| − | 克形连续改变形状时的一些特征和规律的，克莱因瓶和莫比乌斯带变成了拓扑学中最有趣的问题之一。莫比乌斯带的概念被广泛地应用到了建筑，艺术，工业生产中。三维空间里的克莱因瓶 拓扑学的定义编辑 克莱因瓶定义为正方形区域 [0,1]×[0,1] 模掉等价关系(0,y)~(1,y), 0≤y≤1 和 (x,0)~(1-x,1), 0≤x≤1。类似于 Mobius Band, 克莱因瓶不可定向。但 Mobius 带可嵌入 ，而克莱因瓶只能嵌入Café Savarin," said Jack to the chauffeur. The cab started with a jerk, throwing them back on the cushions. "Let me out!" she said—but not very strongly. He affected not to hear. There was a delicious satisfaction in seeing the self-possessed little lady overcome with confusion, if only for a moment. "To-morrow I'll come for you in my own car," he said, nonchalantly. "Are you crazy?" she murmured, really alarmed. He laughed. "Can't I have a car as well as anybody?" " "Please!" He told her at last. The story sounded strange in his own ears. When he came to the end he saw to his astonishment that there were tears in her eyes. "Why—why, what's the matter?" he cried. "I don't know," she said smiling through the rain. "Am I not silly? But I suppose it means change. And I hate changes!" "A change for the better, only. If you knew how I hated poverty!" Her eyes dropped. "I, too," that meant, but she did not care to tell him so, audibly. "If you knew how mean I felt every day when we went to that beanery together, and you had to pay for your own lunch!" "But what was the difference? We both work for our living." "A man feels differently. Why I never would ask you if I could come to see you in the evenings, because I couldn't take you out anywhere. I was afraid I couldn't keep my end up with your gang." "I haven't any gang," she murmured. "Well all that's ended now! Now there's no limit but the sky! And here we are. The lawyer guy told me this was the swellest place down-town." A fresh panic seized her. "I can't eat in a place like this! I'm not fit to be seen!" "Nonsense! You always look like a lady!" Circumstances were too strong for her. She found herself being wafted across the sidewalk, and was delivered into the hands of the maid in the lobby, before she could think of an effective resistance. Indeed they were seated at a snowy little board brightened by an electric candle, before she really got her breath. At Jack's elbow stood a post-graduate waiter with a deferential bend in his back, and at just the right distance an orchestra was discussing the Meditation from Tha?s. A sigh escaped Kate, for after all she was a perfectly human girl. "Oh, this is heavenly!" Jack's eyes sparkled. "Good! I was wondering when you'd begin to let yourself go." He leaned forward. "You should worry! You're the prettiest girl here—and the best dressed!" Which was true—on both counts. There was no doubt about her prettiness; Heaven had attended to that. Eyes of the deepest blue with a glance steady and deep; an adorable little nose, and a mouth at once firm and most kissable. As for her clothes, it may be they were of cheap materials, but the taste that had chosen redeemed them. The hat, most important item, was of Kate's own manufacture, being copied from the window of a milliner whose name is a household word. "Don't be silly," said the wearer severely. "The waiter is waiting." "That's what he's here for! Oh, dear! I wish we could stay all afternoon!" This was put forth really as a proposal rather than a wish. But Kate was relentless. "We'll have to hurry," she said firmly. "Well, we've time for a cup of green turtle, a lobster paté and a coupe St. Jacques," said Jack. A whispered order was added, and one of the yellow backs changed hands. The waiter departed. "One