{"id":42964,"date":"2023-06-06T20:15:51","date_gmt":"2023-06-06T12:15:51","guid":{"rendered":"https:\/\/swarma.org\/?p=42964"},"modified":"2023-06-06T20:15:51","modified_gmt":"2023-06-06T12:15:51","slug":"%e9%95%b6%e5%b5%8c%e5%9b%be%e6%a1%88%ef%bc%9a%e5%a4%8d%e6%9d%82%e6%ba%90%e8%87%aa%e7%ae%80%e5%8d%95%e4%b8%a8%e9%99%88%e5%85%b3%e8%8d%a3","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=42964","title":{"rendered":"\u9576\u5d4c\u56fe\u6848\uff1a\u590d\u6742\u6e90\u81ea\u7b80\u5355\u4e28\u9648\u5173\u8363"},"content":{"rendered":"
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\u201c\u9576\u5d4c\u201d<\/span>\uff08tessellation\uff09<\/span>\uff0c\u4e5f\u79f0\u4e3a\u201c\u5bc6\u94fa\u201d<\/span>\uff08tiling\uff09<\/span>\uff0c\u662f\u4e00\u79cd\u7528\u591a\u5757\u5177\u6709\u76f8\u540c\u51e0\u4f55\u5f62\u72b6\u7684\u74f7\u7816<\/span>\uff08tile\uff09<\/span>\u4e0d\u91cd\u53e0\u4e5f\u4e0d\u7559\u7a7a\u9699\u5730\u94fa\u6ee1\u5730\u9762\u6216\u5899\u58c1\u7684\u6280\u672f\u6216\u827a\u672f\u3002\u751f\u6d3b\u4e2d\u4eba\u4eec\u4f7f\u7528\u5177\u6709\u67d0\u4e9b\u56fa\u5b9a\u51e0\u4f55\u5f62\u72b6\u7684\u7247\u5757\u7ec4\u88c5\u51fa\u5404\u79cd\u5404\u6837\u590d\u6742\u7f8e\u89c2\u7684\u56fe\u6848<\/span>\uff08pattern\uff09<\/span>\uff0c\u5982\u56fe1\u6240\u793a\u3002<\/span><\/p>\n

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\u56fe1 \u201c\u9576\u5d4c\u201d\u6216\u201c\u5bc6\u94fa\u201d\u827a\u672f\u56fe\u6848\u793a\u4f8b<\/span><\/p>\n


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\u201c\u9576\u5d4c\u201d\u7684\u82f1\u6587\u5355\u8bcdtessellation\u6765\u81ea\u5e0c\u814a\u6587\u03c4\u03ad\u03c3\u03c3\u03b5\u03c1\u03b1<\/span>\uff08tessera\uff09<\/span>\uff0c\u610f\u6307\u201c\u56db\u201d\uff0c\u56e0\u4e3a\u7b2c\u4e00\u4ee3\u74f7\u7816\u662f\u56db\u65b9\u5f62\u7684\uff0c\u662f\u65e9\u671f\u4eba\u4eec\u94fa\u5efa\u5730\u677f\u548c\u5899\u58c1\u6240\u4f7f\u7528\u74f7\u7816\u7684\u6700\u7b80\u5355\u51e0\u4f55\u5f62\u72b6\u3002\u9576\u5d4c\u7684\u5386\u53f2\u53ef\u4ee5\u8ffd\u6eaf\u5230\u516c\u5143\u524d4000\u591a\u5e74\uff0c\u90a3\u65f6\u82cf\u7f8e\u5c14\u4eba<\/span>\uff08Sumerians\uff09<\/span>\u5df2\u7ecf\u5f00\u59cb\u7528\u7c98\u571f\u74e6\u7247\u53bb\u88c5\u9970\u4ed6\u4eec\u7684\u623f\u5b50\u548c\u795e\u5e99\u3002\u4e4b\u540e\uff0c\u8fd9\u79cd\u6280\u672f\u548c\u827a\u672f\u7ecf\u5386\u4e86\u57c3\u53ca\u4eba\u3001\u6ce2\u65af\u4eba\u3001\u7f57\u9a6c\u4eba\u548c\u5e0c\u814a\u4eba\u7684\u4f20\u627f\uff0c\u5230\u62dc\u5360\u5ead\u4eba\u3001\u963f\u62c9\u4f2f\u4eba\u3001\u4e2d\u56fd\u4eba\u3001\u65e5\u672c\u4eba\u548c\u6469\u5c14\u4eba\u7684\u53d1\u5c55\uff0c\u671f\u95f4\u4e0d\u65ad\u5730\u88ab\u52a0\u5165\u4e0d\u540c\u65f6\u4ee3\u548c\u4e0d\u540c\u6c11\u65cf\u7684\u4e0d\u540c\u5b97\u6559\u3001\u6587\u5316\u548c\u827a\u672f\u5143\u7d20\u3002\u503c\u5f97\u4e00\u63d0\u7684\u662f\u4f0a\u65af\u5170\u7f8e\u672f\u7279\u522b\u662f\u4e2d\u4e16\u7eaa\u897f\u5317\u975e\u6d32\u7684\u9a6c\u683c\u91cc\u5e03\u548c\u6469\u6d1b\u54e5\u4ee5\u53ca\u5c3c\u65e5\u5229\u4e9a\u3001\u897f\u5357\u6b27\u6d32\u7684\u4f0a\u6bd4\u5229\u4e9a\u534a\u5c9b\u3001\u5730\u4e2d\u6d77\u7684\u897f\u897f\u91cc\u5c9b\u548c\u9a6c\u8033\u4ed6\u7b49\u8fd9\u4e9b\u5730\u533a\u7684\u7a46\u65af\u6797\u6c11\u65cf\u7684\u8d21\u732e\u3002\u4ed6\u4eec\u7684\u8bbe\u8ba1\u79f0\u4e3a\u201czillij\u201d\u6216\u201czellige\u201d\uff0c\u610f\u6307\u8fd9\u662f\u4e00\u95e8\u4ee5\u201c\u5b66\u4e60\u3001\u89c4\u5219\u548c\u4fe1\u4ef0\u201d\u4e3a\u57fa\u7840\u7684\u827a\u672f\u3002<\/span><\/p>\n


