Top Banner
ATHS THE MAGAZINE OF MATHEMATICS December2008, VOL.4 NO. 11 |SSUE4T
4

สามเหลี่ยม Sierpinski (MyMaths47)

Apr 10, 2015

Download

Documents

บทความตีพิมพ์ในนิตยสาร My Maths 47 เรื่องสามเหลี่ยม Sierpinski กับ เกมแห่งความยุ่งเหยิง และ เซลลูลาร์ออโตเมตอน
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: สามเหลี่ยม Sierpinski (MyMaths47)

ATHSTHE MAGAZINE OF MATHEMATICS December2008, VOL.4 NO. 11 |SSUE4T

Page 2: สามเหลี่ยม Sierpinski (MyMaths47)

6a B.Ene. (Dl{trori6 Ensin€rins). [,r.Ens. (Midelccirode Eneimc.ins)

t

f,lrtfifisu $ierpiriskiINUIIhDRC'IUTJDIh€JD

r f

tuaaals oellc1tuc10urr.rni6ar.r Sierpifski tiantasr-D.r Wacta-

Sierpiriski (o.cl. rssz-rsos; finofiooraof

ttrl!*auri {r::ar er;: Jfinuaita.:ar:r $rd u$rrrioq:: cjd

riluouu:n raaralmairiff e.rFi6ulr sierpirski'16lrlu16lqrnar iJ rv6 ur.r 6ru[v]'l

(1) tlt t"uorJ non I n nt :1d i6l t N tlld ur.l

o11,t[vt'lYl.t 3 fl]1-tt1]o1un t5]qglo

n'ltltl46lti.totutlnl1tJill,tlolan 4 fi ]

alnlacrdlu11,ht u915.Jfl 6'tnr4.rA

127 6ou it lrJi tr j i ,r i1d r.v6r erRci rr lny

t,l6?1 1o41fl na511,t6u'toi.tlt 2-4 U.dololl,l

J .q!- )n*!?unro5nurnlir hiliodr lraur.rlFn 3 :1]oirfi z rfio o ;rl o*rfi r; rfio zz ;n o'90;i " ,no :,, grlw a e , t .n-rdlrJtracle.lFl11.r01l9r.isrucr'l'toT1|av l r'}{']Er fi'tuniiD")u-

nllol{L"Snlnori rtu"l4Aftt."dnF uo r1,"rl: vu1t o5\jyt 22

" 7 z).1 ldA. l^U'U ?O LAi YLr ' or 'v 3 O Ueila

l8

, 3) A .y rer oiifr "

acfifllareJua6ererr6n 1 :' :rl rroinvr:]fi

.1ct'llerBl'lotl-ld, _ 141{tel

, )e) ,q-ln€l!'li.l5'11d]&$lAUrlO'lULfl lYlU'llOlUAC a q9tl

"" - . , . t i " ""- - , .t\l,$a[Yt_t] , onuuvluvll'utl a 1d teJ l11d ur.lu aul1 :,]ite

X )A" ) q-(fitlYldorflaidlull.16Uu Sieryjmki) vl[Oa]n

l , r l - lI t : t . / r1 lhm lr ' 'z ' ' l : l im:r3

I I u . ,4n' It l

i ! i y i , lr^ o . t 'hF 6 f fLfr ' , lc sr \ iqpra.Li tv f i r 0 1i c ,14-^,

d-€J$1AUtl SiFrpidski tlllllll? I fl& qlu:r& rn{ 1.58496

!..rsr _tgflarrd rv ru:Jy6 t l&) L.ie.ra.ryu. )!-u l11l [Ln 5

dTl'fl]&ffdl& SierniDski .11,[rtlln hO lLd]tl'L.il ilO

0 luqllld'tl,'lU l lta1lnt tll ! nn l nl: e lir t}.Ll5ale LA:,D, i z< !

6n[q11 fl'tofltt]alut1ri!ou[6u!nt.]'t{l:.luo-.i!!tn '.]lltu1-[v ; ; i

- -i!?ai tolnuoS!11€ll.'[ay u]ou'tfl [50d.11,lollltvu quq-]

:q )e4 "- " , \ ,x^qlu ralluuau nfl11r.[,!ejun]6.16!uwu16l0 uJuea tn llt{

rnu{rr Es (chaos calne)

no{0n

MY MAIHS D**b., ,oos 39

Page 3: สามเหลี่ยม Sierpinski (MyMaths47)

! " ) ,1-Ul 1 AO t|[ A1]!!1,11{.innnoNi9v1liAOt.

