複動態系統的Julia Set http://www.emath.pu.edu.tw/celebrate/celebrate1/p3.htm  

Mandelbrot and Julia sets Make-Your-Own Mandelbrot Set http://library.thinkquest.org/3288/myomand.html Mandelbrot and Julia sets http://www.cut-the-knot.com/blue/julia.html B.Mandelbrot, The Fractal Geometry of Nature, W.H.Freeman and Co., NY, 1977

B.Mandelbrot has discovered a way to index Julia sets for parametric families of functions. The applet below illustrates this concept for a simple function fc(z)=z2+c where z and c are complex. For every c there exists a Julia set; and a related picture appears in the right part of the area. Values for c can be picked from the picture on the left that, loosely speaking, depicts the Mandelbrot set for the family fc(z). To obtain the Mandelbrot set, run iterations zk+1=fc(zk) with z0=0 and c varying in some bounded area (below a rectangle with opposite corners (-2.2, -1.4) and (0.8, 1.4)). It's known that once |zk| becomes greater than 2: |zk|>2, the iterations will eventually escape to infinity. For every c, mark the iteration kc at which this condition first becomes true. Associate with the point c a color number kc from a given palette of colors. This will produce a picture on the left. The Mandelbrot set is the set of c's for which the iterations starting with x0 = 0 are bounded. This is the set that consists of the enterior cardioid-like shape with a circle attached on its left. Each of the two has smaller warts attached which have some more, adding to the ugliness (or is it the beauty?) of the curve.

http://mcasco.com/mset.html  

Sierpinski's Triangle Sierpinski's Triangle is a very famous fractal that's been seen by most advanced math students. This fractal consists of one large triangle, which contains an infinite amount of smaller triangles within. The infinite amount of triangles is easily understood if the fractal is zoomed in many levels. Each zoom will show yet more previously unseen triangles embedded in the visible ones. Creating the fractal requires little computational power. Even simple graphing calculators can easily make this image. The fractal is created pixel by pixel, using random numbers; the fractal will be slightly different each time due to this. Although, if you were to run the program repeatedly, and allow each to use an infinite amount of time, the results would be always identical. No one has an infinite amount of time, but the differences in the finite versions are very small. To generate this fractal, a few steps are involved. First, initial X and Y values should be chosen, either by the program or the user. The values used have little effect on the fractal. Regardless of what's chosen, the same triangle will be created. Next, the program must create a random number, between 0 and 1. Then, three possible routes can be taken.

If the random number is less then 1/3, then the following equations should be applied to X and Y. xn = 0.5 * (xn-1 + 1) yn = 0.5 * yn-1 If the random number is between 1/3 and 2/3, then these equations should be used. xn = xn-1 * 0.5 yn = yn-1 * 0.5 If the number is greater than 2/3, the the following equations should be applied. xn = 0.5 * (xn-1 + 0.5) yn = 0.5 * (yn-1 + 1) Now that X and Y have changed, the point should be plotted on the screen. Finally, loop back to the random number generation and start over again. Only a few hundred iterations are needed to begin to see the triangles. A few thousand pixels will produce a good image.

Sierpinski's Triangle

細胞自動機 (cellular automata; CA ) 最初由馮紐曼(von Neumann)發明(1950 年代) ，Von Neumann（1903-1957），匈裔美籍數學家，生於布達佩斯，卒於華盛頓特區。他是廿世紀少見的數學科學通才，在許多領域都有重要的基本貢獻。 Von Neumann 是猶太人。原姓Neumann，因為父親買下爵位，才加上貴族專稱的「von」。他自幼穎異，記憶力過人，對數學有驚人的天份，但父親希望他從商，幾經折衝，他同時在布達佩斯大學學數學，又在柏林大學學化學（後轉到蘇黎士學化工）。但即使在蘇黎士，他仍與知名數學家 Weyl 與 Polya 交遊。Polya 曾經這樣描述 Von Neumann 「他是我唯一害怕的學生。在課堂如果我提出一個當時未解的問題，通常他在下課後就會直接來找我，給我幾頁完整的解答。」

1926年 Von Neumann 以一篇集合論的論文獲得布達佩斯大學的博士學位，然後以 Rockefeller 獎學金前往哥廷根大學跟隨 Hilbert 作博士後研究，並在柏林，漢堡講學。Von Neumann 在廿餘歲時已經是數學圈中公認的年輕天才。 1930年 Von Neumann 應 Veblen 之邀，到普林斯頓大學客座，1931年普林斯頓大學即授予教授職位，1933年他成為新成立的普林斯頓高等研究院終身職院士。Von Neumann 的家庭宴會在普林斯頓非常熱鬧知名，這在數學家中是很少見的。

綜論 Von Neumann 的數學成就，大致如下： （1）初期工作以數理邏輯（尤其是公設集合論）、測度論、實分析為主。 （2）在《Mathematische Grundlagender Quantenmachanik》（1932）中， Von Neumann 為當時的量子力學打下堅實的數學基礎。 （3）自1929起，Von Neumann 即從事算子代數的先驅性工作，在1930-40年間 Von Neumann 與 Murray 為後來所謂的 Von Neumann 代數寫下系列基本的文章。 （4）Von Neumann 為對局論的發明人，他首先証明零和對局的 minmax 定理，並與 Morgenstern 合著《對局論與經濟行為》，對社會科學、生命科學影響深遠。 （5）Ergdic（遍歷性）定理的証明（1938）。 （6）Von Neumann 對應用數學的興趣，從流體力學始，並對非線性偏微分方程產生莫大的興趣。而對他而言，數值計算是最可能的「實驗」方法，這也使 Von Neumann 成為今日電腦之奠基者，並因此發展 cellular automata 的理論。 另外 Von Neumann 也是氫彈的催生者，1940年起他即熱心 參與美國的各項國防計劃或實驗室，也因此獲得各式各樣的數學或非數學的獎章。 本文參考資料：(1) 大英百科全書。 (2) MacTutor 數學史檔案網站：von Neumann。

