{"id":18525,"date":"2020-02-26T21:43:40","date_gmt":"2020-02-26T13:43:40","guid":{"rendered":"https:\/\/swarma.org\/?p=18525"},"modified":"2020-02-26T21:43:40","modified_gmt":"2020-02-26T13:43:40","slug":"%e5%88%86%e5%bd%a2%e5%87%a0%e4%bd%95%ef%bc%9a%e5%af%bb%e6%89%be%e9%9a%90%e8%97%8f%e7%9a%84%e7%bb%b4%e5%ba%a6-%e9%9b%86%e6%99%ba%e7%99%be%e7%a7%91","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=18525","title":{"rendered":"\u5206\u5f62\u51e0\u4f55\uff1a\u5bfb\u627e\u9690\u85cf\u7684\u7ef4\u5ea6 | \u96c6\u667a\u767e\u79d1"},"content":{"rendered":"
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\u56fe1\uff1a<\/strong>\u7531\u5fd7\u613f\u8005March\u63d0\u4f9b\u7684\u5206\u5f62\u51e0\u4f55\u793a\u4f8b<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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\u4eca\u5929\u5206\u4eab\u590d\u6742\u6027\u79d1\u5b66\u9886\u57df\u91cc\u9762\u4e00\u4e2a\u5178\u578b\u800c\u6709\u8da3\u7684\u6a21\u578b\uff1a\u5206\u5f62\u51e0\u4f55\u3002\u672c\u6587\u5c06\u5c55\u73b0\u5206\u5f62\u7684\u57fa\u672c\u6982\u5ff5\uff0c\u5178\u578b\u6848\u4f8b\uff0c\u77e5\u540d\u5b66\u8005\u3001\u548c\u4e00\u4e9b\u5165\u95e8\u7684\u5b66\u4e60\u8d44\u6e90\u3002<\/section>\n

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\u6211\u4eec\u77e5\u9053\u76f4\u7ebf\u662f1\u7ef4\u7684\uff0c\u5e73\u9762\u662f2\u7ef4\u7684\uff0c\u7a7a\u95f4\u662f3\u7ef4\u7684\uff0c\u65f6\u7a7a\u662f4\u7ef4\u7684\uff0c\u90a3\u8eab\u4f53\u7684\u6bdb\u7ec6\u8840\u7ba1\u3001\u7ef5\u5ef6\u7684\u6d77\u5cb8\u7ebf\u3001\u6676\u83b9\u7684\u96ea\u82b1\u8fd9\u4e9b\u4e0d\u89c4\u5219\u7684\u51e0\u4f55\u4f53\uff0c\u53c8\u662f\u591a\u5c11\u7ef4\u5462\uff1f\u900f\u8fc7\u5206\u5f62\u51e0\u4f55\u7684\u7406\u8bba\uff0c\u6211\u4eec\u53ef\u4ee5\u4e86\u89e3\u5982\u4f55\u8ba1\u7b97\u8fd9\u4e9b\u975e\u89c4\u5219\u7684\u66f2\u7ebf\u7684\u8c6a\u65af\u591a\u592b\u7ef4\u5ea6\uff08\u5373\u5206\u6570\u7ef4\u5ea6\uff09\u3002\u5e76\u4e14\u8fd0\u7528\u8fd9\u79cd\u601d\u7ef4\u53bb\u7406\u89e3\u751f\u6d3b\u4e2d\u90a3\u4e9b\u5e73\u5e38\u4f46\u662f\u5374\u5728\u7ec6\u5c0f\u4e4b\u5904\u53d8\u5316\u7684\u73b0\u8c61\uff0c\u52a0\u6df1\u4f60\u5bf9\u8fd9\u4e2a\u590d\u6742\u4e16\u754c\u7684\u8ba4\u8bc6\u3002<\/section>\n<\/section>\n<\/section>\n<\/section>\n
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\u4e09\u3001\u5206\u578b\u51e0\u4f55\u7684\u51e0\u4e2a\u5178\u578b\u793a\u4f8b<\/strong><\/p>\n

\u56db\u3001\u76f8\u5173\u8d44\u6e90\u63a8\u8350<\/strong><\/p>\n

\u4e94\u3001\u96c6\u667a\u767e\u79d1\u8bcd\u6761\u5fd7\u613f\u8005\u62db\u52df<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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\u5206\u5f62\u51e0\u4f55\u5b66\uff0c\u5c31\u662f\u4e00\u95e8\u4ee5\u4e0d\u89c4\u5219\u5f62\u6001\u4e3a\u7814\u7a76\u5bf9\u8c61\u7684\u51e0\u4f55\u5b66\u3002\u5206\u5f62\u6700\u5927\u7684\u7279\u6027\u662f\u5177\u6709\u81ea\u76f8\u4f3c\u6027\uff0c\u4e0d\u7ba1\u662f\u653e\u5927\u6216\u8005\u7f29\u5c0f\u7814\u7a76\u5bf9\u8c61\uff0c\u4f60\u90fd\u4f1a\u770b\u5230\u5c40\u90e8\u548c\u6574\u4f53\u5177\u6709\u76f8\u4f3c\u7684\u7ed3\u6784\u3002\u56e0\u4e3a\u5206\u5f62\u51e0\u4f55\u7684\u7814\u7a76\u5bf9\u8c61\u666e\u904d\u5b58\u5728\u4e8e\u5927\u81ea\u7136\u4e2d\uff0c\u4f8b\u5982\u6211\u4eec\u719f\u77e5\u7684\u96ea\u82b1\u3001\u6811\u53f6\u3001\u7fa4\u5c71\u3001\u4e91\u6735\u3001\u6d77\u6d6a\u7b49\u7b49\uff0c\u6240\u4ee5\u5206\u5f62\u51e0\u4f55\u5b66\uff0c\u53c8\u88ab\u79f0\u4e3a\u662f\u201c\u5927\u81ea\u7136\u7684\u51e0\u4f55\u5b66\u201d\u3002
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\u4f46\u540c\u65f6\u5206\u5f62\u5728\u73b0\u5b9e\u751f\u6d3b\u4e2d\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u5305\u62ec\u5728\u52a8\u753b\u5236\u4f5c\uff0c3D\u5efa\u6a21\u3001\u75be\u75c5\u76d1\u6d4b\u3001\u80a1\u7968\u5e02\u573a\u4ee5\u53ca\u901a\u4fe1\u7b49\u9886\u57df\u3002<\/span>\u5206\u5f62\u7b2c\u4e00\u6b21\u5e94\u7528\u7684\u7535\u5f71\u662f\u300a\u661f\u9645\u8ff7\u822a2-\u53ef\u6c57\u4e4b\u6012\u300b\uff0c\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6Loren Carpenter\u4ece\u4e00\u4e2a\u5168\u666f\u56fe\u51fa\u53d1\uff0c \u7531\u4e00\u4e9b\u975e\u5e38\u7c97\u7565\u7684\u4e09\u89d2\u5f62\u6784\u6210\uff0c\u7136\u540e\u5bf9\u4e8e\u6bcf\u4e2a\u4e09\u89d2\u5f62\uff0c\u5206\u62104\u4e2a\u5c0f\u4e09\u89d2\u5f62\uff0c\u7ee7\u7eed\u8fd9\u6837\u64cd\u4f5c\u4e0b\u53bb\uff0c\u8fed\u4ee3\uff0c\u5f62\u6210\u4e86\u975e\u5e38\u903c\u771f\u7684\u5c71\u8109\u3002<\/span>
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\u63a8\u8350\u89c6\u9891\uff1a\u3010PBS\u7eaa\u5f55\u7247\u3011\u5bfb\u627e\u9690\u85cf\u7684\u7ef4\u5ea6\u3010\u4e2d\u82f1\u53cc\u5b57\u3011:<\/p>\n