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\u56fe2 \u4f0a\u65af\u5170\u9576\u5d4c\u56fe\u6848\u7f8e\u672f\u8bbe\u8ba1\u793a\u4f8b<\/span><\/p>\n


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\u4ece\u4e2d\u4e16\u7eaa\u5f00\u59cb\uff0c\u7279\u522b\u662f\u5230\u4e8619\u548c20\u4e16\u7eaa\uff0c\u4eba\u4eec\u5f00\u59cb\u4ece\u6570\u5b66\u7406\u8bba\u7279\u522b\u662f\u56fe\u8bba\u7684\u89d2\u5ea6\u6765\u7814\u7a76\u548c\u8bbe\u8ba1\u5404\u79cd\u590d\u6742\u7684\u9576\u5d4c\u56fe\u6848\u3002\u5176\u4e2d\u6700\u6709\u4ee3\u8868\u6027\u7684\u662f\u8377\u5170\u827a\u672f\u5bb6\u83ab\u91cc\u8328\u00b7\u57c3\u820d\u5c14<\/span>\uff08Maurits C. Escher\uff0c1898\u20131972\uff09<\/span>\u53ca\u82f1\u56fd\u6570\u5b66\u7269\u7406\u5b66\u5bb6\u7f57\u6770\u00b7\u5f6d\u7f57\u65af<\/span>\uff08Roger Penrose\uff0c1931\u2013\uff09<\/span>\u548c\u4ed6\u7684\u7236\u4eb2\u3001\u7cbe\u795e\u79d1\u533b\u751f\u83b1\u6602\u5185\u5c14\u00b7\u5f6d\u7f57\u65af<\/span>\uff08Lionel S. Penrose\uff0c1898\u20131972\uff09<\/span>\u7684\u591a\u79cd\u5947\u5999\u8bbe\u8ba1\u3002\u8fd9\u4f4d\u57c3\u820d\u5c14\u5c31\u662f\u90a3\u5e45\u8457\u540d\u900f\u89c6\u9519\u89c9\u7d20\u63cf\u201c\u4e0a\u4e0b\u9636\u68af\u201d<\/span>\uff08Scaffold ascending and descending\uff0c\u56fe3\u5de6\u56fe\uff09<\/span>\u7684\u4f5c\u8005\u3002\u4ed6\u8fd8\u8bbe\u8ba1\u4e86\u8bb8\u591a\u7f8e\u89c2\u7684\u6728\u523b\u3001\u77f3\u7248\u753b\u3001\u94dc\u7248\u753b\u548c\u7d20\u63cf\u9576\u5d4c\u56fe\u6848<\/span>\uff08\u4f8b\u5982\uff0c\u56fe3 \u4e2d\u56fe\u548c\u53f3\u56fe\uff09<\/span>\u3002<\/span><\/p>\n


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\u56fe3  \u57c3\u820d\u5c14\u7684\u201c\u4e0a\u4e0b\u9636\u68af\u201d\u540d\u753b\u548c\u4ed6\u7684\u9576\u5d4c\u56fe\u6848\u793a\u4f8b<\/span><\/p>\n


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\u57c3\u820d\u5c14\u7684\u201c\u4e0a\u4e0b\u9636\u68af\u201d\u8bbe\u8ba1\u662f\u53d7\u5230\u5f6d\u7f57\u65af\u697c\u68af<\/span>\uff08Penrose stairs\uff0c\u56fe4\u5de6\u4e0a\u56fe\uff09<\/span>\u548c\u5f6d\u7f57\u65af\u4e09\u89d2<\/span>\uff08Penrose triangle\uff0c\u56fe4\u5de6\u4e0b\u56fe\uff09<\/span>\u7684\u5f71\u54cd\u5b8c\u6210\u7684\uff0c\u662f\u5f6d\u7f57\u65af\u697c\u68af\u7684\u827a\u672f\u8868\u73b0\u3002\u5f6d\u7f57\u65af\u7236\u5b50\u7559\u4e0b\u4e86\u201c\u5f6d\u7f57\u65af\u5bc6\u94fa\u201d<\/span>\uff08Penrose tiling\uff09<\/span>\uff0c\u5305\u62ec\u5f6d\u7f57\u65af\u697c\u68af\u548c\u5f6d\u7f57\u65af\u4e09\u89d2\u4e3a\u7279\u6b8a\u60c5\u5f62\uff0c\u800c\u540e\u8005\u6700\u65e9\u662f\u7531\u745e\u58eb\u56fe\u5f62\u827a\u672f\u5bb6\u5965\u65af\u5361\u00b7\u8def\u900f\u65af\u74e6<\/span>\uff08Oscar Reutersv\u00e4rd\uff0c1915\u20132002\uff09<\/span>\u57281937\u5e74\u753b\u51fa\u7684\u3002<\/span><\/p>\n


