{"id":43,"date":"2008-10-25T01:23:23","date_gmt":"2008-10-24T17:23:23","guid":{"rendered":"http:\/\/xgkwx.72pines.com\/2008\/10\/25\/43\/"},"modified":"2010-11-13T03:32:22","modified_gmt":"2010-11-12T19:32:22","slug":"43","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/43","title":{"rendered":"\u6211\u7684\u535a\u58eb\u8bfe\u9898\u7684\u79d1\u666e\u4ecb\u7ecd"},"content":{"rendered":"<div class=\"kcite-section\" kcite-section-id=\"43\">\n<h1>\u6211\u7684\u535a\u58eb\u8bfe\u9898\u7684\u79d1\u666e\u4ecb\u7ecd<\/h1>\n<p>\u6211\u5728\u8fd9\u91cc\u4ecb\u7ecd\u4e00\u4e0b\u6211\u7684\u535a\u58eb\u8bfe\u9898\u7684\u80cc\u666f\u3002<\/p>\n<h2>\u81ea\u76f8\u4f3c\u2014\u2014\u5927\u81ea\u7136\u7684\u827a\u672f<\/h2>\n<blockquote><p>Why is geometry often described as &#8216;cold&#8217; and &#8216;dry&#8217;? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.<\/p>\n<p>\u4e3a\u4ec0\u4e48\u51e0\u4f55\u5b66\u5e38\u88ab\u8ba4\u4e3a\u662f\u201c\u51b7\u9177\u201d\u3001\u201c\u67af\u71e5\u201d\uff1f\u5176\u4e2d\u4e00\u4e2a\u539f\u56e0\u5c31\u662f\u5b83\u65e0\u6cd5\u63cf\u8ff0\u4e91\u5f69\u3001\u5c71\u5ce6\u3001\u6d77\u5cb8\u3001\u6811\u6728\u2014\u2014\u6211\u4eec\u7f8e\u4e3d\u7684\u5927\u81ea\u7136\u3002\u4e91\u6735\u4e0d\u662f\u4e00\u4e9b\u7403\u5f62\uff0c\u5c71\u5ce6\u4e0d\u662f\u4e00\u4e9b\u5706\u9525\uff0c\u5c9b\u5c7f\u4e0d\u662f\u4e00\u4e9b\u5706\u5f62\uff0c\u6811\u76ae\u662f\u4e0d\u5149\u6ed1\u7684\uff0c\u95ea\u7535\u4e5f\u4e0d\u6cbf\u4e00\u6761\u76f4\u7ebf\u3002<\/p>\n<p style=\"text-align: right\"><em> &#8211;B. Mandelbrot<\/em><\/p>\n<\/blockquote>\n<p style=\"text-align: center\"><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_690f8v5zjd9_b\" alt=\"\" \/><br \/>\nKoch\u66f2\u7ebf<\/p>\n<p>\u81ea\u76f8\u4f3c\u6027\u662f\u6307\u6bcf\u4e00\u4efb\u610f\u5c0f\u7684\u5c40\u90e8\u7684\u5f62\u72b6\u90fd\u4e0e\u6574\u4f53\u76f8\u540c\uff0c\u4f8b\u5982\uff0cKoch\u66f2\u7ebf\u5c31\u662f\u4e00\u4e2a\u81ea\u76f8\u4f3c\u6027\u7684\u56fe\u5f62\u3002 \u5728\u81ea\u7136\u754c\u4e2d\uff0c\u8bb8\u591a\u81ea\u53d1\u751f\u957f\u7684\u7ed3\u6784\u90fd\u5177\u6709\u81ea\u76f8\u4f3c\u6027\u3002\u4ee5\u4e0b\u662f\u4e00\u4e9b\u81ea\u7136\u754c\u7684\u81ea\u76f8\u4f3c\u7ed3\u6784\uff1a<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_695fz26xxdr_b\" alt=\"\" width=\"285\" height=\"249\" \/><br \/>\n\u6811\u53f6<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_696g5k2gktb_b\" alt=\"\" width=\"243\" height=\"168\" \/><br \/>\n\u9752\u82d4<br \/>\n<!--more--><\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_697c4h6kpcf_b\" alt=\"\" width=\"295\" height=\"221\" \/><br \/>\n\u8702\u871c\u7684\u7ed3\u6676<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_698fhkd8ffm_b\" alt=\"\" \/><br \/>\n\u6811\u51a0<\/p>\n<p>\u8fd9\u4e9b\u81ea\u76f8\u4f3c\u7684\u5f62\u72b6\uff0c\u5728\u6570\u5b66\u4e0a\u53eb\u505a\u5206\u5f62\u3002<\/p>\n<h2>\u5206\u5f62\u2014\u2014\u884c\u8d70\u4e8e\u7ef4\u5ea6\u4e4b\u95f4<\/h2>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_701f9cn75fn_b\" alt=\"\" \/><\/p>\n<p>\u56de\u5230\u6587\u7ae0\u7b2c\u4e00\u6bb5\u63d0\u5230\u7684Koch\u66f2\u7ebf\uff0c\u5b83\u662f\u600e\u6837\u753b\u51fa\u6765\u7684\u5462\uff1f\u9996\u5148\u753b\u4e00\u6bb5\u957f\u5ea6\u4e3a1\u7684\u7ebf\u6bb5\uff0c\u7136\u540e\u64e6\u6389\u4e2d\u95f4\u7684\u4e00\u90e8\u5206\uff0c\u6539\u6210\u4e00\u4e2a\u7b49\u8fb9\u4e09\u89d2\u5f62\uff0c\u5176\u8fb9\u957f\u548c\u5b83\u4e24\u8fb9\u64e6\u5269\u4e0b\u7684\u7ebf\u6bb5\u957f\u5ea6\u76f8\u7b49\uff0c\u5f97\u5230\u56db\u6761\u7ebf\u6bb5\u3002\u7136\u540e\u518d\u5bf9\u8fd9\u56db\u6761\u7ebf\u6bb5\u505a\u540c\u6837\u7684\u4e8b\u60c5\uff0c\u65e0\u6b62\u5883\u5730\u505a\u4e0b\u53bb\u2026\u2026\u8fd9\u5c31\u662fKoch\u66f2\u7ebf\u3002<\/p>\n<p>Koch\u66f2\u7ebf\u7684\u957f\u5ea6\u662f\u591a\u5c11\u5462\uff1f\u6570\u5b66\u5bb6\u91cf\u66f2\u7ebf\u957f\u5ea6\u7684\u529e\u6cd5\u662f\u903c\u8fd1\u6cd5\uff0c\u5982\u4e0b\u56fe\u6240\u793a\uff1a<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_702f3mtv7f5_b\" alt=\"\" \/><\/p>\n<p>\u6211\u4eec\u5148\u7528\u903c\u8fd1\u6cd5\u91cf\u4e00\u4e2a\u534a\u5f84\u4e3a1\u5706\u7684\u5468\u957f\u3002\u6700\u7c97\u7565\u7684\u8fd1\u4f3c\u662f\u5b83\u7684\u5185\u63a5\u4e09\u89d2\u5f62\u7684\u5468\u957f\uff0c\u7136\u540e\u662f\u5185\u63a5\u56db\u8fb9\u5f62\u3001\u5185\u63a5\u4e94\u8fb9\u5f62\u2026\u2026\u5185\u63a5N\u8fb9\u5f62\u3002\u628a\u8fd9\u4e9bN\u8fb9\u5f62\u7684\u5468\u957f\u753b\u5728\u5750\u6807\u56fe\u4e2d\uff0c\u53ef\u89c1\u5f53N\u589e\u5927\u65f6\uff0cN\u8fb9\u5f62\u7684\u5468\u957f\u8d8a\u6765\u8d8a\u903c\u8fd1\u4e00\u4e2a\u786e\u5b9a\u503c\u3002\u5f53N\u65e0\u7a77\u5927\u65f6\uff0c\u8fd9\u4e2aN\u8fb9\u5f62\u7684\u9762\u79ef\u5c31\u7b49\u4e8e\u5706\u7684\u5468\u957f\uff1a2\u03c0\u3002<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_703gs2cwjgz_b\" alt=\"\" \/><br \/>\n<img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_704hs6wqcdv_b\" alt=\"\" \/><\/p>\n<p>\u73b0\u5728\u91c7\u7528\u903c\u8fd1\u6cd5\u91cf\u4e00\u4e0bKoch\u66f2\u7ebf\u7684\u957f\u5ea6\uff1a<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_706wjfc9xgn_b\" alt=\"\" \/><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_707sk54gpf7_b\" alt=\"\" \/><\/p>\n<p>\u6211\u4eec\u91cd\u8d70\u4e00\u6b21\u753bKoch\u66f2\u7ebf\u7684\u8fc7\u7a0b\u6765\u903c\u8fd1\u6700\u540eKoch\u66f2\u7ebf\u7684\u957f\u5ea6\u3002\u8fd8\u8bb0\u5f97Koch\u66f2\u7ebf\u600e\u4e48\u753b\u5417\uff1f\u7b2c\u4e00\u6b21\u753b\u7684\u662f\u957f\u5ea6\u4e3a1\u7684\u7ebf\u6bb5\uff0c\u663e\u7136\uff0cKoch\u7684\u957f\u5ea6\u662f\u5927\u4e8e1\u7684\u3002\u7b2c\u4e8c\u6b21\u753b\u7684\u6298\u7ebf\uff0c\u957f\u5ea6\u662f\u591a\u5c11\u5462\uff1f\u4e2d\u95f4\u7684\u7b49\u8fb9\u4e09\u89d2\u5f62\u8fb9\u957f\u662f1\/3\uff0c\u56e0\u6b64\u6298\u7ebf\u7684\u603b\u957f\u662f4\/3\u3002\u663e\u7136\uff0cKoch\u7684\u957f\u5ea6\u6bd44\/3\u8981\u5927\u3002\u7b2c\u4e09\u6b21\uff0c\u753b\u7684\u6298\u7ebf\u957f\u5ea6\u662f16\/9\uff0c\u5373(4\/3)<sup>2<\/sup>\uff0c\u8bfb\u8005\u53ef\u4ee5\u81ea\u5df1\u7b97\u4e00\u4e0b\u3002\u8fd9\u4ecd\u7136\u5c0f\u4e8eKoch\u7684\u957f\u5ea6\u3002\u7b2c\u56db\u6b21\uff0c(4\/3)<sup>3<\/sup>=64\/27\uff0c\u8fd8\u662f\u4e0d\u591f\u957f\u2026\u2026\u7b2cN\u6b21\uff0c\u957f\u5ea6\u662f(4\/3)<sup>N<\/sup>\u3002\u5f88\u663e\u7136\uff0c\u4e0d\u7ba1\u6211\u4eec\u753b\u7684\u6298\u7ebf\u6709\u591a\u5c11\u6298\uff0cKoch\u66f2\u7ebf\u603b\u662f\u6bd4\u6298\u7ebf\u957f\u5ea6\u8981\u5927\u7684\u3002\u6211\u4eec\u5e0c\u671b\uff0c\u5f53N\u4e3a\u65e0\u7a77\u5927\u65f6\uff0c\u6298\u7ebf\u7684\u957f\u5ea6\u80fd\u81f3\u5c11\u80fd\u903c\u8fd1\u4e00\u4e2a\u786e\u5b9a\u7684\u503c\uff0c\u800c\u5b83\u5c31\u662fKoch\u7684\u957f\u5ea6\u3002\u6211\u4eec\u628a\u6298\u7ebf\u957f\u5ea6\u968fN\u4e0d\u65ad\u589e\u5927\u7684\u8d8b\u52bf\u753b\u5728\u5750\u6807\u56fe\u4e0a\u770b\u770b\u2014\u2014\u5f88\u4e0d\u5e78\uff0c\u968f\u7740N\u4e0d\u65ad\u589e\u5927\uff0c\u6298\u7ebf\u7684\u957f\u5ea6\u4e0d\u662f\u903c\u8fd1\u4e00\u4e2a\u786e\u5b9a\u503c\uff0c\u800c\u662f\u8d8b\u4e8e\u65e0\u7a77\u5927\u3002\u4f8b\u5982\uff0c\u5f53\u6211\u4eec\u753b\u5230\u7b2c128\u6298\u7ebf\u65f6\uff08N=128\uff09\uff0c\u5b83\u7684\u957f\u5ea6\u5df2\u7ecf\u662f(4\/3)<sup>128<\/sup>\u2014\u20141\u5149\u5e74\uff01\u800cKoch\u66f2\u7ebf\u80af\u5b9a\u8981\u6bd4\u8fd9\u6761\u6298\u7ebf\u5927\u3002\u7531\u4e8eKoch\u66f2\u7ebf\u662f\u65e0\u9650\u81ea\u76f8\u4f3c\u4e0b\u53bb\u7684\uff0c\u56e0\u6b64\u5b83\u7684\u957f\u5ea6\u662f\u65e0\u7a77\u5927\u7684\u3002<\/p>\n<p>\u60a8\u6709\u672c\u4e8b\u5728\u8ddd\u79bb\u4e3a1\u4e24\u70b9\u95f4\u753b\u4e00\u6761\u65e0\u7a77\u957f\u7684\u66f2\u7ebf\u5417\uff1f\u53ea\u51c6\u7528\u6709\u9650\u9762\u79ef\u7684\u7eb8\uff01<\/p>\n<p>\u4e5f\u8bb8\u60a8\u8bf4\uff1a\u53ef\u4ee5\uff01\u6211\u5c31\u628a\u8fd9\u5f20\u7eb8\u6d82\u6ee1\uff01<\/p>\n<p>\u662f\u554a\uff0c\u5982\u679c\u4e00\u6761\u66f2\u7ebf\u7ed5\u5f2f\u7ed5\u5f97\u5389\u5bb3\uff0c\u5b83\u6700\u540e\u662f\u80fd\u5360\u4e2a\u9762\u79ef\u7684\u3002\u6211\u4eec\u4e0d\u662f\u5e38\u8bf4\uff0c\u70b9\u52a8\u6210\u7ebf\uff0c\u7ebf\u52a8\u6210\u9762\uff0c\u9762\u52a8\u6210\u4f53\u5417\uff1f\u5982\u679c\u4f60\u628a\u4e00\u5f20\u6709\u9650\u9762\u79ef\u7684\u7eb8\u56fe\u6ee1\uff0c\u60a8\u753b\u7684\u7ebf\u6240\u5360\u7684\u9762\u79ef\u5c31\u662f\u8fd9\u5f20\u7eb8\u7684\u9762\u79ef\u3002\u5c31\u8ba9\u6211\u4eec\u4ece1\u7ef4\u4e0a\u5347\u52302\u7ef4\u8bd5\u8bd5\uff0c\u770b\u770bKoch\u66f2\u7ebf\u5360\u591a\u5927\u7684\u9762\u79ef\u3002<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_708c9z4wf69_b\" alt=\"\" \/><br \/>\n<img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_709cpd6bxkf_b\" alt=\"\" \/><\/p>\n<p>\u6211\u4eec\u8fd8\u662f\u91c7\u7528\u903c\u8fd1\u6cd5\uff0c\u7b2c\u4e00\u6b21\uff0c\u5148\u753b\u8fd9\u4e2aKoch\u66f2\u7ebf\u7684\u5916\u63a5\u4e09\u89d2\u5f62\u3002\u5b83\u7684\u5e95\u662f1\uff0c\u9ad8\u662f\u5c31\u662f\u7b2c\u4e00\u6b21\u6298\u7ebf\u753b\u7684\u7b49\u8fb9\u4e09\u89d2\u5f62\u7684\u9ad8\uff0c\u4e3a\u6839\u53f73\u9664\u4ee56\uff0c\u56e0\u6b64\u9762\u79ef\u662f\u6839\u53f73\u9664\u4ee512\uff08\u8bb0\u4e3aA\uff09\u3002Koch\u66f2\u7ebf\u5360\u7684\u9762\u79ef\u663e\u7136\u6bd4\u8fd9\u5b83\u7684\u5916\u63a5\u4e09\u89d2\u5f62\u8981\u5c0f\u3002\u4e8e\u662f\u6211\u4eec\u505a\u7b2c\u4e00\u6b21\u903c\u8fd1\uff0c\u628aKoch\u5206\u6210\u56db\u4e2a\u76f8\u4f3c\u7684\u90e8\u5206\uff0c\u753b\u56db\u4e2a\u5916\u63a5\u4e09\u89d2\u5f62\u3002\u5b83\u4eec\u7684\u8fb9\u957f\u662f\u662f\u539f\u6765\u5927\u4e09\u89d2\u5f62\u76841\/3\uff0c\u6240\u4ee5\u9762\u79ef\u662f(1\/9)A\uff0c\u56db\u4e2a\u52a0\u8d77\u6765\u5c31\u662f(4\/9)A\uff0cKoch\u66f2\u7ebf\u5360\u7684\u9762\u79ef\u8fd8\u8981\u6bd4\u8fd9\u4e2a\u5c0f\u3002\u7b2c\u4e8c\u6b21\u903c\u8fd1\uff0c\u753b\u4e8616\u4e2a\u4e09\u89d2\u5f62\uff0c\u5b83\u4eec\u7684\u9762\u79ef\u662f\u539f\u6765\u5927\u4e09\u89d2\u5f62\u76841\/81\uff0c\u6240\u4ee5\u603b\u7684\u9762\u79ef\u662f(16\/81)A\uff0c\u6216(4\/9)<sup>2<\/sup>A\u2026\u2026\u7b2cN\u6b21\u753b\u7684\u6240\u6709\u4e09\u89d2\u5f62\u7684\u9762\u79ef\u5c31\u5e94\u8be5\u662f(4\/9)<sup>N<\/sup>A\uff0c\u4f46\u662fKoch\u66f2\u7ebf\u5360\u7684\u9762\u79ef\u8fd8\u8981\u6bd4\u8fd9\u4e2a\u5c0f\u3002\u6211\u4eec\u628a\u7b2cN\u6b21\u753b\u7684\u4e09\u89d2\u5f62\u603b\u9762\u79ef\u5217\u5728\u5750\u6807\u56fe\u4e0a\uff0c\u770b\u770b\u968f\u7740N\u7684\u589e\u5927\uff0c\u9762\u79ef\u4f1a\u4e0d\u4f1a\u903c\u8fd1\u4e8e\u4e00\u4e2a\u6700\u5c0f\u6b63\u503c\u2014\u2014\u5f88\u4e0d\u5e78\uff0c\u968f\u7740N\u7684\u589e\u5927\uff0c\u9762\u79ef\u5237\u7684\u4e00\u4e0b\u5c31\u8dcc\u5230\u96f6\u4e86\u3002\u7b2c100\u6b21\u753b\u7684\u4e09\u89d2\u5f62\u9762\u79ef\u53ea\u6709\u539f\u6765\u76846&#215;10<sup>-36<\/sup>\uff01\u7531\u4e8eKoch\u66f2\u7ebf\u662f\u65e0\u9650\u81ea\u76f8\u4f3c\u4e0b\u53bb\u7684\uff0c\u56e0\u6b64\u5b83\u6240\u5360\u7684\u9762\u79ef\u4e3a\u96f6\u3002<\/p>\n<p>\u5982\u679c\u8bf7\u60a8\u75281\u6bb5\u957f\u5ea6\u4e3a\u65e0\u7a77\u7684\u7ebf\u6765\u6d82\u6ee1\u4e00\u5757\u9762\u79ef\u4e3a0\u7684\u5f62\u72b6\uff0c\u60a8\u80fd\u505a\u5230\u5417\uff1f<\/p>\n<p>\u4e8e\u662f\u6211\u4eec\u53d1\u73b0\uff0cKoch\u66f2\u7ebf\u662f\u4e00\u4e2a\u5f88\u5947\u602a\u7684\u5f62\u72b6\uff1a\u5f53\u5b83\u662f\u7ebf\uff0c\u957f\u5ea6\u65e0\u7a77\uff1b\u5f53\u5b83\u662f\u9762\uff0c\u9762\u79ef\u4e3a\u96f6\u3002\u4f5c\u4e3a\u4e00\u7ef4\u7269\u4f53\uff0cKoch\u66f2\u7ebf\u592a\u201c\u5bc6\u201d\u4e86\uff0c\u4f5c\u4e3a\u4e8c\u7ef4\u5bf9\u8c61\uff0cKoch\u66f2\u7ebf\u53c8\u592a\u201c\u758f\u201d\u4e86\u3002\u4e8b\u5b9e\u4e0a\uff0cKoch\u66f2\u7ebf\u662f\u9ad8\u4e8e1\u7ef4\uff0c\u4f4e\u4e8e2\u7ef4\u7684\u3002\u5b83\u7684\u7ef4\u6570\u57281\u4e0e2\u4e4b\u95f4\uff0c\u662f\u4e00\u4e2a\u975e\u6574\u6570\uff01\u975e\u6574\u6570\u7ef4\u5ea6\uff0c\u662f\u5206\u5f62\u7684\u4e00\u4e2a\u91cd\u8981\u7279\u70b9\u3002<\/p>\n<h2>\u5e42\u5f8b\u2014\u2014\u9690\u85cf\u7684\u5f02\u6b21\u5143<\/h2>\n<p>Koch\u66f2\u7ebf\u7684\u7ef4\u5ea6\u5230\u5e95\u662f\u591a\u5c11\u5462\uff1f\u8981\u8bf4\u6e05\u695a\u8fd9\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u5fc5\u987b\u91cd\u65b0\u5b9a\u4e49\u4e00\u4e0b\u201c\u7ef4\u5ea6\u201d\u7684\u6982\u5ff5\u3002\u5148\u6765\u770b\u770b\u6211\u4eec\u6240\u719f\u6089\u7684\u6574\u6570\u7ef4\u7684\u4e16\u754c\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_731cgmnrcfk_b\" alt=\"\" width=\"443\" height=\"205\" \/><\/p>\n<p>\u5728\u4e00\u7ef4\u4e16\u754c\u91cc\u6709\u4e00\u6761\u7ebf\u6bb5\uff0c\u5b83\u7684\u957f\u5ea6\u4e3a1\uff0c\u6211\u4eec\u7528\u4e00\u4e2a\u8fb9\u957f\u4e3a1\u7684\u6b63\u65b9\u5f62\u628a\u5b83\u76d6\u8d77\u6765\u3002\u5982\u679c\u7528\u8fb9\u957f\u4e3a0.5\u7684\u6b63\u65b9\u5f62\uff0c\u8981\u628a\u8fd9\u6761\u7ebf\u6bb5\u76d6\u8d77\u6765\u5c31\u9700\u8981\u4e24\u4e2a\u3002\u5982\u679c\u7528\u8fb9\u957f\u4e3a1\/3\u7684\u6b63\u65b9\u5f62\uff0c\u5c31\u5f97\u75283\u4e2a\u3002\u4f9d\u6b64\u7c7b\u63a8\uff0c\u5982\u679c\u8981\u7528\u8fb9\u957f\u4e3ar\u7684\u6b63\u65b9\u5f62\uff0c\u5c31\u8981\u7528N\u4e2a\uff0c\u624d\u80fd\u628a\u957f\u5ea6\u4e3a1\u7684\u7ebf\u6bb5\u76d6\u4f4f\u3002\u5f88\u663e\u7136N=1\/r\uff0c\u6216\u8005\u5199\u6210N=r<sup>-1<\/sup>\u3002<\/p>\n<p>\u8fd8\u7b97\u7b80\u5355\u5427\uff0c\u73b0\u5728\u6211\u4eec\u770b2\u7ef4\u4e16\u754c\uff0c\u4e00\u4e2a\u8fb9\u957f\u4e3a1\u7684\u6b63\u65b9\u5f62\uff0c\u8981\u7528\u591a\u5c11\u4e2a\u8fb9\u957f\u4e3ar\u7684\u6b63\u65b9\u5f62\u6765\u8986\u76d6\u5462\uff1f\u7b54\u6848\u662fr<sup>-2<\/sup>\u3002\u5bf9\u4e8e\u6b63\u65b9\u4f53\uff0c\u662fr<sup>-3<\/sup>\u3002<\/p>\n<p>\u7c97\u7565\u5730\u8bf4\uff0c\u5982\u679c\u4e00\u4e2a\u51e0\u4f55\u56fe\u5f62\u603b\u662f\u53ef\u4ee5\u7528N=r<sup>-d<\/sup>\u4e2a\u8fb9\u957f\u4e3ar\u7684\u6b63\u65b9\u5f62\uff08\u4f53\uff09\u6765\u8986\u76d6\uff0c\u90a3\u8fd9\u4e2a\u51e0\u4f55\u56fe\u5f62\u5c31\u662fd\u7ef4\u7684\u3002N=r<sup>-d<\/sup>\u662f\u4e00\u4e2a\u5e42\u51fd\u6570\u3002\u56e0\u6b64\uff0c\u8fd9\u4e00\u89c4\u5f8b\u53eb\u505a\u201c\u5e42\u5f8b\u201d\uff08power