https:\/\/www.bilibili.com\/video\/av13766486?p=1<\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n


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2.\u5206\u5f62\u7684\u91cd\u8981\u6982\u5ff5<\/strong><\/span>
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\u81ea\u76f8\u4f3c\u6027(Self-similarity)<\/strong><\/span><\/section>\n


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\u81ea\u76f8\u4f3c\u662f\u6307\u5c40\u90e8\u4e0e\u6574\u4f53\u5728\u5f62\u6001\u3001\u529f\u80fd\u3001\u4fe1\u606f\u3001\u65f6\u95f4\u3001\u7a7a\u95f4\u7b49\u65b9\u9762\u5177\u6709\u7edf\u8ba1\u610f\u4e49\u4e0a\u7684\u76f8\u4f3c\u6027\uff0c\u6bd4\u5982\u6811\u5e72\u548c\u679d\u6860\u7684\u81ea\u76f8\u4f3c\uff0c\u5927\u7684\u65cb\u6da1\u548c\u5c0f\u65cb\u6da1\u7684\u81ea\u76f8\u4f3c\u7b49\u7b49\u3002<\/span><\/section>\n


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\u65e0\u6807\u5ea6\u6027<\/strong>\uff08<\/strong>Scale-free<\/strong>\uff09<\/strong><\/span><\/p>\n


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\u6240\u8c13\u65e0\u6807\u5ea6\u6027\uff0c\u662f\u6307\u5f53\u6211\u4eec\u53d8\u6362\u4e0d\u540c\u7684\u5c3a\u5ea6\uff0c\u90fd\u4f1a\u53d1\u73b0\u7cfb\u7edf\u8fd8\u662f\u90a3\u6837\uff0c\u5b83\u6ca1\u6709\u53d8\u5316\uff08\u5373\u81ea\u76f8\u4f3c\u6027\uff09\uff0c\u4e5f\u5c31\u662f\u6807\u5ea6\u4e0d\u53d8\u6027\u3002\u6bd4\u5982\u6211\u4eec\u6240\u8bf4\u7684\u5927\u6f29\u6da1\u5957\u7740\u5c0f\u65cb\u6da1\uff0c\u5b83\u4eec\u53d1\u751f\u5728\u4e0d\u540c\u7684\u5c3a\u5ea6\uff0c\u4f46\u662f\u5374\u8868\u73b0\u51fa\u76f8\u4f3c\u6027\uff0c\u8fd9\u5c31\u79f0\u4e3a\u65e0\u6807\u5ea6\u7279\u6027\u3002<\/span><\/p>\n


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\u975e\u7ebf\u6027\uff08Nonlinear\uff09<\/strong><\/span><\/p>\n


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\u975e\u7ebf\u6027\u5c31\u662f\u548c\u7ebf\u6027\u76f8\u5bf9\uff0c\u7ebf\u6027\u662f\u6307\u6210\u6bd4\u4f8b\uff0c\u800c\u975e\u7ebf\u6027\u5219\u6307\u8f93\u5165\u548c\u8f93\u51fa\u4e0d\u6210\u6bd4\u4f8b\uff0c\u6bd4\u5982\u629b\u7269\u7ebf\u5c31\u662f\u975e\u7ebf\u6027\u7684\u3002\u5728\u590d\u6742\u7cfb\u7edf\u4e2d\uff0c\u975e\u7ebf\u6027\u662f\u6700\u91cd\u8981\u7684\u7279\u6027\u4e4b\u4e00\u3002<\/span><\/p>\n


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\u56fe2\uff1a\u6765\u6e90\u4e8e<\/strong>Dave Whyte<\/strong><\/section>\n
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\u65e0\u8fb9\u7684\u5947\u8ff9\u6e90\u81ea\u7b80\u5355\u89c4\u5219\u7684\u65e0\u9650\u91cd\u590d\u3002<\/em><\/section>\n

<\/em><\/section>\n
\u2014\u2014\u5206\u5f62\u4e4b\u7236 <\/em> Benoit B. Mandelbrot<\/em><\/section>\n


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3.\u5206\u5f62\u7684\u51e0\u4e2a\u5178\u578b\u793a\u4f8b<\/strong><\/span><\/p>\n

