¸H§Î

¡]¥»³æ¤¸½d¨Òµ{¦¡¥þ¥Ñ§õ©ú¾Ë¦Ñ®v¼¶¼g¡A¨Ñ¦U¦ì¦P¾Ç°Ñ¦Ò¡^

 

¸H§Î¬O¤°»ò¡H

¸H§Î¬O¥Î¨ÓºÙ¤@ºØ¯S®íªº´X¦ó¹Ï§Î¡A¥¦³B³Bªº§½³¡³£¦³µÛ¤@¼Ëªº¯S¼xµ²ºc©Î¹Ï®×¡C¥Ñ©ó¹q¸£ªº»²§U¡C

¸H§Î±`¨£©ó¤£Â_­«ÂЭ¡¥N¤@­Ó¦¬ÁY©Êªº¨ç¼Æªº·¥­­ÂI¶°¡A¥s°µ§l¤l (attractor)¡A¤@­Ó²³æ¡]¦ý¤£¬O¸H§Î¡^ªº¨Ò¤l¬O f(x) = x2¡A­Y±q x = [0,1) ¥Xµo¡A§â xn+1 = xn2 ¤@ª½¥N¤U¥hªº¸Ü¡A³Ì«áªº­È·|Áͪñ©ó 0¡Ax=0 ³o­ÓÂI´N¬O f(x) = x2 ³o­Ó¤ÏÂЭ¡¥N¬M®gªº§l¤l¡A¤S¦p¥t¤@­Ó¨Ò¤l¬O f(x) = x/2¡A¥¦³Ì«á·íµM¬O¶]¨ì x = 0 ¤W¡C­Y¬O f(x) = -x¡A«h¬O¤@ª½¦b¨â­ÓÂI¤§¶¡¸õ°Ê¡A¥¦­Ìªº§l¤l³£«D±`²³æ¡C

¦³¨Ç¤ÏÂЭ¡¥N¬M®gªº§l¤l´N½ÆÂø±o¦h¡A¤£¬O¤@¨â­Ó©T©wÂI¦Ó¤v¡A¥s©_²§§l¤l (strange attractor)¡A¦Ó¥¦¬J¬Oº¡¨¬«ù©w¨ç¼Æ¤£Â_¬M®gªºµ²ªG¡A·íµM´N¦³¨ä¤@©wªº³W«ß©Ê¡A¤×¨ä¬O¦¬ÁY©Êªº¬M®g¡A´N¬O§âÂI§ë®g¨ì¤ñ­ì¨Ó§ó¤pªº¹v½d³ò¤º¡AµM¦Ó³o­Ó§l¤l¤Wªº©Ò¦³ÂI¨Ì©w¸q¤£·|¦A¨ü¬M®g¹ïÀ³¨ì·sªºÂI¡A¥i¥H·Q¹³¥¦´N·|«O¦³§½³¡µ²ºc¤U³£¤´¤@¦A¨ã¦³¦¹³W«ß©Ê¡C±q³oùؤj®a¥i¥HÅé·|¬°¤°»ò¸g¹L¤£Â_ªº©ñ¤j¥h¬Ý§óºë²Óªººc³y®É¡A¨ä¯S¼xªº¹Ï¼ËÁ`·|¤£Â_¥X²{¡A´N¬O­n³o¼Ë¤~¯àº¡¨¬§l¤lªº©w¸q¡C

¥t¥~¡A¥Ñ©ó¸H§Î©Ò¨ã¦³ªº³oºØ¯S®í©Ê½è¡A¤]¤Þµo¤F¤@ºØ©w¸qºû«×ªº·sÆ[ÂI¡Cºû«×¥i¥H¬Ý§@¬O¤@­ÓªF¦è¨äªø«×ªº¤j¤p»P­«¶qªºÃö«Y¡C¦pªG¨ä­«¶qÀHªø«×ªº¥­¤èÅܤơA«h³o­Ó¥óª«¬O¤Gºûªº¡G­Y¬OÀHªø«×ªº¥ß¤èÅܤơA«h¬O¤Tºûªº¡C¦Ó¥H¸H§Îªº¼Ë¦¡¥X²{ªºª«¥ó¡A¨ä­«¶q»Pªø«×Åܤƪº¤ñ¨Ò¨Ã¤£¬O¾ã¼Æ¡A¥s°µ¬O¸H§Îºû«×¡C