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\u7f57\u6770\u00b7\u5f6d\u7f57\u65af\u8fd8\u6784\u9020\u4e86\u4e00\u4e9b\u975e\u5468\u671f<\/span>\uff08aperiodic\uff09<\/span>\u9576\u5d4c\u56fe\u6848<\/span>\uff08\u4f8b\u5982\uff0c\u56fe4\u4e2d\u4e0a\u56fe\uff09<\/span>\u3002\u4f5c\u4e3a\u5bf9\u6bd4\uff0c\u53e6\u4e00\u7c7b\u9576\u5d4c\u56fe\u6848\u662f\u901a\u8fc7\u5468\u671f<\/span>\uff08periodic\uff09<\/span>\u91cd\u590d\u8fc7\u7a0b\u4ea7\u751f\u7684<\/span>\uff08\u4f8b\u5982\uff0c\u56fe4\u4e2d\u4e0b\u56fe\uff09<\/span>\u3002\u5f6d\u7f57\u65af\u4e09\u89d2\u548c\u5f6d\u7f57\u65af\u697c\u68af\u4e0e\u201c\u83ab\u6bd4\u4e4c\u65af\u5e26\u201d<\/span>\uff08M\u00f6bius strip\uff0c\u56fe4\u53f3\u4e0a\u56fe\uff09<\/span>\u548c\u201c\u4e0d\u53ef\u80fd\u4e09\u53c9\u621f\u201d<\/span>\uff08impossible trident\uff0c\u56fe4\u53f3\u4e0b\u56fe\uff09<\/span>\u7b49\u62d3\u6251\u53ca\u5e7b\u89c9\u89c6\u56fe\u5bc6\u5207\u76f8\u5173\uff0c\u4f46\u8fd9\u91cc\u5c31\u4e0d\u626f\u8fdc\u4e86\u3002\u503c\u5f97\u4e00\u63d0\u7684\u662f\uff0c\u9576\u5d4c\u6280\u672f\u53ea\u662f\u6570\u5b66\u7269\u7406\u5b66\u5bb6\u7f57\u6770\u00b7\u5f6d\u7f57\u65af\u7684\u4e1a\u4f59\u559c\u597d\uff0c\u4ed6\u57282020\u5e7489\u5c81\u9ad8\u9f84\u65f6\u56e0\u201c\u4f7f\u7528\u5de7\u5999\u7684\u6570\u5b66\u65b9\u6cd5\u8bc1\u660e\u4e86\u9ed1\u6d1e\u662f\u7231\u56e0\u65af\u5766\u5e7f\u4e49\u76f8\u5bf9\u8bba\u7684\u76f4\u63a5\u7ed3\u679c\u201d\u800c\u8363\u83b7\u8bfa\u8d1d\u5c14\u7269\u7406\u5b66\u5956\u3002<\/span><\/p>\n


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\u56fe4 \uff08\u5de6\uff09\u5f6d\u7f57\u65af\u697c\u68af\u4e0e\u4e09\u89d2;\uff08\u4e2d\uff09\u975e\u5468\u671f\u4e0e\u5468\u671f\u9576\u5d4c;\uff08\u53f3\uff09\u83ab\u6bd4\u4e4c\u65af\u5e26\u4e0e\u4e0d\u53ef\u80fd\u4e09\u53c9\u621f<\/span><\/section>\n


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\u5e73\u9762\u9576\u5d4c\u4e2d\u4efb\u4f55\u4e00\u79cd\u5f62\u72b6\u7684\u57fa\u672c\u74f7\u7816\u90fd\u79f0\u4e3a\u4e00\u4e2a\u201c\u539f\u578b\u5757\u201d<\/span>\uff08prototile\uff09<\/span>\u3002\u5c31\u4f7f\u7528\u7684\u539f\u578b\u5757\u6570\u91cf\u800c\u8a00\uff0c\u53ea\u7528\u4e00\u79cd\u539f\u578b\u5757\u7684\u9576\u5d4c\u79f0\u4e3a\u5355\u9762\u4f53\u5bc6\u94fa<\/span>\uff08monohedral tiling\uff09<\/span>\u3002\u8fd9\u79cd\u7c7b\u578b\u7684\u74f7\u7816\u7531\u5355\u4e00\u5f62\u72b6\u7ec4\u6210\uff0c\u4e5f\u79f0\u4e3a\u5355\u74f7\u7816<\/span>\uff08monotile\uff09<\/span>\u3002\u56e0\u4e3a\u5b83\u7684\u5fb7\u6587\u662fein Stein\uff0c\u5927\u5bb6\u620f\u79f0\u4e4b\u4e3a einstein<\/span>\uff08\u201c\u7231\u56e0\u65af\u5766\u201d\uff09<\/span>\u3002\u5468\u671f\u6027\u5355\u9762\u4f53\u5bc6\u94fa\u662f\u5f88\u5bb9\u6613\u7684\uff0c\u4f8b\u5982\u53ef\u4f7f\u7528\u6b63\u4e09\u89d2\u5f62\u3001\u6b63\u65b9\u5f62\u6216\u6b63\u516d\u8fb9\u5f62\u7684\u539f\u578b\u5757\u3002\u81f3\u4e8e\u4e94\u8fb9\u5f62\uff0c\u76ee\u524d\u77e5\u9053\u7684\u4ec5\u670915 \u79cd\u539f\u578b\u5757\uff0c\u5982\u56fe5\u6240\u793a\u3002<\/span><\/p>\n