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\u5eb7\u6258\u5c14\u96c6\uff08Cantor set\uff09<\/strong><\/span><\/p>\n


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\u5206\u5f62\u7684\u8d77\u6e90\u662f\u5eb7\u6258\u5c14\u96c6\uff08Cantor set\uff09\u3002\u6211\u4eec\u53d6\u4e00\u4e2a\u7ebf\u6bb5\uff0c\u628a\u5b83\u4e2d\u95f4\u76841\/3\u53bb\u6389\u5f97\u5230\u4e24\u4e2a\u5206\u5f00\u7684\u7ebf\u6bb5\uff0c\u518d\u5bf9\u5269\u4e0b\u7684\u4e24\u6bb5\u8fdb\u884c\u76f8\u540c\u7684\u64cd\u4f5c\uff0c\u5f97\u52304\u4e2a\u7ebf\u6bb5\uff0c\u8fd9\u6837\u91cd\u590d\u8fdb\u884c\u4e0b\u53bb\u76f4\u5230\u65e0\u7a77\uff0c\u6700\u540e\u5f97\u5230\u7684\u56fe\u5f62\u96c6\u5408\u5c31\u662f\u5eb7\u6258\u5c14\u96c6\u3002<\/span><\/p>\n


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\u56fe3\uff1a\u5eb7\u6258\u5c14\u96c6\uff08Cantor set\uff09\u7684\u4e00\u90e8\u5206<\/strong> \u5f62\u4f3c\u9053\u5bb6\u7684\u5366\u56fe<\/strong><\/p>\n


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Mandelbrot \u96c6<\/strong><\/span><\/p>\n


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\u63d0\u5230\u5206\u5f62\uff0c\u6211\u4eec\u5c31\u4e0d\u5f97\u4e0d\u63d0\u66fc\u5fb7\u5e03\u6d1b\u7279\u96c6\u5408\uff08Mandelbrot set\uff09\uff0c\u8fd9\u662f\u4e00\u79cd\u5728\u590d\u5e73\u9762\u4e0a\u7ec4\u6210\u5206\u5f62\u7684\u70b9\u7684\u96c6\u5408\uff0c\u4ee5\u6570\u5b66\u5bb6<\/span>\u4f2f\u52aa\u00b7\u74e6\u66fc\u5fb7\u5e03\u6d1b\u7279<\/span>\uff08Beno\u00eet B. Mandelbrot\uff09<\/span><\/span>\u7684\u540d\u5b57\u547d\u540d\u3002Mandelbrot\u96c6\u5408\u4e0eJulia\u96c6\u5408\u6709\u4e9b\u76f8\u4f3c\u7684\u5730\u65b9\uff0c\u4f8b\u5982\u4ed6\u4eec\u90fd\u4f7f\u7528\u76f8\u540c\u7684\u590d\u4e8c\u6b21\u591a\u9879\u5f0f\u6765\u8fdb\u884c\u8fed\u4ee3\u3002<\/span><\/section>\n

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\u66fc\u5fb7\u5e03\u6d1b\u7279\u96c6\u5408\u53ef\u4ee5\u7528\u590d\u4e8c\u6b21\u591a\u9879\u5f0f\uff1a<\/span>\u3002\u66fc\u5fb7\u5e03\u6d1b\u7279\u96c6\u4e3b\u8981\u662f\u901a\u8fc7\u56fa\u5b9az0=0\uff0c\u4e0d\u65ad\u8fed\u4ee3c\u5f97\u5230\u7684\uff0c\u4ea7\u751f\u4e00\u7ec4\u53d1\u6563\u6570\u5217\uff1a<\/span>\u5982\u679c\u6570\u5217\u53d1\u6563\uff0c\u5219\u5728\u4e8c\u7ef4\u5e73\u9762\u5185\uff0c\u5c06\u6240\u6709\u4e0d\u5c5e\u4e8e\u96c6\u5408\u5185\u7684\u70b9\u6807\u8bb0\u4e3a\u9ed1\u8272\uff0c\u5c5e\u4e8e\u96c6\u5408\u5185\u7684\u70b9\u6309\u7167\u53d1\u6563\u901f\u5ea6\u8d4b\u4e88\u4e0d\u540c\u7684\u989c\u8272\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\u7ecf\u5178\u7684\u66fc\u5fb7\u5e03\u6d1b\u7279\u96c6\u56fe\u50cf\u3002<\/span><\/section>\n


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\u56fe4\uff1aThe Mandelbrot set<\/strong><\/p>\n


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Julia\u96c6<\/strong><\/span><\/p>\n


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Julia\u96c6\u548c Mandelbrot \u96c6\u5177\u6709\u76f8\u540c\u7684\u6570\u5b66\u8868\u793a\u5f62\u5f0f\uff0c\u552f\u4e00\u7684\u4e0d\u540c\u662f\uff0cJulia\u96c6\u662f\u901a\u8fc7\u56fa\u5b9ac\uff0c\u4e0d\u65ad\u8fed\u4ee3\uff0c\u8ba1\u7b97z\u7684\u503c\u5f97\u5230\u7684\uff0c\u4ea7\u751f\u4e00\u7ec4\u53d1\u6563\u5e8f\u5217<\/span>\u540c\u7406\uff0c\u5c06\u6240\u6709\u4e0d\u5c5e\u4e8e\u96c6\u5408\u5185\u7684\u70b9\u6807\u8bb0\u4e3a\u9ed1\u8272\uff0c\u5c5e\u4e8e \u96c6\u5408\u5185\u7684\u70b9\u6309\u7167\u53d1\u6563\u901f\u5ea6\u8d4b\u4e88\u4e0d\u540c\u7684\u989c\u8272\uff0c\u4f1a\u5f97\u5230\u4e0d\u540c\u7684Julia\u96c6\u56fe\u50cf\u3002<\/span><\/section>\n


<\/span><\/p>\n

<\/section>\n

\u56fe5\uff1a\u5f53c\u53d6\u67d0\u4e00\u503c\u7684\u65f6\u5019Julia\u96c6\u5408\u5728\u590d\u5e73\u9762\u4e0a\u7684\u56fe\u50cf<\/strong><\/p>\n