¤j®a¥i¥H³z¹L¦Û¤v°Ê¤â¼gµ{¦¡¡A¨Óµe¥X¦³½ì¤S¦nª±ªº¸H§Î¡C

 

 

¸H§Îµ{¦¡¼g§@­nÂI

(1) ¹Bºâ­¡¥N

½Æ¼Æªº¨Ï¥Î

«Å§iªº¤èªk¬° complex a ©Î complex z(10,10) ¡C

½Æ¼Æªº±`¼Æ¦b Fortran ùجO¼g¦¨ (1.23, 4.56) ³oºØ¼Ë¤l

¦pªG»Ý­n¨ú¥X¨ä¹ê³¡©Îµê³¡¡A¥i¥Î real(c_num) ©Î imag(c_num) ³o¨â­Ó¤º«Ø¨ç¼Æ¡F ¤Ï¤§¡A­Y­n±N¤@­Ó¹ê¼Æ¶ñ¤J½Æ¼Æ¤¤¡A«h¥i¥Î c_num = (1.0,0.0)*r_num ³oºØ¤èªk ¡C

 

ºtºâªk

«ö¦UºØ¸H§Î¨ä¬J©wªº³W«h­¡¥N¦Ó¤v¡A¨Ã«D¨D¸Ñ¼Æ¾Ç°ÝÃD¡C

 

¦¬ÀīקPÂ_

°ò©ó°j°é¦¸¼Æ

¦b§P©w¬°¤£¦¬ÀĤ§«á¡A°O¿ý¤w°õ¦æªº°j°é¦¸¼Æ¡C

°ò©óºë½T«×

¨Ï¥Î¤º«Øªº¹ï¼Æ¨ç¼Æ log( ) ¨ÓÀò±o¦¸¤è

exit «ü¥O

·í§PÂ_¦¨¥ß¡A¨Ã¥B§@¤F¾A·íªº³B²z¤§«á¡A§Ú­Ì·|¦³»Ý­n¸õ¥X¥¼¶]§¹ªº°j°é¡]©ñ±ó³Ñ¤Uªº´`Àô¤£¥²°µ¡^¡A³o®É­Ô¡A¥i¥H¨Ï¥Î exit «ü¥O¨Ó¸õ¥X¤@¼hªº°j°é¡C¥¦ªº»yªk·¥Â²³æ¡A¥u¦³ exit ¦Ó¤v¡C

 

¶Ã¼Æ

IFS Ãþ«¬ªº¸H§Î·|»Ý­n¥Î¨ì¶Ã¼Æ²£¥Í¾¹¡A ´¶¹M¦Ó¨¥¤@­Ó¶Ã¼Æ²£¥Í¾¹¨C¦¸³Q©I¥s´N·|µ¹¥X¤@­Ó 0 ¨ì 1 ¤§¶¡ªº¤£¦P¹ê¼Æ¡A¨ä¥X²{ªº¾÷²v¬O¬Û·í¥­§¡ªº¡C§Ú­Ì³oùبϥΠNumerical Recipes ®Ñ¤¤«Øij´£¨Ñªº ran2¡A¥¦§¡¤Ã«×ªº«~½è¬O³Ì¦nªº¡C¥t¦³¤@­Ó ran3¡A¨ä³t«×¤ñ ran2 §Ö«Ü¦h¡C¡]¦pªG§A¨Ï¥Îªº¬O Fortran90 ©Î§ó·sªºª©¥»¡A´N¦³¤º«Øªº¶Ã¼Æ¨ç¦¡¥i¥Î¡C¡^

 

»¼°j¦¡ªº (recursive) °Æµ{¦¡©I¥s

¦³¤@¨Ç¸H§Î¡A¹³¬O Koch ³·ªá³oºØ³Q¥s°µ¬O regular fractal ªÌ¡A¦pªG¯à¥Î»¼°j¦¡ªº°Æµ{¦¡©I¥s¡A´N¯à§âµ{¦¡¼g±o«D±`²±¶¡C¨Ï¥ÎªÌ­n§ä·sª©¥»ªº Fortran ½sĶ¾¹¡A´N·|¤ä´©¤F¡]¹ê²ßÀô¹Ò¤¤ªº gfortran ¬O¦³¤ä´©ªº¡A¨Æ¹ê¤W¥¦¬O fortran 95¡^¡C

