Predavanje br 4_2010.pdf

1

. .

. .

'' , , , , , , . .

. . ''''''.''.

D'Arcy Thompson

(1860-1948)

:: , 1917, 1917((On Growth and FormOn Growth and Form))

2

, , , , , , , , . . (( , , ) ) , , , , ..

, , , , ..

3 3 , , 5, 5, 8, 8, 13, 13, 34, 55 34, 55 89.89.

, , , , , , , , , , . . . .

3

13 13

221 1 3344

88

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,

: : '''''' '' , , ? ? , , , , , , , , ??

, , ((, , ) ) ::

(())

(Leonardo Pisano, 1170.-1250.)

, , , , . . , , ..

, , (Fibonacci), (Fibonacci), . . Liber AbaciLiber Abaci :: , , , , ??

:: Liber AbaciLiber Abaci

4

, , , , ; ; , , . . . . , , ..

. . , , ((1 + 1 = 2, 2 + 1 = 3, 3 + 2 = 5 1 + 1 = 2, 2 + 1 = 3, 3 + 2 = 5 .), .), ::

-- ((223=6, 13=6, 15=5; 5=5; 335=15, 25=15, 28=16; 58=16; 58 =40, 38 =40, 313=39; 13=39; .),.),

-- 1 1 ((333 3 =9, 5=9, 52=10; 82=10; 888 =64, 5=64, 513=65;13=65;13131313 = 169= 169, , 882121=168=168;; .).),,

-- , , , , (( , , 88, , 2020 , , 2121)),,

-- (( ), ), , , (( 551)/21)/2. . ,, , , ((0,61803...), ..

5

. . '''''' '' , , , , '''''' '' ..

, , . . , , . . , , . .

, , , , . . . .

, , , , . .

, , , , . .

, , 5 5 ((), ), , , 8 8 . . 3 3 . . 3, 5 3, 5 8 8 ..

7

. . , , . . . . ... ... . . . .

, , . . 8 8 13 (55 13 (55 144), 144), . . . . ..

((88) ) (13)(13)

8

((144144))

((5555))

9

. . . . , , . . , , . . , , 34 34 55, 55 55, 55 89, 89, 89 89 144.144.

12

, , . . aa . . , , , , , , , , (( ) ) 1.61803...1.61803...

((aa//bb)) ((((aa bb)/)/bb)) ::

13

, , . . =1.61803=1.61803..

(( )), , : :

, , : : == 0.61803... 0.61803...

, , ,, , , . .

11 : 1: 1 == 1122 : 1: 1 == 2233 : 2: 2 == 11..5555 : 3: 3 == 11..666666666666666688 : 5: 5 == 11..66

13 : 813 : 8 == 11..62562521 :1321 :13 == 11..6153846....6153846....34 :2134 :21 == 11..6190476....6190476....55 :3455 :34 == 11..6176471....6176471....89 :5589 :55 == 11..6181818....6181818....

-

1024 1024

11..6180339887 4989484820 4586834365 6381177203 0917980576 286213546180339887 4989484820 4586834365 6381177203 0917980576 2862135448 48 6227052604 6281890244 9707207204 1893911374 8475408807 538689176227052604 6281890244 9707207204 1893911374 8475408807 5386891752 52 1266338622 2353693179 3180060766 7263544333 8908659593 958290561266338622 2353693179 3180060766 7263544333 8908659593 9582905638 38 3226613199 2829026788 0675208766 8925017116 9620703222 104321623226613199 2829026788 0675208766 8925017116 9620703222 1043216269 69 5486262963 1361443814 9758701220 3408058879 5445474924 6185695365486262963 1361443814 9758701220 3408058879 5445474924 6185695364 4 8644492410 4432077134 4947049565 8467885098 7433944221 254487708644492410 4432077134 4947049565 8467885098 7433944221 2544877066 66 4780915884 6074998871 2400765217 0575179788 3416625624 940758904780915884 6074998871 2400765217 0575179788 3416625624 9407589069 69 7040002812 1042762177 1117778053 1531714101 1704666599 146697987040002812 1042762177 1117778053 1531714101 1704666599 1466979873 73 1761356006 7087480710 1317952368 1761356006 7087480710 1317952368 9427521948 4353056783 0022878569 9427521948 4353056783 0022878569 9782977834 7845878228 9110976250 0302696156 1700250464 3382437769782977834 7845878228 9110976250 0302696156 1700250464 3382437764 4 8610283831 2683303724 2926752631 1653392473 1671112115 881863858610283831 2683303724 2926752631 1653392473 1671112115 8818638513 13 3162038400 5222165791 2866752946 3162038400 5222165791 2866752946 5490681131 7159934323 5973494985 5490681131 7159934323 5973494985 0904094762 1322298101 7261070596 1164562990 9816290555 208524790904094762 1322298101 7261070596 1164562990 9816290555 2085247903 03 5240602017 2799747175 3427775927 7862561943 2082750513 121815625240602017 2799747175 3427775927 7862561943 2082750513 1218156285 85 5122248093 9471234145 1702237358 0577278616 0086883829 5230459265122248093 9471234145 1702237358 0577278616 0086883829 5230459264 4

7878017889 7878017889 9219902707 7690389532 1968198615 1437803149 9741106926 9219902707 7690389532 1968198615 1437803149 9741106926 0886742962 2675756052 3172777520 3536139362 1076738937 645560600886742962 2675756052 3172777520 3536139362 1076738937 645560606060

5922...5922...