<\/p>\n

\u79d1\u8d6b\u66f2\u7ebf Koch curve<\/strong><\/span><\/p>\n


<\/p>\n

\u79d1\u8d6b\u66f2\u7ebf\uff08<\/span>Koch curve<\/span>\uff09\u662f\u4e00\u79cd\u795e\u5947\u7684\u66f2\u7ebf\uff0c\u56e0\u4e3a\u5f62\u6001\u8ddf\u96ea\u82b1\u5f88\u50cf\uff0c\u6240\u4ee5\u4e5f\u79f0\u4e3a\u79d1\u8d6b\u96ea\u82b1\uff0c\u6700\u65e9\u51fa\u73b0\u5728\u6d77\u91cc\u683c\u00b7\u51af\u00b7\u79d1\u8d6b<\/span>\uff08Helge von Koch\uff09<\/span>\u7684\u8bba\u6587\u4e2d\uff0c\u901a\u8fc7\u4e0b\u9762\u8fd9\u4e2a\u52a8\u56fe\uff0c\u4f60\u77e5\u9053\u79d1\u8d6b\u96ea\u82b1\u662f\u5982\u4f55\u5f62\u6210\u7684\u5417\uff1f<\/span><\/p>\n

<\/span><\/p>\n


<\/p>\n

<\/p>\n

\u7b2c\u4e00\u6b65\uff1a\u753b\u4e00\u4e2a\u7b49\u8fb9\u4e09\u89d2\u5f62\uff0c\u5e76\u628a\u6bcf\u4e00\u8fb9\u4e09\u7b49\u5206\uff1b<\/span><\/p>\n

\u7b2c\u4e8c\u6b65\uff1a\u53d6\u4e09\u7b49\u5206\u540e\u7684\u4e00\u8fb9\u4e2d\u95f4\u4e00\u6bb5\u4e3a\u8fb9\u5411\u5916\u4f5c\u6b63\u4e09\u89d2\u5f62\uff0c\u5e76\u628a\u8fd9\u201c\u4e0e\u4e3b\u4e09\u89d2\u5f62\u9760\u8fd1\u7684\u4e00\u8fb9\u201d\u64e6\u6389\uff1b<\/span><\/p>\n

\u7b2c\u4e09\u6b65\uff1a\u91cd\u590d\u4e0a\u8ff0\u6b65\u9aa4\uff0c\u753b\u51fa\u66f4\u5c0f\u7684\u4e09\u89d2\u5f62\uff1b<\/span><\/p>\n

\u7b2c\u56db\u6b65\uff1a\u4e00\u76f4\u91cd\u590d\u3002<\/span><\/p>\n


<\/p>\n

\u7531\u4e0a\u8ff0\u6b65\u9aa4\u6240\u753b\u51fa\u7684\u66f2\u7ebf\u53eb\u505a\u79d1\u8d6b\u66f2\u7ebf\uff08Koch curve\uff09\u3002<\/span><\/p>\n


<\/p>\n

\u8c22\u5c14\u5bbe\u65af\u57fa\u4e09\u89d2\u5f62 Sierpinski triangle<\/strong><\/span><\/p>\n


<\/span><\/p>\n

\u6211\u4eec\u90fd\u77e5\u9053\u4e00\u4e2a\u5e73\u9762\u4e09\u89d2\u5f62\u7684\u7ef4\u5ea6\u662f\u4e8c\u7ef4\u7684\uff0c\u4f46\u662f\u5728\u5206\u5f62\u51e0\u4f55\u4e2d\uff0c\u4e09\u89d2\u5f62\u6709\u4e0d\u540c\u7684\u7ef4\u5ea6\u8ba1\u7b97\u65b9\u6cd5\u3002\u8c22\u5c14\u5bbe\u65af\u57fa\u4e09\u89d2\u5f62\uff08Sierpinski triangle\uff09\u4e5f\u662f\u4e00\u79cd\u5206\u5f62\u5178\u578b\u793a\u4f8b\uff0c\u7531\u6ce2\u5170\u6570\u5b66\u5bb6\u8c22\u5c14\u5bbe\u65af\u57fa\u4e8e1915\u5e74\u63d0\u51fa\u3002\u5b83\u662f\u81ea\u76f8\u4f3c\u96c6\u7684\u4f8b\u5b50\u3002\u5b83\u7684\u8c6a\u65af\u591a\u592b\u7ef4\u6570\u662f<\/span>\u53ef\u4ee5\u901a\u8fc7\u591a\u79cd\u65b9\u6cd5\u5f97\u5230\u8fd9\u4e2a\u4e09\u89d2\u5f62\uff0c\u8fd9\u91cc\u4ecb\u7ecd\u5176\u4e2d\u7684\u4e00\u79cd\u65b9\u6cd5\uff1a<\/span><\/p>\n


<\/span><\/p>\n

\u7b2c\u4e00\u6b65\uff1a\u53d6\u4e00\u4e2a\u5b9e\u5fc3\u7684\u4e09\u89d2\u5f62\u3002(\u591a\u6570\u60c5\u51b5\u4e0b\u4f7f\u7528\u7b49\u8fb9\u4e09\u89d2\u5f62\uff09\uff1b<\/span><\/section>\n
\u7b2c\u4e8c\u6b65\uff1a\u6cbf\u4e09\u8fb9\u4e2d\u70b9\u7684\u8fde\u7ebf\uff0c\u5c06\u5b83\u5206\u6210\u56db\u4e2a\u5c0f\u4e09\u89d2\u5f62\uff1b<\/span><\/section>\n
\u7b2c\u4e09\u6b65\uff1a\u53bb\u6389\u4e2d\u95f4\u7684\u90a3\u4e00\u4e2a\u5c0f\u4e09\u89d2\u5f62\uff1b<\/span><\/section>\n
\u7b2c\u56db\u6b65\uff1a\u5bf9\u5176\u4f59\u4e09\u4e2a\u5c0f\u4e09\u89d2\u5f62\u91cd\u590d\u6b65\u9aa4\u4e00\u3002<\/span><\/section>\n