 

(2) ¹Ï§Î

paper and graph setting

¯È±i¡]¹Ï­±¡^¤j¤p

¹Ï®Ø¡B®y¼Ð»P¼Ðñ¤å¦r

point symbol type

a. coloring

cyclic index

user defined shading

b. magnification

 

repeating

 

(3) Repeating

A simple "GOTO" application

rease graph without asking

 

 

Mandelbrot ¶°¦X

³Ì¨ã¥Nªí©Ê¤]¬O«Ü¦h¤H¤½³Ì»{¬üÄRªº¸H§Î¡A¬O¼Æ¾Ç®a Mandelbrot ¤Î¨ä¥L¤Hªº§V¤O¬ã¨s¤~¤Þ°_¤j®aªº­«µø¡C¥¦¬O½Æ¼Æ¥­­±¤WªºÂI¶°¦X¡A¥s Mandelbrot Set¡AM¡A¨ä©w¸q¬O©Ò¦³½Æ¼Æ¥­­±¤W·|¨Ï¤ÏÂЭ¡¥N¬M®g zn+1 = zn2 + c ¤£·|³y¦¨ | zn | µo´²ªº¨º¨Ç c ­È¡C( z ªºªì©l­È­n¥Î (0,0) )

 

¹ê§@¥Ü½d¡G¤@¨B¤@¨B°Ê¤â¼gµ{¦¡¡]²³æ½d¨Òª©¡A¤£§t©ñ¤j¤Î­«ÂС^

ºtºâªk¤Î§P©w¡]¤U­±½d¨Òµ{¦¡¤¤ªº¬õ¦â³¡¤À¡^

³]©wªì©l z ­È¡]¤@«ß¬°¹s¡^¡B³]©w·sªº c ­È¡]¨C¦¸³£¤£¦P¡^ ¡B¤ÏÂЭ¡¥N¬M®g¡]¨C¦¸³£Àˬd¬O§_µo´²¡A­Y¬O«h c ÂI¼Ð¦¨¥Õ¦â¡A¨Ã¥B¸õ¥X­¡¥N°j°é¡^

 

ø¹Ï¡]¤U­±½d¨Òµ{¦¡¤¤ªºÂŦⳡ¤À¡^

pgopen »P pgclos¡G¶}±Ò¤ÎÃö³¬Ã¸¹Ï¸Ë¸m

papap »P pgenv¡G±±¨î¹Ï­±ªº¤j¤p¥H¤ÎÃä®Ø½d³ò

pgpt¡GµeÂI

pgbbuf »P pgebuf¡G¶}±Ò¤ÎÃö³¬Ã¸¹Ï½w½Ä°Ï

 

¥H¤U¬°½d¨Òµ{¦¡¡G mandel_simple.f mandel_simple.x

program mandelbrot_simple
implicit none
complex c,z_ini,z
integer pgopen,i,j,k,isymbol,n_gen
real c_x_min,c_x_max,c_y_min,c_y_max,c_x,c_y,abs_z

isymbol = -1

write (*,*) 'The program plot Mandelbrot set in a simple way.'
write (*,*)
z_ini = (0.0,0.0)

if ( pgopen('/xwin') .le. 0 ) stop
call pgpap(5.0,1.0)

write (*,*) 'What is the numer of iteration for the mapping ?'
read (*,*) n_gen

c_x_min = -2.0
c_x_max = 0.5
c_y_min = -1.25
c_y_max = 1.25

call pgenv(c_x_min, c_x_max, c_y_min, c_y_max, 1, 0)

do j=1,500
¡@c_x = c_x_min + j*(c_x_max-c_x_min)/500
¡@call pgbbuf
¡@do k=1,500
¡@¡@c_y = c_y_min + k*(c_y_max-c_y_min)/500
¡@¡@c = (1.0,0.0)*c_x + (0.0,1.0)*c_y
¡@¡@z = z_ini

¡@¡@do i=1,n_gen
¡@¡@¡@z = z*z + c
¡@¡@¡@abs_z = conjg(z)*z
¡@¡@¡@if (abs_z.gt.10e+10) then
¡@¡@¡@¡@call pgpt(1,c_x,c_y,isymbol)
¡@¡@¡@¡@exit
¡@¡@¡@endif
¡@¡@
end do
¡@end do
¡@call pgebuf
end do

call pgclos
end


¥H¤U¬O§¹¾ãªºµ{¦¡¡A¨ã¦³©ñ¤j¤Î¸ß°Ý¨Ï¥ÎªÌ¦h¦¸­«µeªº¥\¯à¡A½Ð¤j®a¤ñ¸û°Ñ¦Ò¡G

mandelbrot.f mandelbrot.x mandelbrot.f.txt

 