- ^4, \ . q ;

sN6r nr ao rJd?ld!to qo tr&Ytlll

Acl43n a-l gl tll1,t iO',1 lGlUd lV lll:dU .

s ld t l l tx t { t4r . r !a0nd uAt io

rfi.r r-au 'l a

i . q. ^

a v^1r.GlL ll4trl"llF LqO:.}.ta ld It$4nUtlO ll'YllOlUA0

3 1v -oie: {o"16' I F cs. 6 . r.g i 6, r xi':ti { -fi d&q.! : ,q- :"-

" io l t " " l lL L iOvLUlLf ld! . f lQ"? i6 l u,vd'uutr t i r l r

l "J clLcld 'SFp,l\ tn.lll,iosl'FlL .1311 rl Sl1-{0 d lfill dN

[c" 1r,L)di f {L.e 1. ta i J. x"r . 'x-} 1 lA' d.1] " 1L e) '

, l q. 4 Y - -Li ln L"Xdn.9l l Ln. t t ! ]U6{LOl .1U) Ll l 'U dd: ' 2 1111

vrou.isr;rfi16' ftIdfio : fi}iqodrafi '

hold onl

for n : 115000

dice : rand(l)

if dice<1/3

dot : (x seed)/2+seed

elseif dice>2/3

doi : (z seed)/2+s€edi

else dot = (y-seed)/2+sed!

plot(doi(1) dot(2),'.')i

,z - , ) )sJAV lto .c]!:laifl 6.inrc!'runrrdi..r1'l[14?u.iao lv

l:tnarfiuriuaurv6ur Sierpinski d&eJ6'i1aollrfl f; fl d:lq" ,q , i ; , ]

- - iq , -

tYOql[L Ul 6{ e[vdL&Cfl ]q'11'i1'd1-|1ltvl.]lYldo i61G1dq, i^.

lllUO'.O iO n.!,i l L I lllonn ! ll,lwll?l dlJ lX"t rl l di Ur lM d Ul.lvel

or.,r.;nn,u :ri'lo:rivfi :lu,r rrrilu r.craofio1' nnni q ' q Z ;^ i i4(u'}r.r tt4 qlnal tul,,{uyrdlr'r,)jrandleJt?aElfl Ylt!111-l'tolafra.: [l

d_?rl tof l lvL' lu 3 ?Uirr q

?UflN1l" lc lv]ru ml!J1{

^ -nq. ^. \A - , : -' |lF& [v r U 'aof O u] trqn tL14t"et t

_ nflllt tl-nttt d u 6l ttsl

i t?t ' { vdtot : lutola ' uvcf l r l l l6 l t . ln ldF ur i ldrd. li q ' r 2

r Luel t l l t lLnl 1 do L ?t t 'cn[v ru l Fr loud- rJ.udutr

Sidpiriski ldlaEr) lrqirrqr rl:niu1odltJu16a firmfi r6nA 4 a ,

-X6.1'l l-trlrot :.riio-r1qqnonl'.lounofl .rdr'r'iq'lgl.rt.r'rc.l1{

.t'1t1.14.r" 3l S'Frpinqki tr.8,ll llv rl! ll OiVd'61'lleJl-lJil_

drfion:nrur r{rD,A)\ZvO l!a'l !!AAl{!nAU l!oiol-l 14ulqllan Sierpifski

nt1ldul,lUl tn1{

a = [50*cosd(30)io].b': [100*cosd(30)i50*sind(30)]ic : [100*cosd(30),50+50*sind(30)];d = I50*cosd(30);50+100*sind(30)lie : [0i50+50*sind(30)]if: [0i50*sind(30)]iseed : rmd(2,1)*100,

for D : 1:5000

/ \

fir"lirr u o.rodr,rf :ll.rvhu

ortJrrflunofirir 1 drrvfu 1 uo1) ) ;d lrti-11!!nd!tlll,l lractal rr

ar rrrr; rfir arr rlr6 ua.r

Sierpinskil

{riuuvroaarH MATLAB .i.r6ain,.rd fi'rlo6a.r

oarfi rrora*to':qru{drufiIllruflru MATLAB Aodu}rl

irw*or 1 fir+'ldraal 1ar:.n:orfi ruur::dnnaifirdfiuqi

6rdt Lw.Fotc r[t]5o1rtrlr 'riJ lllllot rli |!ev6ltrh0'1,lou@l

fr16; Grarnriuuono 31o x(0,0), y(50,100) !ra!

z(100,100)

" : 10,01,y: [50i100] l

z: [100;0];