Cellular Automata and Complexity: Collected Papers by Stephen Wolfram Are mathematical equations the best way to model nature? For many years it had been assumed that they were. But in the early 1980s, Stephen Wolfram made the radical proposal that one should instead build models that are based directly on simple computer programs. Wolfram made a detailed study of a class of such models known as cellular automata, and discovered a remarkable fact: that even when the underlying rules are very simple, the behavior they produce can be highly complex, and can mimic many features of what we see in nature. And based on this result, Wolfram began a program to develop what has become A New Kind of Science. The results of Wolfram's work found many applications, from the so-called Wolfram Classification central to fields such as artificial life, to new ideas about cryptography and fluid dynamics. This book is a collection of Wolfram's original papers on cellular automata and complexity. Some of these papers are widely known in the scientific community; others have never been published before. Together, the papers provide a highly readable account of what has become a major new field of science, with important implications for physics, biology, economics, computer science and many other areas. Published (1994): ISBN 0-201-62716-7 (hardcover) ISBN 0-201-62664-0 (paperback)

細胞自動機( cellular automata; CA ) 細胞自動機最初由馮紐曼(von Neumann)發明(1950 年代)，這是一個抽象的圖案產生機制，一旦給定初始值，即可按預先設定的規則，隨時間改變形狀。以人工生命的角度來看，細胞自動機可視為一個讓許多生命生存繁殖的世界(world)，類似地球孕育各種生物一般。 http://math.hws.edu/xJava/CA/

細胞自動機有三個特點： 平行計算 (parallel computation) 局部的 (local)，細胞的狀態變化只受周遭細胞的影響 一致性的 (homogeneous), 每一細胞都受同一組規則控制

設計一個細胞自動機需包含兩部份： (1.)各個細胞的初始狀態(即整個自動機的初始形狀) (2.)根據舊細胞產生新細胞的規則.

以下以一種最為大家所知的細胞自動機來說明，這是由一位數學家康威(John Horton Conway)在 1970年發明的，稱之為 game of Life(生命遊戲)，在這個細胞自動機中，把平面(即二維的空間) 分割成很多方格子(類似圍棋棋盤)，每一格子為一細胞，每一細胞有八個鄰居，細胞有兩種狀態，“生” 或 “死”( 在電腦裡可以 1 或 0 來代表)，細胞自動機的規則如下： [1.] 活細胞如果有二或三個鄰居則可以活到下一世代, 否則就會死於獨居或壅擠 。 [2.] 死細胞處如果恰好有三個活細胞鄰居，則可生出活細胞 ( 就某種意義上來說可視之為"繁殖")。

細胞自動機可以遠較康威所設計的複雜得多，在細胞活動的空間上,可以是一維的，二維的，三維的,，或更高維，細胞的狀態可以有很多種，規則可以非常複雜，透過不同的設計，細胞自動機可以展現無限的多樣性，其中最讓人驚異的是有些細胞自動機可以產生存在於大自然的東西，例如貝殼上的圖案,、雪花的結構,、蜿蜒的河流 ... 等等。事實上，有些研究學者更進一步猜測我們存在的這個宇宙是否就是一種極其複雜的細胞自動機，我們的宇宙的確與理論上的細胞自動機有很多相似的地方，像是上述細胞自動機的三個特點宇宙也都符合，宇宙是平行處理的，宇宙中的每一點受鄰近狀態的影響最大，宇宙各處遵循同樣的自然律(homogeneous)，比較不一樣的地方是在空間上及時間的進行上細胞自動機都是斷續的(discrete)，但是宇宙"似乎"都是連續的(continuous), 不過科學家也還不敢斷定是否如此，也許以後可以證明在極小尺度上空間與時間都是斷續的。不管怎樣，細胞自動機與宇宙有很大的關連，至少可以用它來模擬宇宙的運作及生命的行為模式，而不只是數學上的一個理論。

Cellular Automata http://delta.cs.cinvestav.mx/~mcintosh/oldweb/pautomata.html

潛味音樂網 潛味音樂‧連網 新邏輯藝文網 澀柿子的世界 Radio Art Online Links Eddie's Jazz Page Electronic Music Foundation World Forum for Acoustic Ecology Forced Exposure 私密聆聽 用chaos的algorithm來做音樂應該不是難事， 之前試過先定義七種不同cell， 每個分別代表C D E F G A B， 做出來其實滿無聊的。 我想對cell增加定義和 繁殖/死亡 條件的修改， 應該可以做出很有趣的東西。 http://www.subborg.com/ http://www.sinologic.com/newmusic/sub/board5.html

Artificial Life http://www.alife.org/index.php?page=article&group=news&article=20010810090159

Darwin Pond Darwin Pond is an imaginary gene pool, a primordial puddle of genetic surprises. More technically, Darwin Pond is an Artificial Life Simulation: a virtual world exhibiting the emergence of life-like behaviors. But it's more than just a fun and informative thing to watch, you can participate in this artificial life simulation by building little scenarios and setting up experiments. http://www.ventrella.com/Darwin/darwin.html

(Artificial Intelligent)               人工智慧 AI (Artificial Intelligent) http://www.emls.tku.edu.tw/Student/Multimedia88/16/ai.htm