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\u56fe5  \u76ee\u524d\u77e5\u9053\u768415\u79cd\u4e94\u8fb9\u5f62\u5468\u671f\u5bc6\u94fa<\/span><\/p>\n


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\u81f3\u4e8e\u975e\u5468\u671f\u6027\u5355\u9762\u4f53\u5bc6\u94fa\uff0c\u90a3\u5c31\u56f0\u96be\u5f97\u591a\u4e86\u3002\u7f57\u6770\u00b7\u5f6d\u7f57\u65af\u7b2c\u4e00\u4e2a\u6307\u51fa\uff0c\u7528\u4e24\u79cd\u4e0d\u540c\u7684\u83f1\u5f62\u5c31\u80fd\u5b9e\u73b0\u975e\u5468\u671f\u6027\u5bc6\u94fa<\/span>\uff08\u56fe6\uff0c\u5de6\u56fe\uff09<\/span>\u3002\u591a\u5e74\u4ee5\u540e\uff0c\u5230\u4e862023\u5e745\u6708\uff0c\u4e00\u4e2a\u7531\u82f1\u56fd\u3001\u52a0\u62ff\u5927\u548c\u7f8e\u56fd\u7684\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6\u548c\u6570\u5b66\u5bb6\u7ec4\u6210\u7684\u7814\u7a76\u5c0f\u7ec4<\/span>\uff08David Smith, Joseph S. Myers, Craig S. Kaplan, Chaim Goodman-Strauss\uff09<\/span>\u5ba3\u79f0\uff1a\u4ed6\u4eec\u627e\u5230\u4e86\u53ea\u7528\u4e00\u79cd13\u8fb9\u5f62\u5355\u9762\u4f53\u539f\u578b\u5757\u4fbf\u53ef\u5b8c\u6210\u7684\u975e\u5468\u671f\u6027\u5bc6\u94fa<\/span>\uff08\u56fe6\uff0c\u53f3\u56fe\uff09<\/span>\u3002<\/span><\/p>\n


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\u56fe6  \u975e\u5468\u671f\u5bc6\u94fa\uff1a\u5f6d\u7f57\u65af\u75282\u79cd\u83f1\u5f62; David Smith\u5c0f\u7ec4\u75281\u79cd13\u8fb9\u5f62<\/span><\/p>\n


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\u4e00\u79cd\u7279\u522b\u590d\u6742\u522b\u81f4\u7684\u5355\u9762\u4f53\u5bc6\u94fa\u7c7b\u578b\u662f\u87ba\u65cb\u5355\u9762\u4f53\u5bc6\u94fa<\/span>\uff08spiral monohedral tiling\uff09<\/span>\uff0c\u57281936 \u5e74\u7531\u5fb7\u56fd\u6570\u5b66\u5bb6\u6d77\u56e0\u8328\u00b7\u6c83\u5fb7\u4f2f\u683c<\/span>\uff08Heinz Voderberg\uff0c1911\u20131945\uff09<\/span>\u53d1\u73b0\uff0c\u5982\u56fe7\u6240\u793a\u3002\u5f88\u5feb\uff0c\u5b83\u5c31\u88ab\u63a8\u5e7f\u5230\u53cc\u9762\u4f53\u5bc6\u94fa<\/span>\uff08dihedral tiling\uff09<\/span>\u3001\u4e09\u9762\u4f53\u5bc6\u94fa\uff08trihedral tiling\uff09\u3001\u56db\u9762\u4f53\u5bc6\u94fa<\/span>\uff08tetrahedral tiling\uff09<\/span>\u4ee5\u81f3n<\/em>-\u9762\u4f53\u5bc6\u94fa<\/span>\uff08n<\/em>-hedral tiling\uff09<\/span>\u3002\u4e0d\u8fc7\uff0c\u6211\u4eec\u8fd9\u91cc\u5c31\u4e0d\u8fdb\u4e00\u6b65\u4ecb\u7ecd\u90a3\u4e9b\u7e41\u6742\u7684\u9ad8\u7ef4\u9576\u5d4c\u95ee\u9898\u4e86\u3002<\/span><\/p>\n