¥[¤J¤Fº¥¼h¦â±m«ü¼Ð³]©wªº¶i¶¥¥\¯à¡G

mandel_shade.f mandel_shade.x mandel_shade.f.txt

 

ºû°ò¦Ê¬ì¤Wªººëªö¸ê®Æ
http://en.wikipedia.org/wiki/Mandelbrot_set

 

 

Julia Set

Julia Set ²£¥Íªº¤èªk©M Mandelblort Set ¤@¼Ë¬OÀËÅç­¡¥N f(z) = z*z + c ªºµo´²©Ê¡A¦ý¬O¬O±½¹M¤£¦Pªº z ¦Ó¤£¬O c¡C¤U¹Ï¥ª¥k¤À§O¬O c = (0.2,0.6) ¤Î c = (0.0,1.0) ªºµ²ªG¡G

julia_shade.f julia_shade.f.txt

 

 

¤û¹yªk

¤û¹yªk¬O¤@ºØ§Q¥Î±×²v¤è¦Vªº¤Þ¾É¨Ó¨D¨ç¼Æ®Úªº¤èªk¡A³oºØ¤èªk¦b²q´úÂIÂ÷®Ú°÷ªñ¡]¨ãÅé¦a»¡¬O¦b®Ú¤§³Bªº±×²v»P²q´úÂI³Bªº±×²v¶¡ªºÅܤƬO³æ½Õ©Êªº¡^®É¬O«Ü¦³®Ä²vªº¡AµM¦Ó¹ï©ó¤£¦Pªº²q´úÂI¡A±q¨ºùØ¥Xµo¨Ì±×²v¤è¦V¥h§ä®Ú©Ò·|»Ý­n¥Î¨ìªº­¡¥N¨B¼Æ·|¤ñ§Oªº°_©lÂI¦h¡A¦b¦³¨Ç¨ç¼Æ¬Æ¦Ü·|¶V°lÂ÷®Ú¶V»·¡C¤û¹yªk¥Ñ©ó¨ã¦³©T©wªººtºâ¤½¦¡¡A¦]¦¹­Y¬Oªì©l²q´ú¦ì¸m©T©w¡A³B²zªº¨ç¼Æ¤]¤@¼Ëªº¸Ü¡A

¥H¤û¹yªk¨D½Æ¼Æ¨ç¼Æ f(z) = z3 - 1 ªº®Ú¡A´N¬O¬Û·í©ó¬O¶i¦æ

 

ªº¤ÏÂЭ¡¥N°ÝÃD

z3 - 1 = 0 ­n¨Dº¡¨¬³o­Ó¤èµ{¦¡ªº¸Ñ¡A¤@©w¬O¨ã¦³ 2π/3 ¹ïºÙ©Êªº¡C

newton.f newton.x newton.f.txt

 

newton_shade.f newton_shade.x newton_shade.f.txt

 

 

 

 

Koch ¦±½u

 

koch.f koch.f.txt koch.x

 

koch_snow.f koch_snow.f.txt koch_snow.x

 

½Ð¦Û¤v°µ°µ¬Ý¥H¤Uªº¹Ï§Î¡]¥i§Q¥Î PGPLOT µe¦hÃä§Î»P¶ñ¦âªº¥\¯à¡^

 

 

¿¹¸­ (Fern Leaf)

¥t¤@ºØ²£¥Í¸H§Îªº¤èªk¬O©Ò¿×ªº IFS (Iterated Function Systems) ¡A¥¦ªº§l¤l©Òºc¦¨ªºÂI¶°¡A¥iÅý§Ú­Ì³]­p¦¨Ãþ¦ü¿¹¸­ªº§ÎºA¡A¨äºtºâªk¦p¤U¡G

 

xn+1 = axn + byn + e

yn+1 = cxn + dyn + f

a b c d e f p
0.00 0.00 0.00 0.16 0.00 0.00 0.01
0.85 0.04 -0.04 0.85 0.00 1.60 0.85
0.20 -0.26 0.23 0.22 0.00 1.60 0.07
-0.15 0.28 0.26 0.24 0.00 0.44 0.07