law\uff09<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_728dh5jmnfw_b\" alt=\"\" \/><\/p>\n<p>\u5982\u679c\u76d6\u4e0d\u4e25\uff0c\u6216\u8005\u6709\u591a\u4f59\u7684\u7a7a\u95f4\u600e\u4e48\u529e\uff1f\u6211\u4eec\u6765\u770b\u5982\u4f55\u8986\u76d6\u4e00\u4e2a\u4e09\u89d2\u5f62\uff0c\u53d1\u73b0\u6709\u7684\u5c0f\u6b63\u65b9\u5f62\u5c31\u8981\u76d6\u4f4f\u4e00\u4e9b\u591a\u4f59\u7684\u7a7a\u95f4\u4e86\u3002\u770b\u4e00\u4e0b\u4e00\u4e2a\u8fb9\u957f\u4e3a1\u7684\u4e09\u89d2\u5f62\u8981\u7528\u591a\u5c11\u4e2a\u8fb9\u957f\u4e3ar\u7684\u6b63\u65b9\u5f62\u6765\u8986\u76d6\uff1a<\/p>\n<table border=\"1\" cellspacing=\"2\" cellpadding=\"1\">\n<tbody>\n<tr>\n<td>N(1) = 1<\/td>\n<\/tr>\n<tr>\n<td>N(1\/2) = 3 = 1 + 2<\/td>\n<\/tr>\n<tr>\n<td>N(1\/4) = 10 = 1 + 2 + 3 + 4<\/td>\n<\/tr>\n<tr>\n<td>N(1\/8) = 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8<\/td>\n<\/tr>\n<tr>\n<td>\u2026\u2026<\/td>\n<\/tr>\n<tr>\n<td>N((1\/2)<sup>n<\/sup>) = 1 + 2 + 3 + &#8230; + 2<sup>n<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u53ef\u89c1\uff0cN=(1+r^(-n))*r^(-n)\/2\u3002\u6211\u4eec\u628algN\u4e0elg(1\/r)\u7684\u5173\u7cfb\u4f5c\u56fe\uff0c\u53d1\u73b0\uff0c\u968f\u7740r\u7684\u51cf\u5c0f\uff0c\u56fe\u5f62\u8d8b\u4e8e\u4e00\u6761\u4e0ey=2x\u5e73\u884c\u7684\u76f4\u7ebf\uff0c\u659c\u7387\u662f2\u3002\u56e0\u6b64\uff0c\u867d\u7136\u5728\u4e00\u5f00\u59cb\uff0cd=-lgN\/lgr\u4e0d\u662f\u4e00\u4e2a\u5b9a\u503c\uff0c\u4f46\u662f\u5f53r\u2014\u2014\u4e5f\u5c31\u662f\u5c0f\u6b63\u65b9\u5f62\u7684\u8fb9\u957f\u2014\u2014\u975e\u5e38\u5c0f\u7684\u65f6\u5019\uff0cd\u5c31\u7b49\u4e8e2\u3002\u5728\u6570\u5b66\u4e0a\u7684\u8868\u8fbe\u5c31\u662f\u6781\u9650lim<sub>r-&gt;0<\/sub>(-lgN\/lgr)\u5b58\u5728\u4e14\u7b49\u4e8e2\u3002<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_730hr67bxdb_b\" alt=\"\" \/><\/p>\n<p>\u4e3a\u4ec0\u4e48\u5f53\u6b63\u65b9\u5f62\u8fb9\u957f\u8f83\u5927\u7684\u65f6\u5019\u6709\u6240\u504f\u79bb\u5462\uff1f\u539f\u56e0\u5c31\u662f\u6709\u4e9b\u6b63\u65b9\u5f62\u8986\u76d6\u4e86\u591a\u4f59\u7684\u9762\u79ef\u3002\u5f53\u6b63\u65b9\u5f62\u8fb9\u957f\u51cf\u5c0f\u7684\u65f6\u5019\uff0c\u8fd9\u90e8\u5206\u591a\u8986\u76d6\u4e86\u7684\u9762\u79ef\u4e5f\u5c31\u5c0f\u4e86\u3002\u53ef\u4ee5\u60f3\u8c61\uff0c\u5982\u679c\u6b63\u65b9\u5f62\u7684\u8fb9\u957f\u65e0\u7a77\u5c0f\uff0c\u8fd9\u90e8\u5206\u591a\u8986\u76d6\u7684\u9762\u79ef\u5c31\u6ca1\u4e86\uff0c\u6240\u6709\u6b63\u65b9\u5f62\u6b63\u597d\u8986\u76d6\u4e86\u90a3\u4e2a\u5927\u4e09\u89d2\u5f62\u3002<\/p>\n<p>\u6240\u4ee5\uff0c\u51c6\u786e\u7684\u7ef4\u5ea6\u5b9a\u8bae\u5e94\u8be5\u662f\uff0c\u5f53r\u975e\u5e38\u5c0f\u65f6\uff0c\u5982\u679c\u9700\u8981N=r<sup>-d<\/sup>\u4e2a\u8fb9\u957f\u4e3ar\u7684\u6b63\u65b9\u5f62\u8986\u76d6\u4e00\u4e2a\u51e0\u4f55\u56fe\u6848\uff0c\u90a3\u8fd9\u4e2a\u51e0\u4f55\u56fe\u6848\u5c31\u662fd\u7ef4\u7684\u3002\u5728\u6570\u5b66\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u6781\u9650\u6765\u5b9a\u4e49\uff0c\u56fe\u5f62\u7684\u7ef4\u6570d=lim<sub>r-&gt;0<\/sub>(-lgN\/lgr)\u3002\u8fd9\u5c31\u662f\u8ba1\u76d2\u7ef4\u6570\uff08box counting dimension\uff09\uff0c\u53c8\u53eb\u505aMinkowski-Bouligand\u7ef4\u6570\u3002<\/p>\n<p>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u62ff\u6b63\u65b9\u5f62\u6765\u8986\u76d6\u4e00\u4e0bKoch\u66f2\u7ebf\uff0c\u5b83\u662f\u591a\u5c11\u7ef4\u7684\uff08d=?