<\/span><\/section>\n


<\/p>\n

<\/p>\n

\u56fe5\uff1a\u8c22\u5c14\u5bbe\u65af\u57fa\u4e09\u89d2\u5f62<\/strong>\uff08Sierpinski triangle\uff09<\/strong><\/p>\n


<\/section>\n

\n
\n
\n
\n
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<\/p>\n

4.\u8457\u540d\u5b66\u8005\u7b80\u4ecb<\/strong><\/p>\n

<\/strong><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h1>\n

 <\/span><\/section>\n

\u4f2f\u52aa\u00b7\u74e6\u66fc\u5fb7\u5e03\u6d1b\u7279 Beno\u00eet B. Mandelbrot<\/strong><\/span><\/p>\n


<\/strong><\/span><\/p>\n

<\/section>\n

\u56fe6\uff1a\u4f2f\u52aa\u00b7\u74e6\u66fc\u5fb7\u5e03\u6d1b\u7279 Beno\u00eet B. Mandelbrot<\/strong><\/p>\n


<\/p>\n

\u6cd5\u56fd\u3001\u7f8e\u56fd\u6570\u5b66\u5bb6\uff08\u53cc\u56fd\u7c4d\uff09\uff0c\u6700\u5927\u7684\u6210\u5c31\u662f\u521b\u7acb\u4e86\u5206\u5f62\u51e0\u4f55\u30021967\u5e74\uff0c<\/span>\u66fc\u5fb7\u5e03\u6d1b\u7279<\/span>\u5728\u7f8e\u56fd\u300aScience\u300b\u6742\u5fd7\u4e0a\u53d1\u8868\u9898\u76ee\u4e3a\u300a\u82f1\u56fd\u7684\u6d77\u5cb8\u7ebf\u6709\u591a\u957f\u300b\u7684\u5212\u65f6\u4ee3\u8bba\u6587\uff0c\u6807\u5fd7\u7740\u5176\u5206\u5f62\u601d\u60f3\u7684\u840c\u82bd\u3002<\/span><\/p>\n

 <\/span><\/p>\n

\u6731\u5c14\u00b7\u4ea8\u5229\u00b7\u5e9e\u52a0\u83b1 Jules Henri Poincar\u00e9<\/strong><\/span><\/p>\n


<\/strong><\/span><\/p>\n

<\/p>\n

\u56fe7\uff1a\u6731\u5c14\u00b7\u4ea8\u5229\u00b7\u5e9e\u52a0\u83b1 Jules Henri Poincar\u00e9<\/strong>
<\/p>\n


<\/span><\/p>\n

\u6cd5\u56fd\u6700\u4f1f\u5927\u7684\u6570\u5b66\u5bb6\u4e4b\u4e00\uff0c\u7406\u8bba\u79d1\u5b66\u5bb6\u548c\u79d1\u5b66\u54f2\u5b66\u5bb6\u3002\u5e9e\u52a0\u83b1\u88ab\u516c\u8ba4\u662f19\u4e16\u7eaa\u540e\u548c20\u4e16\u7eaa\u521d\u7684\u9886\u8896\u6570\u5b66\u5bb6\uff0c\u662f\u7ee7\u9ad8\u65af\u4e4b\u540e\u5bf9\u4e8e\u6570\u5b66\u53ca\u5176\u5e94\u7528\u5177\u6709\u5168\u9762\u77e5\u8bc6\u7684\u6700\u540e\u6570\u5b66\u5bb6\u3002\u65e9\u572819\u4e16\u7eaa\u521d\uff0c\u5e9e\u52a0\u83b1\u5728\u7814\u7a76\u4e09\u4f53\u95ee\u9898\u4e2d\uff0c\u4f7f\u7528\u4e86\u65b0\u7684\u51e0\u4f55\u65b9\u6cd5\u3002<\/span><\/p>\n

 <\/span><\/p>\n

\u683c\u5965\u5c14\u683c\u00b7\u5eb7\u6258\u5c14Georg Cantor<\/strong><\/span><\/p>\n


<\/strong><\/span><\/p>\n

<\/p>\n

\u56fe8\uff1a\u683c\u5965\u5c14\u683c\u00b7\u5eb7\u6258\u5c14Georg Cantor<\/strong>
<\/p>\n


<\/span><\/p>\n

\u5fb7\u56fd\u6570\u5b66\u5bb6\uff0c\u521b\u7acb\u4e86\u73b0\u4ee3\u96c6\u5408\u8bba\uff0c\u662f\u5b9e\u6570\u7cfb\u4ee5\u81f3\u6574\u4e2a\u5fae\u79ef\u5206\u7406\u8bba\u4f53\u7cfb\u7684\u57fa\u7840\uff0c\u6784\u9020\u4e86\u4e09\u5206\u5eb7\u6258\u5c14\u96c6\u3002<\/span><\/p>\n

 <\/span><\/p>\n

\u6d77\u91cc\u683c\u00b7\u51af\u00b7\u79d1\u8d6b Helge von Koch<\/strong><\/span><\/p>\n


<\/strong><\/span><\/p>\n

<\/p>\n

\u56fe9\uff1a\u6d77\u91cc\u683c\u00b7\u51af\u00b7\u79d1\u8d6b Helge von Koch<\/strong>
<\/p>\n


<\/span><\/p>\n

\u745e\u5178\u6570\u5b66\u5bb6\uff0c\u5728\u4ed61904\u5e74\u7684\u4e00\u7bc7\u8bba\u6587\u201c\u5173\u4e8e\u4e00\u4e2a\u53ef\u7531\u57fa\u672c\u51e0\u4f55\u65b9\u6cd5\u6784\u9020\u51fa\u7684\uff0c\u65e0\u5207\u7ebf\u7684\u8fde\u7eed\u66f2\u7ebf\u201d\uff08\u539f\u6587\u7684\u6cd5\u6587\u6807\u9898\u4e3a\uff1a\u201cSur une courbe continue sans tangente, obtenue par une construction g\u00e9om\u00e9trique \u00e9l\u00e9mentaire<\/em>\u201d\uff09\u4e2d\uff0c\u4ed6\u63cf\u8ff0\u4e86\u79d1\u8d6b\u66f2\u7ebf\u7684\u6784\u9020\u65b9\u6cd5\u3002<\/span><\/p>\n

