FRACTAL_2

英國的海岸線有多長？ http://atlas-zone.com/complex/fractals/dimension/coastline.htm 碎形幾何學之父 Benoit Mandelbrot 於 1967 年，在一篇幾乎算是他思想轉捩點的論文中，如此地發問：「英國的海岸線有多長？」他之所以會想到海岸線的問題，靈感來自於英國數學家 Lewis Fry Richardson 遺稿中一篇晦澀的論文，其中他所摸索的一大堆爭議性主題，後來成為混沌理論（Chaos Theory）的一部份。

當初 Lewis Fry Richardson 為了想要瞭解一些國家鋸齒形的海岸線長度，所以翻閱西班牙、葡萄牙、比利時與荷蘭的百科全書，他發現書上在估計同一個國家的海岸線長度時，竟然有百分之二十的誤差，Lewis Fry Richardson 指出：這種誤差是因為他們使用不同長度的量尺所導致的。

他同時發現海岸線長度 L 與測量尺度 s 的關係如下，其中，值得注意的是 log(1/s) 與 log(L) 呈線性關係，其斜率為一定值 d：

如果我們試圖實際測量英國海岸線的長度，我們可以拿著兩腳器，先撐開一碼長，然後沿著英國的海岸線行進，會測量出某個海岸線長度值，如果我們將兩腳器的距離調小，再次測量海岸線，那麼必定會得到比原先所測量的更長的海岸線長度，那是因為，較小尺寸的量度可以掌握更多海岸線的細節，我們可以這樣推論：用越小的量尺來量度海岸線長度，所量出的結果會越長 。

Benoit Mandelbrot 說，其實任何海岸線的長度在某個意義下皆為無限長 ，或者說，海岸線的長度是依量尺的長短而定。實際量度英國的海岸線實在是太麻煩了，我們可以嘗試用不同的量尺去測量 Koch Curve 的長度，Koch Curve 同樣具有海岸線般的扭曲與轉折。如下圖所示，當我們用越長的量尺去測量 Koch Curve 的長度，就會有越多的細節無法量到，而當 s 趨近於無限小時，L 顯然也會趨近於無限長 ，但是，1/s 與 L 並不是正比關係， 而是呈現指數關係 ，如左下方的關係圖所示。其中，值得注意的是， L 相當於量尺的測量次數（我們定義做 N(s)）乘上量尺的長度，可以寫成 L=N(s)*s。

Exploring Patterns in Nature Tutorials http://physionet.ph.biu.ac.il/tutorials/epn/ Center for Polymer Studies at Boston University

"Walking" Along a Coastline

EXERCISE:測量英國海岸線長 用ruler method測量英國地圖海岸線長，畫圖找出1/s 與 L 的關係 You will need: a map of the coastline to be measured and a pair of calipers(測徑器) .                                                     

EXERCISE:畫出海岸線 Roll the die. If you get 1, push the midpoint left by 4 centimeters, hold it with the thumb tack. If you get 2, push the midpoint left by 2 centimeters, hold it with the thumb tack. If you get 3, leave the midpoint where it is, but hold it with a thumb tack. If you get 4, push the midpoint right by 2 centimeters, hold it with the thumb tack. If you get 5, push the midpoint right by 4 centimeters, it with the thumb tack. If you get 6, roll again.

IRELAND

IRELAND

Grid method Box Counting Method An alternative method is called the grid method or box counting method or covering method. The grid method is a bit more versatile than the ruler method, and can be used for different kinds of fractals.

Box Counting Method Covering the Coastline with Boxes 邊長 預測 測量 方塊數目 16 8 4 2 1

Fractal Dimension How many disks does it take to cover the Koch coastline? Well, it depends on their size of course. 1 disk with diameter 1 is sufficient to cover the whole thing, 4 disks with diameter 1/3, 16 disks with diameter 1/9, 64 disks with diameter 1/27, and so on. In general, it takes 4n disks of radius (1/3)n to cover the Koch coastline. If we apply this procedure to any entity in any metric space we can define a quantity that is the equivalent of a dimension. The Hausdorff-Besicovitch dimension of an object in a metric space is given by the formula

where N(h) is the number of disks of radius h needed to cover the object. Thus the Koch coastline has a Hausdorff-Besicovitch dimension which is the limit of the sequence Is this really a dimension? Apply the procedure to the unit line segment. It takes 1 disk of diameter 1, 2 disks of diameter 1/2, 4 disks of diameter 1/4, and so on to cover the unit line segment. In the limit we find a dimension of This agrees with the topological dimension of the space.