\uff09\u3002<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_712w27pxdcj_b\" alt=\"\" \/><\/p>\n<p>\u53ef\u89c1\uff0cN=3*4^n\u3002d=lim<sub>r-&gt;0<\/sub>(-lgN\/lgr)\uff0c\u6240\u4ee5\u6211\u4eec\u62fflgN\u5bf9lg(1\/r)\u4f5c\u56fe\uff0c\u5f97\u5230\u7684\u662f\u4e00\u6761\u76f4\u7ebf\uff0c\u5b83\u7684\u659c\u7387\u662flg4\/lg3\uff0c\u8fd9\u4e2a\u6570\u5927\u7ea6\u7b49\u4e8e1.26\u3002\u56e0\u6b64Koch\u66f2\u7ebf\u662f1.26\u7ef4\u7684\u3002<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_734sw8fgndn_b\" alt=\"\" \/><\/p>\n<p>\u4e0b\u56fe\u662f\u4e00\u4e2aGasket\u2014\u2014\u53e6\u4e00\u4e2a\u8457\u540d\u7684\u5206\u5f62\u3002\u5b83\u7684\u7ef4\u6570\u662flg3\/lg2\uff0c\u7ea6\u7b49\u4e8e1.58\u3002\u56e0\u6b64Gasket\u662f1.58\u7ef4\u7684\u3002<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_7359db24s5p_b\" alt=\"\" \/><br \/>\n<img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_732fhdvnhdt_b\" alt=\"\" \/><\/p>\n<h2>\u968f\u673a\u5206\u5f62\u2014\u2014\u5076\u7136\u6027\u80cc\u540e\u7684\u5fc5\u7136<\/h2>\n<p>\u8fd9\u5806\u4e71\u4e03\u516b\u7cdf\u7684\u6b63\u65b9\u5f62\u56fe\u6848\u662f\u4e0d\u662f\u81ea\u76f8\u4f3c\u7684\uff1f<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_713dcf3bpdk_b\" alt=\"\" \/><br \/>\n<img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_714czkt6n6v_b\" alt=\"\" \/><\/p>\n<p>\u5982\u679c\u8bd5\u56fe\u5728\u91cc\u9762\u627e\u5230\u4ec0\u4e48\u5c40\u90e8\u80fd\u8ddf\u6574\u4f53\u76f8\u4f3c\uff0c\u90a3\u5fc5\u7136\u662f\u5f92\u52b3\u65e0\u529f\u7684\u3002\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u6362\u4e00\u4e2a\u89d2\u5ea6\uff0c\u7edf\u8ba1\u4e00\u4e0b\u4e0d\u540c\u5927\u5c0f\u7684\u6b63\u65b9\u5f62\u6570\u91cf\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\uff1a<\/p>\n<ul>\n<li>\u4e00\u4e2a\u8fb9\u957f\u4e3a1\/2\u7684\u6b63\u65b9\u5f62\uff08\u7ea2\uff09<\/li>\n<li>3\u4e2a\u8fb9\u957f\u4e3a1\/4\u7684\u6b63\u65b9\u5f62\uff08\u7eff\uff09<\/li>\n<li>9\u4e2a\u8fb9\u957f\u4e3a1\/8\u7684\u6b63\u65b9\u5f62\uff08\u6df1\u84dd\uff09<\/li>\n<li>27\u4e2a\u8fb9\u957f\u4e3a16\/1\u7684\u6b63\u65b9\u5f62\uff08\u6d45\u84dd\uff09<\/li>\n<li>\u2026\u2026<\/li>\n<\/ul>\n<p>\u53ef\u89c1\uff0c\u8fb9\u957f\u4e3ar\u7684\u6b63\u65b9\u5f62\u4e2a\u6570N\u4e4b\u95f4\u5448\u5e42\u5f8b\u5173\u7cfbN=lg3*r<sup>-(lg3\/lg2)<\/sup>\u3002\u540c\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u770b\u5230\uff0c\u53d6\u5de6\u4e0b\u89d2\u56db\u5206\u4e4b\u4e00\u7684\u533a\u57df\uff0c\u6b63\u65b9\u5f62\u4e2a\u6570\u5448\u76f8\u540c\u7684\u7edf\u8ba1\u89c4\u5f8b\uff0c\u518d\u53d6\u5de6\u4e0b\u89d2\u516b\u5206\u4e4b\u4e00\u7684\u533a\u57df\uff0c\u76f8\u9762\u7684\u7edf\u8ba1\u89c4\u5f8b\u4ecd\u7136\u6210\u7acb\u3002\u8fd9\u5c31\u662f\u8bf4\uff0c\u8fd9\u5806\u4e4d\u770b\u8d77\u6765\u4e71\u4e03\u516b\u7cdf\u7684\u6b63\u65b9\u5f62\u56fe\u6848\uff0c\u5728\u7edf\u8ba1\u5206\u5e03\u4e0a\u662f\u81ea\u76f8\u4f3c\u7684\uff0c\u5b83\u662f\u4e00\u4e2a\u968f\u673a\u5206\u5f62\u3002\u5b83\u7684\u4efb\u4e00\u5c40\u90e8\u7684\u67d0\u9879\u7edf\u8ba1\u5206\u5e03\u6027\u8d28\u90fd\u4e0e\u6574\u4f53\u76f8\u4f3c\u3002\u5b83\u7684\u7ef4\u6570\u662flg3\/lg2\uff0c\u7ea6\u7b49\u4e8e1.58\u3002<\/p>\n<p>\u81ea\u7136\u754c\u4e2d\u7684\u5f88\u591a\u770b\u4f3c\u65e0\u5e8f\u7684\u968f\u673a\u8fc7\u7a0b\u90fd\u5c5e\u4e8e\u968f\u673a\u5206\u5f62\u3002<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_717fs62rmdx_b\" alt=\"\" width=\"202\" height=\"185\" \/><br \/>\n\u7ec6\u83cc\u7e41\u6b96\u56fe\u6848\uff08\u517b\u6599\u7a00\u5c11\u4e0b\uff09<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_718dkqrj7f2_b\" alt=\"\" \/><br \/>\n\u4eba\u57fa\u56e0\u5e8f\u5217\u5206\u5e03<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_719whpf2zxn_b\" alt=\"\" width=\"236\" height=\"201\" \/><\/p>\n<p>\u63c9\u76b1\u7684\u7eb8\u56e2\u4e2d\u6240\u542b\u7684\u7a7a\u9699\u5927\u5c0f\u5206\u5e03\u662f\u81ea\u76f8\u4f3c\u7684<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_720cvvwmdgk_b\" alt=\"\" \/><br \/>\n\u6591\u9a6c\u7684\u6761\u7eb9\u7684\u9762\u79ef\u5206\u5e03<\/p>\n<h2>\u805a\u5408\u7269\u2014\u2014\u5206\u5f62\u7684\u4e13\u5bb6<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_722g6qf47hd_b\" alt=\"\" width=\"194\" height=\"184\" \/><\/p>\n<p>\u805a\u5408\u7269\u662f\u7531\u5f88\u591a\u7ed3\u6784\u5355\u5143\u9996\u5c3e\u76f8\u63a5\u800c\u5f62\u6210\u7684\u957f\u94fe\u5206\u5b50\u3002\u6ca1\u6709\u9ad8\u5206\u5b50\uff0c\u81ea\u7136\u754c\u4e0d\u4f1a\u5b58\u5728\u751f\u547d\u3002\u56e0\u4e3a\u86cb\u767d\u8d28\u662f\u805a\u5408\u7269\u3001\u9057\u4f20\u7269\u8d28DNA\u548cRNA\u662f\u805a\u5408\u7269\u3001\u690d\u7269\u7684\u7ea4\u7ef4\u7d20\u662f\u805a\u5408\u7269\u3002\u5728\u6211\u4eec\u7684\u751f\u6d3b\u4e2d\uff0c\u6240\u6709\u5851\u6599\u3001\u6a61\u80f6\u548c\u7eba\u7ec7\u54c1\u90fd\u662f\u805a\u5408\u7269\uff0c\u7eb8\u548c\u6728\u6750\u6765\u6e90\u4e8e\u690d\u7269\u7684\u7ea4\u7ef4\u7d20\uff0c\