14

'''' '' '' . .

(Luca Paccioli), (Luca Paccioli), , , '''' '' (1509 '' (1509 .), .), (( ))''... ''... , , ''. ''. ..

((FraFra Luca Luca PaccioliPaccioli))

(1445.-1517.)

. . , , . .

(Phidias) (490 (Phidias) (490 430 430 ....) ) . . (().).

Phidias Phidias

(490 (490 432 432 ....))

15

. .

. .

16

Raphael's Raphael's CrucifixionCrucifixion

Michelangelo's Michelangelo's Holy FamilyHoly Family ... ... (1475-1564)

(1483-1530) (())

. . . . , , , , . . , , . . , , , , '''''' '' . . . .

18

, , --

19

--

. . . . (( ) ) 1.61804 ...1.61804 ..., , ..

20

Fibonacci Inspired British National Wildflower Center Addition by Ian Simpsons Architects

, , . . , , ..

. . ..

1979. 1979. . . (Horst Vogel) (Horst Vogel) . . : : , , . ''. '' '' '' ?? , , , , . . . . '''' ''.''.

21

, , '''''' '' . . , , . . 34/55 34/55 360360

oo

222,5222,5oo. . , ,

, , 222,5222,5oo 180180

oo, ,

360360oo, , 137,5137,5

oo. .

34/5534/55, , , , 137,50776137,50776

oo.

'''''' '' , , , , ..

22

Melvyn Bragg probes the Fibonacci sequence

23

. . , , , , ..

-- , , , , , , (( ) ) . . , , , , ,, , , , , . . , , , , , , ... ...

'''''', '', -- , , . . ,, ..

24

(Euclid) ''(Euclid) '''', '', 300 300 . . ., ., , , . . '''''' '' ''''''.''.

((Benoit MandelbrotBenoit Mandelbrot, , 1924. 1924. , , , , .) .) (( fractus fractus , , ). ).

Euclid Euclid (325 BCE - 265 BCE)

Special Collections & Archives, University Libraries, George Mason University, Fairfax County

Benoit B. Mandelbrot

1924

, , , , , , , , . . '''' '', '', ScienceScience (1967.), (1967.), , , . . . . . . '''''' '' (( ).).

, , . . . . (() ) . . . .

25

Fractal Landscape Fractal Landscape

(())

30

(( ) ) 300 300

32

(())

33

Benoit MandelbrotBenoit Mandelbrot

19781978

34

Georg Cantor1845 1845 19181918

. . (Cantor) (Cantor) (( , 1845 , 1845 1918, 1918, , , ). ). , , . . , , ..

33 , ,

1904. 1904. (Helge von Koch, 1870. (Helge von Koch, 1870. 1924.). 1924.).

, , , , , , , , . . . . , , . . , , (( , , , , ) ) (() ) ..

Helge von Koch

(1870-1924)

35

: : (( ) )

Sierpinski, Waclaw Franciszek

(1882-1969)

37

Karl Menger(1902-1985)

, , . . , , , , . . , , , , . .

-- , , ; ;

-- , , , , ; ; -- , , , , . . (( ) )

--, , , , (( ) ) ..

38

(() ) . .

: : -- , , DDEE = = 00; ; -- (( , , , ,

) ) .. , , : : DDEE = = 11; ;

-- :: , , . . DDEE = 2= 2; ;

-- . . , , : : , , ((), ), DDEE = 3= 3..

. . ? ? ,, ,, (0,5)?(0,5)?

. .

, , : : , , , , , , , , (( . . ) ) ..

, , . . , , ,, , , DDEE = 2= 2. . ((), ), DDEE = 3= 3. . , , , , . . , , ??

39

, , . . . .

..

: :

, , . . --. . , , , , (( --)). .

, , , , ..

, , (( ) ) . . , , ..

40

, , (() ) ..

,, ,, , , . . , , . . ::

, , . . , , . . , , ..

, , , , . . : : ??

1968. 1968. ((AristidAristid LindenmayerLindenmayer) ) , , ..

41

(Aristid Lindenmayer)

(1925 1989)

LL-- , , , , , , . . LL-- '''''''' . . (( -- ), ), ..

LL--, , , , , , , , (() ) , , LL-- , , . .

) ) , , ) ) . . ..

) ) LL--

)) LL-- , ,

42

'''' '', '', (Przemyslaw (Przemyslaw Prusinkiewicz) Prusinkiewicz) (1990.) (1990.) LL--. . ..

..

(Przemyslaw Prusinkiewicz)

The AlgorithmicBeauty of

Plants,Springer,1990

Spiral GalaxySpiral Galaxy

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure true /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /NA /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /LeaveUntagged /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice

Documents

Predavanje br 4_2010.pdf