The problem now is, how do we interpret a result like 1. 261859507 The problem now is, how do we interpret a result like 1.261859507...? This does not agree with the topological dimension of 1 but neither is it 2. The Koch coastline is somewhere between a line and a plane. Its dimension is not a whole number but a fraction. It is a fractal. Actually fractals can have whole number dimensions so this is a bit of a misnomer. A better definition is that a fractal is any entity whose Hausdorff-Besicovitch dimension strictly exceeds its topological dimension (D > DT). Thus, the Peano space-filling curve is also a fractal as we would expect it to be. Even though its Hausdorff-Besicovitch dimension is a whole number (D = 2) its topological dimension (DT = 1) is strictly less than this. →The monster has been tamed.

Surrounding the Koch Coastline with Boxes (a way to determine its dimension)

                log (1/h) log N(h)  0  7.60837 -0.693147  7.04054 -1.38629  6.32972 -2.56495  4.85981 -3.09104  4.21951 -3.49651  3.52636 -3.78419  3.29584 -4.00733  3.04452 -4.18965  2.99573 -4.34381  2.70805 -4.47734  2.56495 -5.17615  1.60944 Koch Coastline dimension (experimental) = 1.18 dimension (analytical) = 1.26 deviation = 6.35%

San Marco Dragon log (1/h) log N(h) 0 8.02355 -0.693147 7.29438  0  8.02355 -0.693147  7.29438 -1.38629  6.52209 -2.63906  5.03044 -3.21888  4.29046 -3.61092  4.00733 -3.91202  3.52636 -4.12713  3.3322 -4.31749  2.94444 -4.46591  2.83321 -4.60517  2.63906 -5.29832  2.07944 San Marco Dragon dimension (experimental) = 1.16 dimension (analytical) = ??? deviation = ???

log (1/h) log N(h)  0 11.0904 -0.693147  9.71962 -1.38629  8.31777 -2.63906  5.99146 -3.21888  4.79579 -3.61092  4.15888 -3.91202  3.58352 -4.12713  3.4012 -4.31749  3.21888 -4.46591  2.77259 -4.60517 -5.29832  1.38629                                        dimension (experimental) = 1.82 dimension (analytical) = 2.00 deviation = 9.00% 正方形

練習題 使用Box Counting Method計算IRELAND海岸線的碎形維度 log (1/h) log N(h) 64 32 16 8 4 2 1

IRELAND 1 x 1

IRELAND 2 x 2

IRELAND 4 x 4

IRELAND 8 x 8

IRELAND 16 x 16

IRELAND 32 x 32

FRACTAL_3

1970年代左右，數學家 Benoit Mandelbrot 在一篇幾乎算是他思想轉捩點的論文「英國的海岸線有多長？」中，發展出了新的維度觀念 ─ 幾何學：碎形(分形)。 三十年間，碎形幾何，與混沌理論，複雜性科學共同匯合，試圖解釋過去科學家們所忽略的非線性現象，與大自然的複雜結構，把觸角伸入，除了物理、化學之外的生理學、經濟學、社會學、氣象學，乃至於天文學所談及的星體分布。 搖身一變，碎形幾何已經變成了主要能描述大自然的幾何學了。這些研究開拓了人們對於維度、尺度、結構的新看法，大致歸納如下：

◆碎形具有分數維度：不同於整數維度的一維線段，二維矩形，碎形所具有的維度是分數的，例如無窮擴張三分之四的卡區曲線，其維度是 1.2618…。 ◆碎形具有尺度無關性：對於「同一個」碎形結構，以不同大小的量尺來量度「可觀察的區域」，碎形會具有一致的碎形維度。例如，如果我們不同程度地放大或縮小 Mandelbrot Set，我們會發現圖形的複雜度，或摺疊程度，或粗糙程度並未因此而改變。