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<\/p>\n

\u56fe7 \u6c83\u5fb7\u4f2f\u683c\u7684\u87ba\u65cb\u5355\u9762\u4f53\u5bc6\u94fa<\/span><\/p>\n


<\/p>\n

\u57fa\u4e8e\u6240\u7528\u591a\u9762\u4f53\u539f\u578b\u5757\u7684\u79cd\u7c7b\uff0c\u9576\u5d4c\u53ef\u5206\u4e3a\u89c4\u5219\u3001\u534a\u89c4\u5219\u548c\u4e0d\u89c4\u5219\u7684\uff0c\u5982\u56fe8\u6240\u793a\u3002<\/span><\/p>\n


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<\/p>\n

\u56fe8  \u89c4\u5219\uff08\u5de6\uff09\u3001\u534a\u89c4\u5219\uff08\u4e2d\uff09\u548c\u4e0d\u89c4\u5219\uff08\u53f3\uff09\u7684\u9576\u5d4c\u56fe\u6848<\/span><\/p>\n


<\/p>\n

\u4e00\u4e2a\u6709\u8da3\u7684\u534a\u89c4\u5219\u9576\u5d4c\u56fe\u6848\u662f\u201c\u62ff\u7834\u4ed1\u5bc6\u94fa\u201d<\/span>\uff08Napoleon tiling\uff09<\/span>\u3002\u5982\u56fe9\u6240\u793a\uff0c\u4ece\u7ea2\u8272\u4e09\u89d2\u5f62\u5f00\u59cb\uff0c\u5728\u5176\u4e09\u6761\u8fb9\u5916\u5404\u81ea\u751f\u6210\u4e00\u4e2a\u7b49\u8fb9\u4e09\u89d2\u5f62\u3002\u5f53\u5e74\u62ff\u7834\u4ed1\u53d1\u73b0\uff0c\u4ee5\u4e09\u4e2a\u65b0\u4e09\u89d2\u5f62\u7684\u4e2d\u5fc3\u9ed1\u70b9\u4f5c\u4e3a\u9876\u70b9\u7ec4\u6210\u7684\u4e09\u89d2\u5f62\u662f\u4e2a\u6b63\u4e09\u89d2\u5f62\u3002\u6211\u4eec\u8fd9\u91cc\u4e0d\u53bb\u8ba8\u8bba\u8fd9\u4e2a\u62ff\u7834\u4ed1\u6b63\u4e09\u89d2\u5f62\uff0c\u800c\u662f\u4ee5\u56db\u4e2a\u62fc\u63a5\u5728\u4e00\u8d77\u7684\u4e09\u89d2\u5f62\u4f5c\u4e3a\u57fa\u672c\u539f\u578b\u5757\uff0c\u4e0d\u65ad\u91cd\u590d\u76f8\u540c\u7684\u89c4\u5219\u5411\u5916\u6269\u5c55\uff0c\u4ece\u800c\u83b7\u5f97\u4e00\u4e2a\u5bc6\u94fa\u3002<\/span><\/p>\n


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\u56fe9  \u62ff\u7834\u4ed1\u5bc6\u94fa<\/span><\/p>\n


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\u663e\u800c\u6613\u89c1\uff0c\u56fe\u5f62\u7684\u201c\u5bf9\u79f0\u6027\u201d<\/span>\uff08symmetry\uff09<\/span>\u5728\u8fd9\u91cc\u5927\u6709\u7528\u6b66\u4e4b\u5730\u3002\u5728\u9576\u5d4c\u4e2d\u4f5c\u6570\u5b66\u5206\u6790\u548c\u8ba1\u7b97\u65f6\uff0c\u901a\u5e38\u5047\u5b9a\u5e73\u9762\u662f\u65e0\u9650\u7684\uff0c\u5373\u5728\u4efb\u4f55\u65b9\u5411\u90fd\u6ca1\u6709\u8fb9\u754c\u3002\u4ece\u800c\uff0c\u5728\u4e00\u4e2a\u65e0\u9650\u5e73\u9762\u4e0a\u4fbf\u53ef\u4ee5\u8003\u8651\u5404\u79cd\u5bf9\u79f0\u51e0\u4f55\u53d8\u6362\uff0c\u4e3b\u8981\u5305\u62ec\uff1a\u5e73\u79fb\u53d8\u6362\uff0c\u5373\u628a\u56fe\u5f62\u5411\u4e00\u5b9a\u7684\u65b9\u5411\u5e73\u79fb\u4e00\u5b9a\u7684\u8ddd\u79bb\uff1b\u65cb\u8f6c\u53d8\u6362\uff0c\u5373\u628a\u56fe\u5f62\u76f8\u5bf9\u4e8e\u67d0\u4e2a\u53c2\u8003\u70b9\u8f6c\u52a8\u4e00\u5b9a\u7684\u89d2\u5ea6\uff1b\u6ed1\u7fd4\u53cd\u5c04\u53d8\u6362\uff0c\u5373\u628a\u56fe\u5f62\u76f8\u5bf9\u4e8e\u67d0\u6761\u53c2\u8003\u8f74\u7ebf\u4f5c\u955c\u9762\u53cd\u5c04\u7136\u540e\u518d\u8fdb\u884c\u4e00\u6b21\u4e00\u5b9a\u8ddd\u79bb\u7684\u5e73\u79fb\u3002\u8fd9\u4e09\u79cd\u51e0\u4f55\u53d8\u6362\u5f97\u5230\u7684\u7ed3\u679c\u56fe\u5f62\u548c\u539f\u6765\u7684\u56fe\u5f62\u662f\u7b49\u957f\u7b49\u5ea6<\/span>\uff08isometric\uff09<\/span>\u7684\uff0c\u5c31\u662f\u8bf4\u539f\u6765\u56fe\u5f62\u7684\u4efb\u4f55\u8fb9\u957f\u548c\u89d2\u5ea6\u90fd\u6ca1\u6709\u6539\u53d8\uff0c\u53ea\u662f\u53d8\u6362\u4e86\u4f4d\u7f6e\u548c\u65b9\u4f4d\uff0c\u5982\u56fe10\u6240\u793a\u3002<\/span><\/p>\n