 <\/span><\/p>\n

\u74e6\u8328\u74e6\u592b\u00b7\u8c22\u5c14\u5bbe\u65af\u57fa Wac\u0142aw Sierpi\u0144ski<\/strong><\/span><\/p>\n


<\/strong><\/span><\/p>\n

<\/p>\n

\u56fe10\uff1a\u74e6\u8328\u74e6\u592b\u00b7\u8c22\u5c14\u5bbe\u65af\u57fa Wac\u0142aw Sierpi\u0144ski<\/strong><\/p>\n


<\/p>\n

\u6ce2\u5170\u6570\u5b66\u5bb6\uff0c\u56e0\u5bf9\u96c6\u5408\u8bba\u3001\u6570\u8bba\u3001\u51fd\u6570\u7684\u7406\u8bba\u548c\u62d3\u6251\u5b66\u7684\u51fa\u8272\u8d21\u732e\u800c\u95fb\u540d\u3002\u4e24\u4e2a\u8457\u540d\u7684\u5206\u5f62\u662f\u6839\u636e\u4ed6\u7684\u540d\u5b57\u547d\u540d\uff1a\u8c22\u5c14\u5bbe\u65af\u57fa\u4e09\u89d2\u5f62\u548c\u8c22\u5c14\u5bbe\u65af\u57fa\u5730\u6bef\uff1b\u53e6\u5916\u8fd8\u6709\u8c22\u5c14\u5bbe\u65af\u57fa\u6570\u548c\u8c22\u5c14\u5bbe\u65af\u57fa\u95ee\u9898\u4e5f\u662f\u4ee5\u4ed6\u7684\u540d\u5b57\u547d\u540d\u3002<\/span><\/p>\n


<\/p>\n

\u8d39\u5229\u514b\u65af\u00b7\u8c6a\u65af\u591a\u592b<\/strong> Felix Hausdorff<\/strong><\/span><\/p>\n


<\/strong><\/span><\/p>\n

<\/p>\n

\u56fe11\uff1a\u8d39\u5229\u514b\u65af\u00b7\u8c6a\u65af\u591a\u592b Felix Hausdorff<\/strong><\/p>\n


<\/p>\n

\u5fb7\u56fd\u6570\u5b66\u5bb6\uff0c\u4ed6\u662f\u62d3\u6251\u5b66\u7684\u521b\u59cb\u4eba\u4e4b\u4e00\uff0c\u4ed6\u5b9a\u4e49\u548c\u7814\u7a76\u504f\u5e8f\u96c6\u3001\u8c6a\u65af\u591a\u592b\u7a7a\u95f4\u548c\u8c6a\u65af\u591a\u592b\u7ef4\uff0c\u8bc1\u660e\u8c6a\u65af\u591a\u592b\u6781\u5927\u5b9a\u7406\uff08Hausdorff maximality theorem\uff09\u3002\u63d0\u51fa\u4e86\u8c6a\u65af\u591a\u592b\u7ef4\u6570\u6765\u8ba1\u7b97\u5206\u5f62\u7ef4\u6570\u3002<\/span><\/p>\n


<\/p>\n

\n
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\n
\n
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<\/p>\n

5.\u76f8\u5173\u8d44\u6e90\u63a8\u8350<\/strong>
<\/strong><\/p>\n

<\/strong><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h1>\n


<\/section>\n

\u4e66\u7c4d\uff1a<\/strong><\/span><\/p>\n

\n


<\/p>\n

\u5206\u5f62\u5bf9\u8c61\uff1a<\/strong>\u5f62\u3001\u673a\u9047\u548c\u7ef4\u6570<\/strong><\/span> <\/span><\/p>\n

Fractals:From,Chance,and Dimension<\/em><\/p>\n

 <\/span><\/p>\n

\u672c\u4e66\u8003\u5bdf\u548c\u7814\u7a76\u51fa\u73b0\u5728\u81ea\u7136\u754c\u4e2d\u7684\u82e5\u5e72\u5178\u578b\u5206\u5f62\u5bf9\u8c61\uff0c\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4e00\u4e2a\u5173\u4e8e\u5206\u5f62\u7684\u5185\u5bb9\uff0c\u610f\u4e49\u53ca\u65b9\u6cd5\u7684\u627c\u8981\u4ecb\u7ecd\u3002\u5c3d\u7ba1\u81ea\u8be5\u4e66\u7b2c\u4e00\u7248\uff08\u6cd5\u6587\u7248\uff09\u95ee\u4e16\u4ee5\u6765\uff0c\u5206\u5f62\u7684\u7406\u8bba\u53ca\u5176\u5e94\u7528\u53d1\u5c55\u6781\u4e3a\u8fc5\u901f\uff0c\u5e76\u51fa\u73b0\u4e86\u5927\u91cf\u7684\u6709\u5173\u8457\u4f5c\uff0c\u4f46\u6b64\u4e66\u4ecd\u4e0d\u5931\u4e3a\u5206\u5f62\u7406\u8bba\u6700\u597d\u7684\u5165\u95e8\u4e66\u4e4b\u4e00\u3002<\/span><\/p>\n

\n

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\n
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\u5206\u5f62\u5bf9\u8c61\uff1a<\/span>\u5f62\u3001\u673a\u9047\u548c\u7ef4\u6570 <\/span><\/p>\n

Fractals:From,Chance,and Dimension<\/span><\/p>\n

http:\/\/rrd.me\/gfyYz<\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n<\/section>\n