¤Wªí¦@¦³¥|²Õ¡A³Ì«á¤@¦Cªº p ¬O¥Nªí­n¨Ï¥Î¸Ó²Õºtºâ¦¡ªº¾÷²v¡C¤]¥i¥Hªí¥Ü¦¨¤U¦¡¡G

 

µe¥X¨Óªºµ²ªG¬O³o­Ó¼Ë¤l¡G

fern.f fern.x fern.f.txt ran2.f

 

¥t¤@ºØ¡A«Ü¹³¬O¸ôÃ䤣ª¾¦WÂø¯óªø¥X¨ÓªºÁJªº³¡¤À¡G

weed.f weed.x weed.f.txt ran2.f

 

¥t¦³·¬¸­ªº IFS¡]½Ð¦Û¤v°µ°µ¬Ý¡^

a b c d e f p
0.14 0.01 0.00 0.51 -0.08 -1.31 0.10
0.43 0.52 -0.45 0.50 1.49 -0.75 0.35
0.45 -0.49 0.47 0.47 -1.62 -0.74 0.35
0.49 0.00 0.00 0.51 0.02 1.62 0.20

maple.f ran2.f

 

 

¬°¤°»ò·|¦³¦p¹Ï¤¤³o¯ëªº¦Û§ÚÃþ¦ü¡H¦]¬°³o¨ÇÂI³£¬Oº¡¨¬­¡¥N¨ç¼Æ¨t²Îªº§l¤l¡A¤]´N¬O»¡¡A³o¨ÇÂI³£º¡¨¬ "¥N¤F«Ü¦h¦¸¤]¤£·|¶]±¼ªº¯S©Ê"¡C¯à°÷¦b³o¼Ëªº±ø¥ó³Q¿z¿ï¥X¨ÓªºÂI¡A§e²{¦bµe­±¤W¡A§Î¦¨¤F½à¤ß®®¥Øªº®ÄªG¡C

¥H¿¹¸­ÂI¶°¬°¨Ò¡A¥¦¬O¥Ñ¥|­Ó¦¬ÁY©Êªº¬M®g©Òºc¦¨¡A·í¤@­Ó°_©lÂI³Q¬Y­Ó¬M®g¨ç¼Æ³B²z¤§«á¡A´N¸¨¤J¸û¤pªº¯S©w½d³ò¤§¤¤¡A¦ý¦Aºò±µ¤U¨Ó¡A¥¦¦³¥i¯à³Q¥t¤@­Ó¬M®g¨ç¼Æ³B²z¡A©ó¬O¸¨¤J¦b­ì½d³ò¤§¤Uªº§ó¤pªº¥t¤@ºØÂI¤À§G±Æ¦C¼Ò¦¡ªº½d³ò¡C§A¥i¥H·Q¹³¦p¦¹Ä~Äò­¡¥N¤U¥h¡]¹Lµ{¤¤¤£¦Pªº¥|­Ó¨ç¼Æ³£¦³¥Î¨ì <°Ý¡A¨º¾÷²v¬O°µ¤°»ò¥Îªº¡H>¡^¡A«h¨C¤@­Ó·L¤p§½³¡¤@©w³£¨ã¦³Ãþ¦üªº¹Ï§Î¡A¤]´N¬O¦P®É¨ã¦³º¡¨¬¥|ºØ¨ç¼Æ¤§§l¤lªººc³y¡C

¾÷²v¥i¥H¤£¦P


¨Ó·½¡Ghttp://zeuscat.com/andrew/chaos/spleenwort.fern.html

 

The Collage Theorem
http://www.cut-the-knot.org/ctk/ifs.shtml

Fern Functions
http://mathworld.wolfram.com/BarnsleysFern.html

¸­¤lªº§Îª¬
http://www.fukuoka-edu.ac.jp/~fukuhara/jikken/leaf_shape.html

¸­¯ßºôµ¸
http://www.nibb.ac.jp/math/work_fujita.html

 

 