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\u56fe10  \u5e73\u79fb\u53d8\u6362\uff08\u5de6\uff09\u3001\u65cb\u8f6c\u53d8\u6362\uff08\u4e2d\uff09\u548c\u6ed1\u7fd4\u53cd\u5c04\u53d8\u6362\uff08\u53f3\uff09<\/span><\/p>\n


<\/p>\n

\u5728\u9576\u5d4c\u8bbe\u8ba1\u4e2d\uff0c\u4eba\u4eec\u5f80\u5f80\u901a\u8fc7\u5404\u79cd\u4e0d\u540c\u7ec4\u5408\u7684\u5bf9\u79f0\u51e0\u4f55\u53d8\u6362\u6765\u83b7\u5f97\u7f8e\u89c2\u7684\u6548\u679c\u3002\u8fd9\u91cc\u5f53\u7136\u8981\u8bb0\u4f4f\uff0c\u4e0d\u7ba1\u9576\u5d4c\u5982\u4f55\u8bbe\u8ba1\uff0c\u5b83\u4e00\u5b9a\u8981\u5bc6\u94fa\uff0c\u5373\u94fa\u6ee1\u6574\u4e2a\u5e73\u9762\u5e76\u4e14\u6ca1\u6709\u91cd\u53e0\u4e5f\u4e0d\u7559\u7a7a\u9699\u3002\u65e9\u57281891\u5e74\uff0c\u4fc4\u7f57\u65af\u6570\u5b66\u5bb6\u53f6\u592b\u683c\u62c9\u592b\u00b7\u8d39\u5965\u591a\u7f57\u592b<\/span>\uff08Evgraf S. Fedorov\uff0c1853\u20131919\uff09<\/span>\u5c31\u8bc1\u660e\u4e86\uff0c\u4f7f\u7528\u4efb\u4f55\u4e00\u79cd\u539f\u578b\u5757\uff0c\u6700\u591a\u53ef\u4ee5\u670917\u79cd\u4e0d\u540c\u7684\u7ec4\u5408\u65b9\u5f0f\u53bb\u5bc6\u94fa\u4e00\u4e2a\u5e73\u9762\u30021924\u5e74\uff0c\u5308\u7259\u5229\u88d4\u7f8e\u56fd\u6570\u5b66\u5bb6\u4e54\u6cbb\u00b7\u6ce2\u5229\u4e9a<\/span>\uff08George P\u00f3lya\uff0c1887\u20131985\uff09<\/span>\u72ec\u7acb\u5730\u8bc1\u660e\u4e86\u540c\u6837\u7684\u7ed3\u679c\u3002\u6570\u5b66\u5bb6\u4eec\u53d1\u73b0\uff0c\u8fd9\u4e9b\u7ec4\u5408\u53ef\u4ee5\u6784\u6210\u67d0\u79cd\u6570\u5b66\u201c\u5bf9\u79f0\u7fa4\u201d<\/span>\uff08symmetry group\uff09<\/span>\uff0c\u79f0\u4e3a\u201c\u5899\u7eb8\u7fa4\u201d<\/span>\uff08wallpaper group\uff09<\/span>\u3002\u53d7\u6ce2\u5229\u4e9a\u7684\u542f\u53d1\uff0c\u57c3\u820d\u5c14\u8fd8\u8bbe\u8ba1\u4e8643\u79cd\u5305\u542b\u8fd917\u79cd\u4e0d\u540c\u7ec4\u5408\u7684\u6f02\u4eae\u9576\u5d4c\u56fe\u6848\u3002<\/span><\/p>\n


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

\u8fd8\u6709\u4e00\u79cd\u57fa\u4e8e\u201c\u975e\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55\u201d\u6765\u8bbe\u8ba1\u7684\u6709\u8da3\u9576\u5d4c\u56fe\u6848\uff0c\u5982\u56fe11\u6240\u793a\uff0c\u5176\u4e2d\u5de6\u56fe\u5c5e\u4e8e\u53cc\u66f2\u51e0\u4f55<\/span>\uff08hyperbolic geometry\uff09<\/span>\u56fe\u6848\uff0c\u53f3\u56fe\u5c5e\u4e8e\u7403\u9762\u51e0\u4f55<\/span>\uff08spherical geometry\uff09<\/span>\u56fe\u6848\u3002<\/span><\/p>\n