u56e0\u6b64\u4e5f\u662f\u805a\u5408\u7269\u3002\u5176\u5b9e\u805a\u5408\u7269\u5206\u5b50\u672c\u8eab\u5c31\u5b58\u5728\u7740\u5f88\u591a\u968f\u673a\u5206\u5f62\u7684\u884c\u4e3a\u3002<\/p>\n<h3>\u65e0\u89c4\u7ebf\u56e2<\/h3>\n<p>\u751f\u6d3b\u4e2d\uff0c\u5f88\u5c11\u89c1\u5230\u4e00\u6761\u5f88\u957f\u5f88\u8f6f\u7684\u4e1c\u897f\u81ea\u5df1\u5904\u4e8e\u4f38\u76f4\u7684\u72b6\u6001\uff0c\u5b83\u4eec\u603b\u662f\u4ee5\u8fd9\u6837\u6216\u90a3\u6837\u7684\u5f62\u5f0f\u5f2f\u66f2\u8d77\u6765\u3002\u51c6\u786e\u5730\u8bf4\uff0c\u4e00\u6839\u957f\u7ebf\u6070\u597d\u5448\u4f38\u76f4\u7684\u6837\u5b50\uff0c\u4e0d\u662f\u4e0d\u53ef\u80fd\uff0c\u53ea\u662f\u673a\u7387\u975e\u5e38\u5730\u4f4e\u3002\u4e0d\u4fe1\uff0c\u4f60\u53ef\u4ee5\u62ff\u4e00\u6839\u6bdb\u7ebf\uff08\u8981\u957f\u4e00\u70b9\u7684\uff09\uff0c\u8ba9\u4ece\u4e00\u5b9a\u9ad8\u5ea6\u81ea\u7531\u5730\u843d\u5230\u5730\u4e0a\u3002\u4e0d\u65ad\u91cd\u590d\uff0c\u770b\u770b\u91cd\u590d\u51e0\u6b21\u8fd9\u6839\u7ebf\u56e2\u6389\u5730\u4e0a\u4e4b\u540e\u662f\u76f4\u7684\u3002\u5982\u679c\u4e0d\u8003\u8651\u91cd\u529b\u548c\u6469\u64e6\u529b\uff0c\u4e00\u6839\u8f6f\u7684\u957f\u7ebf\u5448\u4f38\u76f4\u72b6\u6001\u7684\u673a\u7387\u5c31\u66f4\u5c0f\u4e86\uff0c\u57fa\u672c\u53ef\u4ee5\u8bf4\u5982\u679c\u4f60\u4e0d\u53bb\u62c9\u5b83\u7684\u8bdd\uff0c\u5b83\u51e0\u4e4e\u4e0d\u53ef\u80fd\u662f\u4f38\u76f4\u7684\u3002\u805a\u5408\u7269\u5206\u5b50\u90fd\u662f\u957f\u94fe\u5206\u5b50\uff0c\u8ddf\u5b83\u7684\u76f4\u5f84\u76f8\u6bd4\uff0c\u5b83\u7684\u957f\u5ea6\u53ef\u8fbe\u51e0\u5341\u5230\u51e0\u767e\u4e07\u500d\u3002\u56e0\u6b64\uff0c\u805a\u5408\u7269\u5206\u5b50\u94fe\u603b\u662f\u5448\u65e0\u89c4\u5377\u66f2\u7684\u72b6\u6001\uff0c\u79d1\u5b66\u5bb6\u5f62\u8c61\u5730\u628a\u5b83\u79f0\u4f5c\u201c\u65e0\u89c4\u7ebf\u56e2\u201d\u3002<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_721gfm9s83j_b\" alt=\"\" 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loading=\"lazy\" decoding=\"async\" src=\"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_723ct84c3hn_b\" alt=\"\" width=\"270\" height=\"251\" 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Seddon\u9ed8\u9ed8\u5e72\u4e86\u4e8c\u5341\u5e74\u624d\u53d1\u7b2c\u4e00\u7bc7\u6587\u7ae0\uff08\u4e3a\u4ec0\u4e48\uff09\u3002\u987a\u4fbf\u63d0\u4e00\u4e0b\uff0c\u5173\u4e8e\u5206\u5f62\uff0c\u6211\u66fe\u7ecf\u4f5c\u4e3a\u81ea\u5df1\u535a\u58eb\u8bfe\u9898\u5f00\u9898\u7684\u8d44\u6599\u603b\u7ed3\u4ecb\u7ecd\u8fc7\uff0c\u73b0\u5728\u6211\u4e8b\u5b9e\u4e0a\u4f5c\u7684\u8bfe\u9898\u8ddf\u5206\u5f62\u5173\u7cfb\u5df2\u7ecf\u4e0d\u5927\u4e86\u3002\u5173\u4e8ede Genne\uff0c\u6211\u201c\u7eaa\u5ff5\u201d\u8fc7\u4e24\u6b21\u3002\u5173\u4e8eK. Seddon\uff0c\u6211\u5728\u8c08publish or perish\u95ee\u9898\u65f6\u987a\u5e26\u63d0\u5230\u8fc7\u3002\u5e42\u7387\u3001\u81ea\u76f8\u4f3c\u3001\u5206\u5f62\u848b\u654f\u5f3a\u535a\u58eb\u662f\u7814\u7a76\u65e0\u673a\u6750\u6599\u7684\uff0c\u8ddf\u9ad8\u5206\u5b50\u5dee\u5f88\u8fdc\u3002\u4f46\u662f\u5bf9\u4ed6\u7684\u535a\u5ba2\u91cc\u4ecb\u7ecd\u7684\u5206\u5f62\u73b0\u8c61\u5374\u5f88\u773c\u719f\u3002\u6587\u5b57\u53e3\u53e3\u58f0\u58f0\u5730\u8bf4\u81ea\u76f8\u4f3c\u3001\u5206\u5f62\uff0c\u4f46\u662f\u56fe\u53ea\u662f\u7ed9\u51fa\u4e86\u4e00\u4e2a\u5e42\u7387\u5173\u7cfb\u3002\u4f46\u662f\uff0c\u5bf9\u4e8e\u5927\u4f17\u800c\u8a00\uff0c\u5206\u5f62\u660e\u660e\u662f\u90a3\u4e2a\u201c\u5927\u5c41\u80a1\u201d\u56fe\u6848\uff08Mandelbrot set\uff09\u3002\u4f60\u8ddf\u4eba\u5bb6\u8bf4\u8fd9\u6750\u6599\u91cc\u6709\u5206\u5f62\uff0c\u90a3\u6750\u6599\u91cc\u53c8\u6709\u5206\u5f62\uff0c\u5149\u753b\u4e00\u6761\u76f4\u7ebf\u600e\u4e48\u884c\uff1f\u4eba\u5bb6\u542c\u4e86\u8fd9\u8bdd\u5fc3\u91cc\u4ee5\u4e3a\u4f60\u6750\u6599\u91cc\u6709\u597d\u591a\u5927\u5c41\u80a1\u5462\u3002\u6211\u5f00\u9898\u65f6\u6d89\u53ca\u5230\u7684\u5206\u5f62\u5b58\u5728\u4e8e\u6e17\u6e1d\u95ee\u9898\uff0c\u8bf4\u51dd\u80f6\u5316\u70b9\u5904\u6574\u4e2a\u51dd\u80f6\u4f53\u7cfb\u662f\u4e00\u4e2a\u904d\u5e03\u7684\u81ea\u76f8\u4f3c\u7f51\u7edc\u3002\u4e8e\u662f\u66fe\u7ecf\u6709\u4eba\u95ee\u6211\uff0c\u53bb\u505a\u7535\u955c\u80fd\u4e0d\u80fd\u770b\u5230\u7c7b\u4f3c\u53f6\u8109\u7684\u90a3\u79cd\u81ea\u76f8\u4f3c\u56fe\u6848\uff1f\u8fd9\u5c31\u662f\u827a\u672f\u4e0e\u5546\u4e1a\u7684\u95ee\u9898\u3002\u5927\u4f17\u975e\u8981\u901a\u8fc7\u94fa\u5929\u76d6\u5730\u7684\u5927\u5c41\u80a1\u624d\u8ba4\u8bc6\u5206\u5f62\u662f\u5565\uff0c\u4f