would think you had been coming here all your life," said Kate demurely. This was delicious flattery. "I've planned it in dreams," he said. Presently the waiter returned, smiling from ear to ear, and bearing a bunch of violets almost as big as a cart wheel. Their delicious fragrance filled all the air. With a flourish he placed them before Kate. She gasped. "Oh! How wonderful! For me!" "Who do you think?" said Jack. "But—but what shall I do with them?" "Put them on. Any woman can wear violets without hurting." "But what will they think when I get back to the office." "The worst!" said Jack solemnly. "Oh, Mr. Norman!" "Why go back to the office?" asked Jack very offhand. "Oh, Mr. Norman!" she said again, with a scandalized air. "My name is Jack," he said unabashed. She made believe not to hear. "I can't bear to think of you working even for a day longer in that stuffy hole! Why, my first thought when I heard the news was I can take her out of that! What fun will it be for me to fluff around town spending money when you are still jailed there, punishing the alphabet." "What do you mean?" she said, trying to look indignant. "You know what I mean. Or if you don't, look at me and you'll see!" She did not avail herself of the invitation. "You don't seem to have thought much of me. What I might like. Am I nothing to you, but a sort of little follower, a hanger-on to help you spend money!" "Oh, Katy, that's unjust. Look at me! Katy darling, I love you. Will you marry me?" "Somebody will hear you," she murmured glancing nervously around. "That's no answer." "Why—I scarcely know you!" "Time will fix that." "You're not in earnest." "I am! Look at me! I know you well! For months I have thought of you night and day. Oh, I tried to cut you out at first; I thought I was only storing up trouble for myself. Poor devil of a stool-warmer like me. What chance did I have? But I couldn't help myself! Every time I saw your face at the window I forgot my hard-headed resolutions. You see you had me at a disadvantage. I had an ideal of what a lady was, that I got from my mother—but knocking round in cheap boarding houses, well you don't meet that kind. It was just plumb luck my meeting you. First time I heard your voice you just knocked me out. That was what I had wanted—all my life. Look at me! Don't you think I'm in earnest now?" "Please, not here!" she murmured. He suddenly realized that a girl is entitled to a certain degree of privacy in receiving a proposal. "Oh! I clean forgot where we were!" he said contritely. "I'm sorry. The two things are so mixed up in my mind, I felt I couldn't tell you quick enough." A silence fell between them. He studied her face wistfully, but could read nothing in the closed lips and downcast eyes. "Katy, dear, can't you give me one word to go on?" She shook her head. "Nothing definite, Katy—but just a hint I can't stand the suspense." She murmured softly: "My answer is no." "Oh, Katy!" he said brokenly. "Sometimes I thought you looked at me as if—my mistake, I suppose. Don't you like me, Katy?" "One doesn't marry on liking. I used to like you as a poor boy; But money changes people's characters. I'll have to wait and see." 四维（或更高维）空间。莫比乌斯带编辑 把一条纸带的一段扭180°，再和另一端粘起来就得到一条莫比乌斯带的模型。这也是一个只有莫比乌斯带、一个面的曲面，但是和球面、轮胎面和克莱因瓶不同的是，它有边（注意，它只有一条边）。如果我们把两条莫比乌斯带沿着它们唯一的边粘合起来，你就得到了一个克莱因瓶 莫比乌斯带 莫比乌斯带 （当然不要忘了，我们必须在四维空间中才能真正有可能完成这个粘合，否则的话就不得不把纸撕破一点）。同样地，如果把一个克莱因瓶适当地剪开来，我们就能得到两条莫比乌斯带。除了我们上面看到的克莱因瓶的模样，还有一种不太为人所知的“8字形”克莱因瓶。它看起来和上面的曲面完全不同，但是在四维空间中它们其实就是同一个曲面－－克莱因瓶。实际上，可以说克莱因瓶是一个3°的莫比乌斯带。我们知道，在平面上画一个圆，再在圆内放一样东西，假如在二度空间中将它拿出来，就不得不越过圆周。超新星爆发是某些恒星在演化接近末期时经历的一种剧烈爆炸。一般认为质量小于9倍太阳质量左右的恒星，在经历引力坍缩的过程后是无法形成超新星的。[75] 在大质量恒星演化到晚期，内部不能产生新的能量，巨大的引力将整个星体迅速向中心坍缩，将中心物质都压成中子状态，形成中子星，而外层下坍的物质遇到这坚硬的“中子核”反弹引起爆炸。这就成为超新星爆发，质量更大时，中心更可形成黑洞。[76] 在超新星爆发的过程中所释放的能量，需要我们的太阳燃烧900亿年才能与之相当。[77] 超新星研究有着关乎人类自身命运的深层意义。如果一颗超新星爆发的位置非常接近地球，目前国际天文学界普遍认为此距离在100光年以内，它就能够对地球的生物圈产生明显的影响，这样的超新星被称为近地超新星。有研究认为，在地球历史上的奥陶纪大灭绝，就是一颗近地超新星引起的，这次灭绝导致当时地球近60%的海洋生物消失。[78]