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

<\/p>\n

\u56fe11  \u201c\u975e\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55\u201d\u9576\u5d4c\u56fe\u6848<\/span><\/p>\n


<\/p>\n

\u663e\u7136\uff0c\u5728\u5404\u79cd\u7ec4\u5408\u5206\u7c7b\u4e2d\uff0c\u989c\u8272\u4e5f\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u56e0\u7d20\uff0c\u5373\u5f62\u72b6\u76f8\u540c\u4f46\u989c\u8272\u4e0d\u540c\u7684\u74f7\u7816\u662f\u5426\u88ab\u8ba4\u4e3a\u662f\u4e0d\u540c\u7684\uff1f\u8fd9\u5728\u827a\u672f\u8bbe\u8ba1\u4e2d\u901a\u5e38\u90fd\u6ca1\u6709\u4e89\u8bae\uff0c\u4f46\u5728\u6570\u5b66\u7814\u7a76\u548c\u8ba8\u8bba\u4e2d\u662f\u9700\u8981\u4e8b\u5148\u754c\u5b9a\u7684\u30021961 \u5e74\uff0c\u534e\u88d4\u7f8e\u7c4d\u903b\u8f91\u5b66\u5bb6\u738b\u6d69<\/span>\uff081921\u20131995\uff09<\/span>\u6ce8\u610f\u5230\u51b3\u7b56\u95ee\u9898\u4e0e\u9576\u5d4c\u95ee\u9898\u53ef\u6309\u67d0\u79cd\u65b9\u5f0f\u5bf9\u5e94\u8d77\u6765\u3002\u4e3a\u65b9\u4fbf\u8868\u8fbe\u4ed6\u5f15\u5165\u4e86\u5e26\u4e0d\u540c\u989c\u8272\u7684\u6b63\u65b9\u5f62\u539f\u578b\u5757\uff0c\u540e\u4eba\u79f0\u4e4b\u4e3a\u201c\u738b\u6d69\u591a\u7c73\u8bfa\u9aa8\u724c\u201d<\/span>\uff08Wang domino\uff09<\/span>\u3002\u738b\u6d69\u8bc1\u660e\u4e86\uff1a\u56fe\u7075\u673a\u53ef\u8f6c\u5316\u4e3a\u4e00\u7ec4\u738b\u6d69\u591a\u7c73\u8bfa\u9aa8\u724c\u3002\u4ed6\u8fd8\u63d0\u51fa\u4e86\u4e00\u4e2a\u201c\u591a\u7c73\u8bfa\u9aa8\u724c\u95ee\u9898\u201d\uff1a\u662f\u5426\u5b58\u5728\u4e00\u7ec4\u738b\u6d69\u591a\u7c73\u8bfa\u9aa8\u724c\uff0c\u901a\u8fc7\u9002\u5f53\u7684\u653e\u7f6e\u8ba9\u90bb\u63a5\u9aa8\u724c\u8fb9\u7f18\u4e24\u4fa7\u5177\u6709\u4e0d\u540c\u989c\u8272\uff0c\u4f7f\u5f97\u5b83\u4eec\u5bc6\u94fa\u6574\u4e2a\u5e73\u9762\uff1f\u4ed6\u89c2\u5bdf\u5230\uff0c\u5982\u679c\u8fd9\u4e2a\u95ee\u9898\u662f\u903b\u8f91\u5b66\u4e0d\u53ef\u5224\u5b9a\u7684\uff0c\u90a3\u4e48\u5c31\u7b49\u4e8e\u8bf4\u5fc5\u5b9a\u5b58\u5728\u4e00\u7ec4\u975e\u5468\u671f\u738b\u6d69\u591a\u7c73\u8bfa\u9aa8\u724c\u6765\u5b9e\u73b0\u5bc6\u94fa\u3002\u5f53\u5e74\uff0c\u8fd9\u4e2a\u903b\u8f91\u5b66\u4e0d\u53ef\u5224\u5b9a\u95ee\u9898\u4f3c\u4e4e\u662f\u96be\u4ee5\u7f6e\u4fe1\u7684\uff0c\u6240\u4ee5\u738b\u6d69\u63a8\u6d4b\u8be5\u95ee\u9898\u5728\u903b\u8f91\u4e0a\u5e94\u8be5\u662f\u53ef\u5224\u5b9a\u7684\uff0c\u56e0\u800c\u4e0d\u5b58\u5728\u8fd9\u79cd\u975e\u5468\u671f\u56fe\u5f62\u96c6\u5408\u3002<\/span><\/p>\n