¸H§Î»P¦ÛµM

¥Ñ©ó¸H§Î©Ò¨ã¦³ªº¦Û§ÚÃþ¦üªº¯S©Ê¡A¨Ï¥¦»P³\¦h¦ÛµM¬É¤¤¦s¦bªº´X¦ó¹Ï§Î¦³¨Ç¬Û¹³¤§³B¡C¨Ò¦pªK¸­ªº¤À¤e¡Bªe¬y»P¸­¯ßªº¤À¤ä¡B¤sÀ®ªº½ü¹ø¡B¶³ªº§Îª¬¡BÂq°_ªº®ü®ö¡B¯¿¬y»P¶Ã¬y¡Aµ¥µ¥¡C¥i¿×¤@¨F¤@¥@¬É¡B¤@ªá¤@¤Ñ°ó¡C¡]·PÁºô¸ô¤W¤§·Ó¤ù¨Ó·½¡^

 

 

¸H§ÎªºÀ³¥Î

¼v¹³³B²z

 

http://fractalfoundation.org/OFC/OFC-12-1.html

http://fractalfoundation.org/OFC/OFC-12-2.html

http://fractalfoundation.org/OFC/OFC-12-3.html

 

 

¸H§ÎÃÀ³N

ºô¸ô¤W¥i¨£³\¦hµ²¦X¸H§Î»PÃÀ³Nªº§@«~¡C

http://sprott.physics.wisc.edu/carlson/

http://www.fractalus.com/gumbycat/

²V¨PÃÀ´Y
http://www-chaos.umd.edu/gallery.html

ICM 2006 Benoit Mandelbrot ¸H§ÎÃÀ³NÄvÁÉ
http://www.fractalartcontests.com/2006/

´Ý§Î¬ü¾Ç»P°êµe
http://members.tripod.com/gia_5/fractal/aesth.htm

 

 

 

¨ä¥Lª`·N¨Æ¶µ

¥»¸`½d¨Òµ{¦¡¬Ò¨Ï¥Î³æºë«×¡A­Y­n¹F¨ì¸û¨Îªº©ñ¤j¯Å¼Æ¡A«hÀ³¥ÎÂùºë«×¡C

 

 

°Ñ¦Ò®ÑÄy

¤@¥»«Ü¦nªº®Ñ¬O "The Beauty of Fractals"¡A¥Ñ H.-O. Petigen »P P.H. Richter ©ÒµÛ¡ASpringer-Verlag ¥Xª©¡C

 

ºô¸ô¸ê·½

Fractal Mathematics
http://www.hiddendimension.com/Mathematics_Main.html

Cynthia Lanius' Lessons: A Fractals Lesson - Introduction
http://math.rice.edu/~lanius/fractals/

Fractals
http://www.ocf.berkeley.edu/~wwu/fractals/fractals.html

«Ü´Îªº°Êµe
http://www.stanford.edu/~willywu/downloads/xaos1.mpg

¥Î XaoS ³nÅé°µªº
http://wmi.math.u-szeged.hu/xaos/doku.php?id=main

Fractal Art
http://www.fractalus.com/info/manifesto.htm

http://math.rice.edu/~lanius/frac/

Fractals in Nature and How to Measure Them
http://www.fbmn.fh-darmstadt.de/home/sandau/biofractals/abstract_sfi.html

Julia and Mandelbrot Sets (Clark University)
http://aleph0.clarku.edu/~djoyce/julia/index.html

ºû°ò¦Ê¬ì¡GJulia Set
http://en.wikipedia.org/wiki/Julia_set

ºû°ò¦Ê¬ì¡GDe Rham curve
http://en.wikipedia.org/wiki/De_Rham_curve

ºû°ò¦Ê¬ì¡GIterated Function System (IFS)
http://en.wikipedia.org/wiki/Iterated_function_system

 

§Q¥Î°ö°ò (BASIC) »y¨¥¼g¸H§Îªººô­¶¡]¥~³¡¸ê·½¡^

ªì¾ÇªÌ®e©ö¤Jªùªº BASIC »y¨¥¡GDecimal BASIC ©xºô¡B¤U¸ü¡B­^¤åºô§}¡B¤â¥U¡B§Ö³t¤Jªù

BASIC »P¦Û§ÚÃþ¦ü (Self-similarity)
http://www.geocities.jp/thinking_math_education/self-sim/self-sim.htm

BASIC »P¸H§Î
http://www.geocities.jp/thinking_math_education/fractal/fractals.htm

True BASIC
http://www.truebasic.com/

Free BASIC Compilers and Interpreters
http://www.thefreecountry.com/compilers/basic.shtml