<\/p>\n

\u5927\u81ea\u7136\u7684\u5206\u5f62\u51e0\u4f55<\/strong><\/span> <\/span><\/em><\/p>\n

The Fractal Geometry of Nature<\/span><\/em><\/p>\n


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\u5927<\/span>\u81ea\u7136\u7684\u5206\u5f62\u51e0\u4f55 The Fractal Geometry of Nature<\/span><\/p>\n

http:\/\/rrd.me\/gfzjk<\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n

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\u5e02\u573a\u7684\uff08\u9519\u8bef\uff09\u884c\u4e3a\uff1a<\/strong>\u98ce\u9669\u3001\u7834\u4ea7\u4e0e\u6536\u76ca\u7684\u5206\u5f62\u89c2\u70b9<\/strong><\/span><\/p>\n

The Misbehavior of Markets: A Fractal View of Financial Turbulence<\/em><\/p>\n


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\u300a\u5e02\u573a\u7684\uff08\u9519\u8bef\uff09\u884c\u4e3a\u300b\u4ee5\u5206\u5f62\u89c6\u89d2\u89c2\u5bdf\u91d1\u878d\u5e02\u573a\u7684\u884c\u4e3a\uff0c\u63a8\u7ffb\u4e86\u4f5c\u4e3a\u5f53\u4ee3\u91d1\u878d\u5206\u6790\u57fa\u7840\u7684\u201c\u968f\u673a\u6e38\u8d70\u201d\u7406\u8bba\u3002\u901a\u8fc7\u5206\u5f62\u6a21\u578b\uff0c\u5e02\u573a\u8868\u73b0\u88ab\u91cd\u65b0\u9610\u91ca\u3002\u672c\u4e66\u662f\u73b0\u4ee3\u91d1\u878d\u7406\u8bba\u6807\u51c6\u5de5\u5177\u548c\u6a21\u578b\u7684\u4e00\u6b21\u9769\u547d\u6027\u91cd\u4f30\uff0c\u4e66\u4e2d\u7684\u89c2\u70b9\u98a0\u8986\u4e86\u6210\u5343\u4e0a\u4e07\u6295\u8d44\u8005\u7684\u65e2\u6709\u89c2\u5ff5\u3002<\/span><\/p>\n<\/section>\n

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\u5e02\u573a\u7684\uff08\u9519\u8bef\uff09\u884c\u4e3a\uff1a<\/span>\u98ce\u9669\u3001\u7834\u4ea7\u4e0e\u6536\u76ca\u7684\u5206\u5f62\u89c2\u70b9The Misbehavior of Markets: A Fractal View of Financial Turbulence<\/span><\/p>\n

http:\/\/rrd.me\/gfz56<\/span><\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n


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\u8bfe\u7a0b<\/strong><\/span>\uff1a<\/strong><\/span><\/p>\n


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\u5206\u5f62\u4e0e\u5947\u5f02\u5438\u5f15\u5b50\u7684\u51e0\u4f55\u5b66<\/span><\/strong><\/span>
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\u6b64\u8bfe\u7a0b\u96b6\u5c5e\u4e8e\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u4e0e\u6df7\u6c8c<\/span><\/p>\n

\u672c\u8bfe\u7a0b\u5bf9\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u548c\u6df7\u6c8c\u8fdb\u884c\u4e86\u8be6\u7ec6\u7684\u8bb2\u89e3\uff0c\u5f3a\u8c03\u5206\u6790\u65b9\u6cd5\u3001\u5177\u4f53\u5b9e\u4f8b\u548c\u51e0\u4f55\u76f4\u89c9\u3002\u4e3b\u8bb2\u4eba\u4e3aSteven Strogatz\u3002<\/span><\/p>\n


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\u5206\u5f62\u4e0e\u5947\u5f02\u5438\u5f15\u5b50\u7684\u51e0\u4f55\u5b66<\/span><\/p>\n

https:\/\/campus.swarma.org\/play\/coursedetail?id=10621<\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n


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TED\u5206\u4eab\u89c6\u9891\uff1aBenoit Mandelbrot: Fractals and the Art of Roughness <\/span>\u4f2f\u52aa\u00b7\u74e6\u66fc\u5fb7\u5e03\u6d1b\u7279: \u5206\u5f62\u548c\u7c97\u7cd9\u7684\u827a\u672f\uff1a<\/span><\/p>\n

http:\/\/www.ted.com\/talks\/benoit_mandelbrot_fractals_the_art_of_roughness.html<\/span><\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n

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\u7eaa\u5f55\u7247\uff1a<\/span>
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Hunting the Hidden Dimension \u5bfb\u627e\u9690\u85cf\u7684\u7ef4<\/span><\/p>\n

https:\/\/www.bilibili.com\/video\/av13766486\/?p=2<\/span><\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n

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\u5173\u4e8e\u5206\u5f62\u7684\u66f4\u591a\u4ecb\u7ecd\uff1a<\/span><\/p>\n

http:\/\/webarchive.loc.gov\/all\/20011116035117\/http:\/\/dmoz.org\/science\/math\/chaos_and_fractals\/<\/span>
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\u77e5\u8bc6\u6982\u5ff5\uff1a<\/strong><\/span><\/p>\n

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\u65e0\u6807\u5ea6\u7279\u6027<\/p>\n

\u5eb7\u6258\u5c14\u96c6\uff08Cantor Set\uff09<\/p>\n

\u66fc\u5fb7\u5e03\u6d1b\u7279\u96c6\u5408\uff08Mandelbrot set\uff09<\/p>\n

Julia\u96c6<\/p>\n

\u8c22\u5c14\u5bbe\u65af\u57fa\u4e09\u89d2\u5f62\uff08Sierpinski triangle\uff09<\/p>\n

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\u4e66\u7c4d\uff1a<\/strong><\/span><\/p>\n


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\u5206\u5f62\u5bf9\u8c61\uff1a\u5f62\u3001\u673a\u9047\u548c\u7ef4\u6570 <\/p>\n

Fractals:From,Chance,and Dimension<\/p>\n


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\u5927\u81ea\u7136\u7684\u5206\u5f62\u51e0\u4f55 <\/p>\n