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

1964\u5e74\uff0c\u738b\u6d69\u7684\u5b66\u751f\u7f57\u4f2f\u7279\u00b7\u4f2f\u6770<\/span>\uff08Robert Berger\uff0c1938\u2013\uff09<\/span>\u5728\u4ed6\u7684\u535a\u58eb\u5b66\u4f4d\u8bba\u6587\u4e2d\u8bc1\u660e\u4e86\uff1a\u738b\u6d69\u591a\u7c73\u8bfa\u9aa8\u724c\u95ee\u9898\u662f\u903b\u8f91\u4e0d\u53ef\u5224\u5b9a\u7684\uff0c\u56e0\u6b64\u5bfc\u5e08\u63d0\u51fa\u7684\u95ee\u9898\u662f\u5bf9\u7684\u4f46\u4ed6\u731c\u6d4b\u7684\u7b54\u6848\u662f\u9519\u7684\u3002\u4f2f\u6770\u5177\u4f53\u6784\u9020\u4e86\u4e00\u7ec420426 \u4e2a\u975e\u5468\u671f\u6027\u738b\u6d69\u591a\u7c73\u8bfa\u9aa8\u724c\u3002\u4ed6\u8fd8\u8bf4\uff0c\u8fd9\u4e2a\u6570\u5b57\u53ef\u4ee5\u51cf\u5c11\u5230 104\uff0c\u4f46\u5176\u8bc1\u660e\u6ca1\u6709\u6b63\u5f0f\u53d1\u8868\u30021968 \u5e74\uff0c\u7f8e\u56fd\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6\u5510\u7eb3\u5fb7\u00b7\u9ad8\u5fb7\u7eb3<\/span>\uff08Donald E. Knuth\uff0c1938\u2013\uff09<\/span>\u4fee\u6539\u4e86\u4f2f\u6770\u7684\u6784\u9020\u7a0b\u5e8f\uff0c\u53d1\u73b0\u53ea\u9700\u8981 92 \u4e2a\u738b\u6d69\u591a\u7c73\u8bfa\u9aa8\u724c\u5c31\u53ef\u4ee5\u4e86\u30021971\u5e74\uff0c\u7f8e\u56fd\u6570\u5b66\u5bb6\u62c9\u6590\u5c14\u00b7\u7f57\u5bbe\u900a<\/span>\uff08Raphael M. Robinson\uff0c1911\u20131995\uff09<\/span>\u8fdb\u4e00\u6b65\u7b80\u5316\u4e86\u4f2f\u6770\u7684\u6784\u9020\u7a0b\u5e8f\uff0c\u6307\u51fa\u53ea\u9700\u898156\u4e2a\u591a\u7c73\u8bfa\u9aa8\u724c\u30021996\u5e74\uff0c\u6377\u514b\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6\u5361\u96f7\u5c14\u00b7\u5e93\u5229\u514b<\/span>\uff08Karel Culik II, 1934\u2013\uff09<\/span>\u628a\u8fd9\u4e2a\u6570\u5b57\u51cf\u5c11\u523013\u4e2a\u30022015\u5e74\uff0c\u6cd5\u56fd\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6 Emmanuel Jeandel \u548c Michael Rao \u8bc1\u660e\u4e86\uff0c\u53ea\u970011\u4e2a\u738b\u6d69\u591a\u7c73\u8bfa\u9aa8\u724c\u5e76\u4f7f\u75284\u79cd\u989c\u8272\u4fbf\u8db3\u591f\u4e86\uff0c\u5982\u56fe12\u6240\u793a\u3002<\/span><\/p>\n


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\u56fe12   Jeamdel \u548cRao\u768411\u4e2a\u738b\u6d69\u591a\u7c73\u8bfa\u9aa8\u724c\u548c4\u79cd\u989c\u8272\u5206\u914d<\/span><\/p>\n


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\u56fe14  \u5206\u5f62\u9576\u5d4c\u56fe\u6848\u793a\u4f8b<\/span><\/p>\n


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\u590d\u6742\u79d1\u5b66\u6700\u65b0\u8bba\u6587<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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1. <\/strong>\u56e0\u4e3a\u5979\uff0c\u201c\u79d1\u5b66\u5bb6\u201d\u6210\u4e3a\u4e00\u4e2a\u79f0\u8c13\u4e28\u9648\u5173\u8363<\/a><\/strong><\/span><\/strong><\/section>\n
2. <\/strong>\u8d6b\u5179\uff0c\u4ed6\u8bc1\u5b9e\u4e86\u7535\u78c1\u6ce2\u7684\u5b58\u5728\u4e28\u9648\u5173\u8363<\/a><\/strong><\/span><\/section>\n
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3. <\/strong><\/strong>\u5206\u5f62\u2014\u2014\u6545\u4e8b\u4e4b\u5916\u4e28\u9648\u5173\u8363<\/a><\/strong><\/strong><\/span><\/section>\n
4. \u300a\u5f20\u6c5f\u00b7\u590d\u6742\u79d1\u5b66\u524d\u6cbf27\u8bb2\u300b\u5b8c\u6574\u4e0a\u7ebf\uff01<\/strong><\/a><\/strong><\/strong><\/strong><\/section>\n
5. \u6210\u4e3a\u96c6\u667aVIP\uff0c\u89e3\u9501\u5168\u7ad9\u8bfe\u7a0b\uff0f\u8bfb\u4e66\u4f1a<\/span><\/strong><\/a><\/strong><\/strong><\/strong><\/strong><\/section>\n
6. <\/strong><\/strong><\/strong>\u52a0\u5165\u96c6\u667a\uff0c\u4e00\u8d77\u590d\u6742\uff01<\/a><\/strong><\/strong><\/strong><\/span><\/section>\n


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<\/mp-style-type><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

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