[rr9 di.,unl- L" n $.,] x{l Lt'le \3,1d i c],.r.1,1d ueJo'utv l

qr.r 16 on1qfl{q al]ur.i'r 11f)

seed : rand(2,1)*100i

l,dr lC 5.n00 6:: hltal,l{)t !14odtl1^ r.sldlCiJQ Ull-lit:

tLd y l0.r l !61..r j ' r ix" 3 n4 60 a aqltL{ l1 t0. , /3). b

o / 2 l q r , I ",rr "uri"r | . |""a::. oL.u"lrr : | 1rid f i t tronfl l l

' \3 3t " 13

40 MY MATHS DEmbef 2oo3

Page 4: สามเหลี่ยม Sierpinski (MyMaths47)

dice : rmd(l),

iJ dicc<t/6

dot : 2*(a-seed)/3+s.cdi

erseif (dice>1/6) L (dice<2/6)

dot : 2+(b seed)/3+seed;

clseif (dice>2/6) & (dicc<3/6 )doi : 2*(c se€d)/3 + seed,

elseif (dice>3/6) & (dice<a/6 )dot : 2*(d-seed)/3+seed,

elseif (dice>4/6) & (dice<5/6 )dot : 2*(e s€€d)/3 +seed;

elseif (dice>5/6)

dot : 2*(f seed)/3+seed;

plot(dot(1) dot(2),r.),

ud'r"r I a: 'L ' t [ ' r . ,v ' r ) rL ' .1r t ' . : 's NlvF, I

' Iorarqor.rsierpi!sl<i [orrd r nr5[dun!l{6c61t 00

(Cellular Automaionl fr iSorurnrrtf luarr tvF lr

sp-. 'o. , r 16ufrn" iu"r . | l "Ddda i rc" l u,rr6o i

finuru:,rff urqad vfiio'no.rrtadorarilugtJn:i'lofr'16' urqi

i_rlr nr6.r1ionir{aarqa I nulinlcr 6J lqaa tdaa?l.i'ru?r

^a ^ : i6JtF?C.udir d[y6rU3,vL " i1,1rd Udl f l l lL 2 r ,9 sel

9'l'l-1.i Fllll 'ooLsll qau' (Autonaton) ul.l'lufl.l vl-lEl ql

14t '3I I c11 t . la!r6rLC)lAt t { r .11 t6 i l t l r [XlYl1,{6ldtn

-. idl lr .!Lhl. p r'flt1.i rlt i utM-i 1,1 6 11{1-ltl.l d$l6 a l

r .9 lAda t p p |Gl"r .GrpL |1," f l ro cr(J ldol l4 ld lYlul I ] l l

OOIJl'!1lqlOl llO0Ul'1.i.:lUrqOOA!?1n16iA!lO'lO'),)olurnl:5 , l^

"1. :' . lc .l-9 dL1 8,d Utr.{: .1lLtA'eF (l-116 U tdFFF

5 i : i ,d'l4.eq,rx-t"urs d r..odu Iq50 lu!rL:i iu" .{":.F1,

u!r.r,iraailv!!141i6iirar-ririruviru"loau'lqdnl9r rfi u

t6al16atnrfil6a A New Kind of science 1o.t Stephen

wolfrm) r{64615 oalq!}]qa1{ lu!]n5fl!6lrr]r-n6{fln

i !6n:r!r: Eu lovfit ld' i ie.u r "itf lr."

rtvFua.r

biFfp,nsK, , r ,9te )erLr v u: ! !

. r r .ouO:Lo 1l l 32 ̂ ,12 !Ol l fd l6{LO !&F l [Yl ]n!

r riarfr u,r6ofi cirwirir o uavrirvuono6:.rdufrriarqlll 'fiel onSu

r' rtr1tno (i, i)

stltttiuxtnu x-y ivll

siruvli.!1io.: fi €irltgdr

1io.J s(i, j) lir6'!rnri

116r.lulntta3F l,t Ll 1,141,

1lu (stcpl ' i jnoanl l

r 6 ! udrr o.:r ia{ qrr.. l

n4 r ?run4lanlitoa

s,( i , j) : s, r( i , j)+

q l i I i \+q / i i r i /mft i r l

x .4ynfl10.!lg!!ul,\o to.tlcl'.]'tf o11lo.t1o.t to I ait

J-"111,1'tlq.tutn1tL l, tl| n! r€114r !oai nfttl,t'tt 6l't8.l1to.i o1

1?l l | "u ta l l lLLl1- -dl" t re!vLeYl l l { L C4 '11U1)ql ' l tqe :

-nautm-{t (t-1) o':t6an fitntl1l,to lvnonN9ru r : {li "

6lr[r"^rar1a0$,!a]5ruaq?ilfl t : 1, 2,3 qlr&aro!

l * lo t = 31 YIA

r lhou l l , lsr l r r

t1a.l lnltul 31 1r1,r

46 fi14 tOfiffOlO3

?11o tia'tua-tq

aastau lo Ft t l aF u

na 11!'rrl]u s.(i. j)

: s, , ( i - 1, j )+

s. ,(i, j t) (nod 2)

6lnto {114L]]" tpr | 3 l 5. .1udlr i t1 l?11411rO TAUULLL

dr:..r r14ff u:.r Sierpinsk' lriufru +

MY MATHS D-".h., roffi 4l