46\u771f\u6b63\u7684\u7814\u7a76\u5b9e\u8df5\u4e2d\u5374\u662f\u5728\u627e\u76f4\u7ebf\u3002\u6750\u6599\u5b66\u91cc\u7684\u524d\u6cbf\u6709\u5f88\u5927\u4e00\u5757\u662f\u7814\u7a76\u65e0\u5e8f\u95ee\u9898\uff0c\u65b9\u6cd5\u65e0\u975e\u662f\u4ece\u65e0\u5e8f\u4e2d\u627e\u6709\u5e8f\u3002\u5206\u5f62\u5c31\u662f\u65e0\u5e8f\u4e2d\u80fd\u627e\u5230\u4e00\u79cd\u6709\u5e8f\u3002\u4f46\u4ece\u65e0\u5e8f\u4e2d\u627e\u5230\u7684\u7edd\u4e0d\u4f1a\u662f\u51e0\u4f55\u5206\u5f62\uff0c\u53ea\u4f1a\u662f\u968f\u673a\u5206\u5f62\u2014\u2014\u4e5f\u5c31\u662f\u8bf4\uff0c\u5b83\u7684\u81ea\u76f8\u4f3c\u6027\u4e0d\u662f\u51e0\u4f55\u610f\u4e49\u4e0a\u7684\uff0c\u800c\u662f\u7edf\u8ba1\u610f\u4e49\u4e0a\u7684\u3002\u9664\u4e86\u628a\u5206\u5e03\u51fd\u6570\u5bf9\u5206\u5e03\u6240\u5728\u7a7a\u95f4\u5c3a\u5ea6\u4f5c\u56fe\u80fd\u770b\u5230\u5e42\u7387\u5173\u7cfb\u4e4b\u5916\uff0c\u4f60\u518d\u4e5f\u770b\u4e0d\u5230\u201c\u66f4\u52a0\u5206\u5f62\u201d\u7684\u4e1c\u897f\u4e86\uff0c\u4f8b\u5982\u90a3\u4e2aMandelbrot\u5927\u5c41\u80a1\u3002\u848b\u654f\u5f3a\u535a\u58eb\u7684\u535a\u5ba2\u91cc\u5217\u4e3e\u4e86\u8bb8\u591a\u65e0\u673a\u6750\u6599\u7814\u7a76\u4e2d\u7684\u5206\u5f62\u73b0\u8c61\u3002\u6750\u6599\u5b66\u7814\u7a76\u91cc\u6700\u901a\u5e38\u7684\u60c5\u51b5\u662f\u5728\u9759\u6001\u6563\u5c04\u5b9e\u9a8c\uff08\u5149\u3001\u4e2d\u5b50\uff09\u4e2d\u6563\u5c04\u5149\u5f3a\u7684\u89d2\u5ea6\u4f9d\u8d56\u6027\u5728\u4e00\u5b9a\u89d2\u5ea6\u8303\u56f4\u5185\u9075\u5faa\u67d0\u79cd\u5e42\u5f8b\u5173\u7cfb\uff1aS(q)~q-D\uff0c\u5176\u4e2dD\u662f\u5206\u5f62\u7ef4\u6570\uff0cq\u662f\u6ce2\u77e2\u3002\u6309\u6211\u7684\u4e13\u4e1a\u80cc\u666f\u6211\u4e5f\u5c31\u53ea\u6709\u8d44\u683c\u8bf4\u8bf4\u9ad8\u5206\u5b50\u7684\u4f8b\u5b50\u3002\u4e00\u4e2a\u6559\u79d1\u4e66\u7ea7\u522b\u7684\u4f8b\u5b50\u5c31\u662f\u94fe\u6784\u8c61\u7edf\u8ba1\u3002\u4ece\u81ea\u76f8\u4f3c\u7684\u6982\u5ff5\u51fa\u53d1\uff0c\u4e00\u67613\u7ef4\u7406\u60f3\u9ad8\u65af\u94fe\u7684\u81ea\u76f8\u4f3c\u6027\u662f\u6307\u5176\u5747\u65b9\u56de\u8f6c\u534a\u5f84<Rg2>~n2\u3002\u8981\u201c\u770b\u201d\u5230\u8fd9\u4e2a\u81ea\u76f8\u4f3c\u73b0\u8c61\uff0c\u5c31\u8981\u505a\u9759\u6001\u6563\u5c04\u5b9e\u9a8c\u3002\u4e00\u5b9a\u6ce2\u957f\u7684\u5149\u6700\u6613\u88ab\u4e00\u5b9a\u5c3a\u5bf8\u7684\u539f\u5b50\u6216\u7535\u5b50\u5bc6\u5ea6\u6563\u5c04\uff08\u539f\u5b50\u8fd8\u662f\u7535\u5b50\u8981\u770b\u4f60\u7684\u5149\u6e90\u662f\u4e2d\u5b50\u8fd8\u662fX\u5c04\u7ebf\uff09\u3002\u6563\u5c04\u51fd\u6570S(q)\u662f\u5728q-1\u5c3a\u5ea6\u5c0f\u4e8eRg\u7684\u524d\u63d0\u4e0b\uff0c\u53ef\u4ee5\u201c\u770b\u201d\u5230\u94fe\u6bb5\u5bc6\u5ea6\u7684\u7a7a\u95f4\u5206\u5e03\uff0c\u6570\u5b66\u4e0a\u7528\u5bc6\u5ea6\u81ea\u76f8\u5173\u51fd\u6570\u0393\u8868\u793a\u3002\u0393\u7684Fourier\u53d8\u6362\u5c31\u662f\u5149\u6563\u5c04\u51fd\u6570S(q)\u3002\u5982\u679c\u6709<Rg2>~n2\uff0c\u5149\u6563\u5c04\u5b9e\u9a8c\u4e2d\u5c31\u80fd\u201c\u770b\u5230\u201dS(q)~q-2\uff0c\u5373\u5206\u5f62\u7ef4\u6570D=2\u3002\u8fd9\u662f\u7406\u60f3\u94fe\u3002Flory\u8003\u8651\u4f53\u79ef\u6392\u9664\u6548\u5e94\u7b97\u51faD=6\/5\u3002\u4ee5\u4e0a\u8fd9\u4e9b\u6750\u6599\u5b66\u7684\u4f8b\u5b50\u4f30\u8ba1\u7ed9\u75f4\u8ff7\u4e8e\u5927\u5c41\u80a1\u7684\u5927\u4f17\u8bfb\u8005\u6cfc\u4e86\u51b7\u6c34\u3002\u4e0d\u662f\u5206\u5f62\u4e48\uff1f\u600e\u4e48\u4e0d\u89c1\u5927\u5c41\u80a1\u5462\uff1f\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u6570\u5b66\u89c6\u9891\u7cfb\u5217Dimensions\u7684\u7b2c\u516d\u96c6\u8db3\u4ee5\u6ee1\u8db3\u6b64\u7c7b\u89c2\u4f17\uff08\u8981\u7ffb\u5899\uff09\uff1a\u6211\u662f\u4ece\u8fd9\u4e2a\u89c6\u9891\u91cc\u7b2c\u4e00\u6b21\u77e5\u9053\u539f\u6765Mandelbrot\u96c6\u5408\u662f\u63cf\u8ff0Julia\u96c6\u5408\u51fa\u73b0\u5206\u5f62\u7684\u533a\u57df\u800c\u5df2\u3002Julia\u96c6\u5408\u5206\u5f62\u662fT(z)=z2+c\u64cd\u4f5c\u65e0\u9650\u8fed\u4ee3\u7684\u7ed3\u679c\u3002\u53ea\u6709\u590d\u5e38\u6570c\u5728\u4e00\u5b9a\u53d6\u503c\u8303\u56f4\u5185\uff0c\u8fed\u4ee3\u7684\u7ed3\u679c\u624d\u662f\u4e00\u4e2a\u80fd\u770b\u5230\u4e00\u7247\u9762\u79ef\u7684\u5206\u5f62\uff08\u8fde\u901a\uff09\u2014\u2014\u5154\u5b50\uff01\u3002\u5728\u6b64\u8303\u56f4\u5916\uff0cJulia set\u5c31\u53d8\u6210\u65e0\u9650\u7cbe\u7ec6\u7684\u56fe\u6848\uff08\u975e\u8fde\u901a\uff09\u2014\u2014\u5565\u90fd\u770b\u4e0d\u89c1\u4e86\u3002\u8fd9\u4e2a\u8303\u56f4\u5728\u590d\u5e73\u9762\u4e0a\u7684\u56fe\u5f62\u5c31\u662fMandelbrot\u5206\u5f62\uff0c\u5b83\u754c\u5b9a\u4e86\u8fde\u901a\u548c\u975e\u8fde\u901a\u7684Julia