◆碎形具有自我模仿性：對於「同一個」碎形結構，自我模仿就是尺度一層一層縮小的結構重複性，它們不僅在越來越小的尺度裡重複細節，而且是以某種固定的方式將細節縮小尺寸，造成某種循環重現的複雜現象。 ◆碎形代表有限區域的無限結構：例如，卡區的雪花曲線，是一條無限長，而結構不斷重複的線段，被限制在最初三角形的正圓區域內。例如，原本是一固定線段的 Cantor Set，最後變成一系列數量無窮，但總長度卻為零的點集合。

◆碎形隱含一種整體性：我們可以從某一尺度的碎形，來推知另一尺度的「同一個」碎形的大致樣子，這意味著一種整體性，小細節的 傾向可以透露大細節的傾向，大細節的絲毫改變可以令所有小細節全面改觀，再造成整個碎形圖形的變化。 ◆碎形是觀察手段的相對結果：回到 Mandelbrot 的那篇論文「英國的海岸線有多長？」，作為碎形結構的海岸線本身，在某種意義下是無限長，但是對於不同的觀察者而言，海岸線長度卻端視其手中的量尺（不同的觀察手段）而定，Mandelbrot 說：「數據結果是依觀察者與其對象而改變。」也正是這個觀念，才促使他發展出不同於過去科學家的維度量度的新理論。

◆碎形是非線性動力過程的結果：大自然的外貌、結構是非線性動力過程所造成的結果，我們也只能在非線性現象中，才能找到碎形的蹤跡，於是碎形幾何與非線性動力學有著密不可分的關係。

典型的碎形 　　 　　 Pythagorean Trees 　　　　 Cantor Set 　　　　 Sierpinski Gasket And Carpet 　　　　 Koch Curve 　　　　 Cesaro Curve 　　　　 Levy Curve 　　　　 Dragon Curve 　　　　 Peano Curve 　　　　 Hilbert Curve 　　　　 H-Fractal 　　　　 Tree Fractal http://www.atlas-zone.com/complex/fractals/index.html

繪製碎形的方法 　起始元與生成元疊代法（Initiator and Generator Iteration Method） 幾何變換疊代函數系統（Deterministic Iterated Function System）　　 隨機疊代函數系統（Random Iterated Function Sysytems） 規範疊代函數系統（Formula Iterated Function Sysytems） 　　　 　

Mandelbrot and Julia Sets The definition of the Mandelbrot set is : the set of all the complex numbers, c, such that the iteration of is bounded (starting with z =0 ). The Mandelbrot set is the graph of all the complex numbers c, that do not go to infinity when iterated in , with a starting value of z =0 .

Mandelbrot set Named after Benoit B. Mandelbrot. A fractal generated by iterating: , and plotting how fast it diverges to infinity for different values of the complex number c (speed represented as colours). The black set represents the "prisoner" points that do not diverge: it is the Mandelbrot set.

Julia sets Named after Gaston Julia (1893--1978). A fractal generated by iterating: and plotting how fast it diverges to infinity for different values of the complex number z (speed represented as colours) for a set value of c. The black set represents the "prisoner" points that do not diverge: the border of this set is the Julia set. Values of c that lie within the Mandelbrot set result in connected Julia sets; values of c from outside result in disconnected Julia sets. We can draw an array of Julia sets for various values of c, and map out the Mandelbrot set.

Julia set A Julia set is almost the same thing. It is defined to be : the set of all the complex numbers, z , such that the iteration of is bounded for a particular value of c. Again, more simply put it is the graph of all the complex numbers z, that do not go to infinity when iterated in , where c is constant.

Mandelbrot Set Explorer http://math.bu.edu/DYSYS/explorer/page1.html

Mandelbrot Explorer Gallery

Julia and Mandelbrot Sets Mandelbrot and Julia Sets Julia Set Fractal (2D) Color Cycling on the Mandelbrot Set The Mandelbrot/Julia Set Applet　 http://aleph0.clarku.edu/~djoyce/julia/julia.html http://www.cut-the-knot.org/blue/julia.shtml http://local.wasp.uwa.edu.au/~pbourke/fractals/juliaset/ http://www.cut-the-knot.org/Curriculum/Magic/MandelCycle.shtml http://math.bu.edu/DYSYS/applets/JuliaIteration.html

Java Julia Set Generator

複動態系統的Julia Set http://www.emath.pu.edu.tw/celebrate/celebrate1/p3.htm http://www.ibiblio.org/e-notes/MSet/Anim/ManJuOrb.htm

碎形與藝術 http://www.fractalus.com/