 200214 M .att+( , >#n'%e+ Fractal Cities( >#nd#n: Acade"ic ,re**( 19941 8 rivedi indu te"p%e*: "#de%* #$ a $racta% univer*e The +is!al Com!ter : 243-

2@( 19@9

17 N Sa%a he pre*ence #$ the Se%$-*i"i%arit+ in Architecture: *#"e ea"p%e*

&"er'ent Nature - ,attern*( 6r#;th and Sca%in' in the Science*( #r%d Scienti$ic23-2@2( 2002

1 C .r#;n( R it*che+ he $racta% 'e#"etr+ #$ ancient Ma+a *ett%e"ent

*o!rnal o Archaeolo&ical $cience 30: 1719-1732( 20031@ N Sa%a racta% "#de%* in architecture: a ca*e #$ *tud+

http://"athunipait/U'ri"/J*a%a;#r5*h#p,!19 & >#renK Fractals and Fractal Architect!re( Mater he*i*( Pienna( 2003

20 , Ruin#;icK Cha#* and 'e#"etric #rder in architecture and de*i'n *o!rnal orGeometry and Grahics 4: 19-20( 2000

21 http://en;i5ipedia#r'/;i5i/I*%a"ic_architecture

22 JS C#;en Mu*%i"* in China: he "#*?ue $a!di Aramco World( etrie#ed .//0-

/1-/2( 30-3( 19@23 CI e**i#* A $racta% *tudi# ACAD%A 324 Worksho 5roceedin&s( 19@24 S !ur"i*evic( Ci$tci#'%u racta%* in architectura% de*i'n )athematics and

Desi&n Javier .arra%%# the univer*it+ #$ the .a*?ue C#untr+( 19@@

2 8-C en( -N 8a# An ana%+tic *tud+ #$ architectura% de*i'n *t+%e + $racta%di"en*i#n "eth#d( ..nd %nternational $ymosi!m on A!tomation and o6otics in

Constr!ction %$AC .//7( Ita%+( 200(http://;;;iaarc#r'/pu%icati#n*/$u%%tet/i*arc200-73;enpd$

Fractal Geometry and Architecture Design Case Review Study

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