The Fractal Geometry of Nature<\/p>\n

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\u5e02\u573a\u7684\uff08\u9519\u8bef\uff09\u884c\u4e3a\uff1a\u98ce\u9669\u3001\u7834\u4ea7\u4e0e\u6536\u76ca\u7684\u5206\u5f62\u89c2\u70b9<\/p>\n

The Misbehavior of Markets: A Fractal View of Financial Turbulence<\/p>\n

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\u671f\u520a\uff1a<\/strong><\/span><\/p>\n

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Chaos\uff0cSoliton and Fractals<\/p>\n


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[1]\u5206\u5f62\uff0c\u767e\u5ea6\u767e\u79d1 :http:\/\/baike.baidu.com\/view\/83243.htm<\/span><\/strong><\/span><\/p>\n

[2]\u5206\u5f62\uff0c\u7ef4\u57fa\u767e\u79d1:<\/span><\/strong><\/span><\/p>\n

http:\/\/zh.wikipedia.org\/wiki\/%E5%88%86%E5%BD%A2<\/span><\/strong><\/span><\/p>\n

[3]Mandelbrot set, WIKI :http:\/\/en.wikipedia.org\/wiki\/Mandelbrot_set<\/span><\/strong><\/span><\/p>\n

[4]Julia set, WIKI :http:\/\/en.wikipedia.org\/wiki\/Julia_set<\/span><\/strong><\/span><\/p>\n

[5]\u795e\u5947\u7684\u5206\u5f62\u827a\u672f\uff08\u56db\uff09\uff1a<\/span><\/strong><\/span><\/p>\n

Julia \u96c6\u548c Mandelbrot \u96c6\uff0cMatrix67 http:\/\/www.matrix67.com\/blog\/archives\/292<\/span><\/strong><\/span><\/p>\n

[6] \u8c6a\u65af\u591a\u592b\u7ef4\u6570<\/span><\/strong><\/span><\/p>\n

http:\/\/zh.wikipedia.org\/wiki\/%E8%B1%AA%E6%96%AF%E5%A4%9A%E5% A4%AB%E7%BB%B4%E6%95%B0<\/span><\/strong><\/span><\/p>\n

[7] \u8ba1\u76d2\u4f4d\u6570 <\/span><\/strong><\/span><\/p>\n

http:\/\/zh.wikipedia.org\/wiki\/%E8%AE%A1%E7%9B%92%E7%BB%B4%E6%9 5%B0<\/span><\/strong><\/span><\/p>\n

[8] \u79d1\u8d6b\u66f2\u7ebf\uff0c\u7ef4\u57fa\u767e\u79d1<\/span><\/strong><\/span><\/p>\n

http:\/\/zh.wikipedia.org\/wiki\/%E7%A7%91%E8%B5%AB%E6%9B%B2%E7%B 7%9A<\/span><\/strong><\/span><\/p>\n

[9]  \u8c22\u5c14\u5bbe\u65af\u57fa\u4e09\u89d2\u5f62<\/span><\/strong><\/span><\/p>\n

http:\/\/zh.wikipedia.org\/wiki\/%E8%AC%9D%E7%88%BE%E8%B3%93%E6%96%AF%E5%9F %BA%E4%B8%89%E8%A7%92%E5%BD%A2<\/span><\/strong><\/span><\/p>\n

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[10]<\/span><\/strong>\u9648\u5173\u8363\u8001\u5e08\u5173<\/span><\/strong><\/span>\u4e8e\u5206\u5f62\u7684\u4fe1\u606f\u6574\u7406<\/span><\/strong><\/span>\uff1a<\/span><\/strong><\/p>\n

https:\/\/www.ee.cityu.edu.hk\/~gchen\/pdf\/FractalsStories.pdf<\/span><\/strong><\/span><\/p>\n<\/section>\n

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[11]<\/span><\/strong>\u5206\u5f62\u7684\u4ee3\u7801\u6f14\u793a<\/span><\/strong><\/span>\uff1a<\/span><\/strong><\/p>\n

http:\/\/www.fxysw.com\/thread-4982-1-1.html<\/span><\/strong><\/span><\/p>\n<\/section>\n

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[12]<\/span><\/strong>IBM\u7684\u5206\u5f62<\/span><\/strong><\/span>\uff1a<\/span><\/strong><\/p>\n

https:\/\/www31.ibm.com\/ibm\/cn\/ibm100\/icons\/fractal\/impacts.shtml <\/span><\/strong><\/span><\/p>\n<\/section>\n

[13]<\/span><\/strong>\u6d45\u8c08\u5206\u5f62\u51e0\u4f55\u5b66 <\/span><\/strong><\/span>http:\/\/www.yingzhenli.net\/home\/pdf\/%E5%88%86%E5%BD%A2%E4%B8%8E%E6%B7%B7%E6%B2%8C.pdf <\/span><\/strong><\/span><\/section>\n
[14]<\/span><\/strong>youtube\u4e0a\u7684\u89c6\u9891\uff1a<\/span><\/strong><\/span><\/section>\n
https:\/\/www.youtube.com\/watchv=PD2XgQOyCCk<\/span><\/strong><\/span><\/span><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n
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\u5982\u4f55\u7528 Mathematica  \u7aa5\u63a2\u8d28\u6570\u5206\u5e03\u4e2d\u9690\u85cf\u7684\u5206\u5f62\u89c4\u5f8b<\/a><\/span><\/strong><\/section>\n
\u7cfb\u5217\u516c\u5f00\u8bfe\u9884\u544a\uff1a\u638c\u63e1\u590d\u6742\u6027\u601d\u7ef4\uff0c\u5e94\u5bf9\u77ac\u606f\u4e07\u53d8\u7684\u4e16\u754c<\/a><\/span><\/strong><\/section>\n
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\u5546\u52a1\u5408\u4f5c\u53ca\u6295\u7a3f\u8f6c\u8f7d\uff5cswarma@swarma.org
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