set\u3002\u97f3\u4e50\u4e0e\u5316\u5b66\u56de\u5e94\u5b8c\u5bf9Mandelbrot\u7684\u7eaa\u5ff5\u4e4b\u540e\uff0c\u518d\u56de\u5e94\u674e\u6dfc\u7814\u7a76\u5458\u7684\u201c\u827a\u672f\u5174\u8da3\u201d\u8bdd\u9898\u3002\u674e\u6dfc\u7814\u7a76\u5458\u5bf9\u7269\u7406\u5b66\u5bb6\u6216\u6570\u5b66\u5bb6\u4e2d\u7684\u8bd7\u4eba\u6bd4\u8f83\u719f\u6089\uff0c\u6211\u662f\u53e4\u5178\u97f3\u4e50\u7231\u597d\u8005\uff0c\u56e0\u6b64\u6211\u80fd\u4e3e\u51fa\u4e00\u4e9b\u5316\u5b66\u5bb6\u4e2d\u7684\u97f3\u4e50\u5bb6\uff0cor vise versa\u3002\u4f8b\u5982\uff0cAlexander Borodin\u662f\u4fc4\u7f57\u65af\u4f5c\u66f2\u5bb6\uff0c\u6240\u8c13\u201c\u4e94\u4eba\u56e2\u201d\u4e4b\u4e00\u3002\u4ed6\u521b\u4f5c\u7684\u6700\u4f18\u7f8e\u7684\u65cb\u5f8b\u5e94\u8be5\u662f\u5f26\u4e50\u56db\u91cd\u594f\u7684\u7b2c\u4e8c\u4e50\u7ae0\u201c\u591c\u66f2\u201d\u3002\u4ee5\u4ed6\u547d\u540d\u7684\u53cd\u5e94\uff1a E. Elgar\uff0c\u82f1\u56fd\u4f5c\u66f2\u5bb6\u3002\u7535\u5f71\u300a\u5979\u6bd4\u70df\u82b1\u5bc2\u5bde\u300b\u91cc\u7684\u5973\u4e3b\u89d2\uff0c\u5927\u63d0\u7434\u624bdu Pre\u7684\u6210\u540d\u4f5c\u5c31\u662fElgar\u7684\u5927\u63d0\u7434\u534f\u594f\u66f2\u3002\u800c\u6211\u8ba4\u4e3aElgar\u6700\u4f18\u7f8e\u7684\u65cb\u5f8b\u5e94\u8be5\u662f\u5c0f\u63d0\u7434\u534f\u594f\u66f2\u7684\u6162\u4e50\u7ae0\u3002\u4ed6\u53d1\u660e\u4e86\u4e00\u4e2a\u5236\u5907\u786b\u5316\u6c22\u7684\u88c5\u7f6e\u5e76\u6210\u529f\u7533\u8bf7\u4e86\u4e13\u5229\u3002\u8fd9\u4e24\u4e2a\u4eba\u4e5f\u8bb8\u5728\u97f3\u4e50\u754c\u6bd4\u5728\u5316\u5b66\u754c\u6709\u540d\u3002\u90a3\u597d\uff0cJohannes Br\u00f8nsted\u8fd9\u4e2a\u540d\u5b57\uff0c\u6240\u6709\u5316\u5b66\u7cfb\u7684\u5b66\u751f\u90fd\u5e94\u8be5\u8ba4\u8bc6\u2014\u2014\u9178\u78b1\u7684\u8d28\u5b50\u7406\u8bba\u3002\u4ed6\u662f\u4e1a\u4f59\u7537\u4e2d\u97f3\u6b4c\u5531\u5bb6\u548c\u753b\u5bb6\u3002 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W. Anderson, 1972\uff09\u3002\u4e8b\u5b9e\u4e0a\uff0cemergence\u4e5f\u597d\uff0c\u590d\u6742\u6027\u4e5f\u597d\uff0c\u5927\u5bb6\u7684\u6f5c\u610f\u8bc6\u90fd\u662f\u60f3\u641e\u6e05\u695a\u751f\u547d\uff0c\u5c24\u5176\u662f\u751f\u547d\u7684\u90a3\u4e9b\u5947\u5999\u4e4b\u5904\uff08\u800c\u4e0d\u662f\u201c\u5408\u7406\u201d\u4e4b\u5904\uff09\u3002\u800c\u5047\u5982\u5173\u6ce8\u201c\u5947\u5999\u201d\u4e4b\u5904\uff0c\u5927\u81ea\u7136\u4e5f\u4e0d\u4ec5\u5728\u751f\u547d\u91cc\u4e3a\u6211\u4eec\u5c55\u73b0\u5947\u5999\u3002\u8bb8\u591a\u8f6f\u7269\u8d28\u7684\u5947\u5999\u73b0\u8c61\uff0c\u548c\u751f\u547d\u7c7b\u4f3c\uff08\u751f\u547d\u5c31\u662f\u4e00\u56e2\u8d85\u7ea7\u8f6f\u7269\u8d28\uff09\uff0c\u4e5f\u662f\u6765\u6e90\u4e8e\u5404\u79cd\u7a0b\u5ea6\u7684\u590d\u6742\u6027\u7684\u3002\u9ad8\u5206\u5b50\u7684\u53d1\u73b0\uff0c\u4e3a\u6211\u4eec\u5c55\u73b0\u4e86\u201c\u6b7b\u7269\u201d\u7684\u201c\u6d3b\u201d\u7684\u4e00\u9762\uff0c\u548c\u77f3\u5934\u548c\u91d1\u5c5e\u76f8\u6bd4\uff0c\u9ad8\u5206\u5b50\u6750\u6599\u7684\u7c98\u5f39\u6027\uff0c\u4ee5\u53ca\u8868\u73b0\u51fa\u6765\u7684\u5c48\u670d\u3001\u7ec6\u9888\u3001\u8bb0\u5fc6\uff0c\u4ee5\u81f3\u65f6\u6e29\u7b49\u6548\u73b0\u8c61\u7b49\u7b49\uff0c\u7ed9\u4eba\u611f\u89c9\u6709\u4e00\u80a1\u201c\u7075\u6c14\u201d\uff0c\u4e00\u79cd\u201c\u6027\u683c\u201d\uff0c\u5176\u5bf9\u523a\u6fc0\u54cd\u5e94\u4e0d\u50cf\u77f3\u5934\u6216\u91d1\u5c5e\u90a3\u6837\u5343\u7bc7\u4e00\u5f8b\u3002\u9ad8\u5206\u5b50\u7684\u7406\u8bba\uff0c\u4e5f\u8d85\u8d8a\u4e86\u4e00\u822c\u7684\u5316\u5b66\u7ed3\u6784\u5c42\u9762\uff0c\u800c\u662f\u5fae\u89c2\u3001\u4ecb\u89c2\u3001\u5b8f\u89c2\u591a\u5c42\u6b21\u53d1\u5c55\uff0c\u5728\u65f6\u7a7a\u4e0a\u63a2\u7d22\u9ad8\u5206\u5b50\u7684\u591a\u5c3a\u5ea6\u6027\uff0c\u7814\u7a76\u624b\u6bb5\u4ece\u4e2d\u5b50\u81ea\u65cb\u56de\u6ce2\u5230\u5149\u5b66\u663e\u5fae\u955c\uff0c\u5b9e\u9645\u4e0a\u4e0d\u4ec5\u662f\u9ad8\u5206\u5b50\u7814\u7a76\u672c\u8eab\uff0c\u66f4\u662f\u63ed\u793a\u4e86\u81ea\u7136\u754c\u201c\u7075\u6c14\u201d\u7684\u7531\u6765\uff0c\u4e3a\u751f\u547d\u73b0\u8c61\u8fdb\u4e00\u6b65\u53bb\u9b45\uff0c\u6307\u51fa\u4e86\u7814\u7a76\u7684\u65b9\u5411\u3002\u5173\u4e8e\u6df7\u6c8c\u3001\u6709\u5e8f\u548c\u751f\u547d\u73b0\u8c61\u7684\u6570\u7406\u8ba8\u8bba\u65e9\u5c31\u5f00\u59cb\u4e86\uff0c\u6700\u8457\u540d\u7684\u83ab\u8fc7\u4e8eJ. 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