| + | <center><font size="5" color=blue;">'''突发！两面开战，伊朗又在波斯湾动手了，美国向联合国紧急抗议！” |
| − | 克莱因瓶是一个不可定向的二维紧流形，而球面或轮胎面是可 克莱因瓶 克莱因瓶 定向的二维紧流形。如果观察克莱因瓶，有一点似乎令人困惑－－克莱因瓶的瓶颈和瓶身是相交的，换句话说，瓶颈上的某些点和瓶壁上的某些点占据了三维空间中的同一个位置。我们可以把克莱因瓶放在四维空间中理解：克莱因瓶是一个在四维空间中才可能真正表现出来的曲面。如果我们一定要把它表现在我们生活的三维空间中，我们只好将就点，把它表现得似乎是自己和自己相交一样。克莱因瓶的瓶颈是穿过了第四维空间再和瓶底圈连起来的，并不穿过瓶壁。用扭结来打比方，如果把它看作平面上的曲线的话瓶定义为正方形区域 [0,1]×[0,1] 模掉等价关系(0,y)~(1,y), 0≤y≤1 和 (x,0)~(1-x,1), 0≤x≤1。类似于 Mobius Band, 克莱因瓶不可定向。但 Mobius 带可嵌入 ，而克莱因瓶只能嵌入四维（或更高维）空间。莫比乌斯带编辑 把一条纸带的一段扭180°，再和另一端粘起来就得伽利略·伽利雷证明了哥Café Savarin," said Jack to the chauffeur. The cab started with a jerk, throwing them back on the cushions. "Let me out!" she said—but not very strongly. He affected not to hear. There was a delicious satisfaction in seeing the self-possessed little lady overcome with confusion, if only for a moment. "To-morrow I'll come for you in my own car," he said, nonchalantly. "Are you crazy?" she murmured, really alarmed. He laughed. "Can't I have a car as well as anybody?" "But I thought—that is—you always said——" "That I was as poor as Job's turkey, eh? Only a stall. I just worked for Fisher for the sociological experience. I don't have to work really." She looked at him with troubled eyes. He couldn't resist the temptation to tease her a little. "My old man's a multi-millionaire," he rattled on. "Of course I get sick of that life sometimes, and scout about a bit." Her eyes became so reproachful his heart smote him. "Oh, that's only a joke," he said quickly. "Lord knows the poverty was real enough—but it's over for good!" "For both of us," he would have liked to add, but did not quite dare. "Look!" he cried, drawing his hand out of his pocket with the great roll of yellow-backed bills. "My income for half an hour!" "Where did you get it?" she said aghast He laughed again. "Honest, I didn't steal it." "Please!" He told her at last. The story sounded strange in his own ears. When he came to the end he saw to his astonishment that there were tears in her eyes. "Why—why, what's the matter?" he cried. "I don't know," she said smiling through the rain. "Am I not silly? But I suppose it means change. And I hate changes!" "A change for the better, only. If you knew how I hated poverty!" Her eyes dropped. "I, too," that meant, but she did not care to tell him so, audibly. "If you knew how mean I felt every day when we went to that beanery together, and you had to pay for your own lunch!" "But what was the difference? We both work for our living." "A man feels differently. Why I never would ask you if I could come to see you in the evenings, because I couldn't take you out anywhere. I was afraid I couldn't keep my end up with your gang." "I haven't any gang," she murmured. "Well all that's ended now! Now there's no limit but the sky! And here we are. The lawyer guy told me this was the swellest place down-town." A fresh panic seized her. "I can't eat in a place like this! I'm not fit to be seen!" "Nonsense! You always look like a lady!" Circumstances were too strong for her. She found herself being wafted across the sidewalk, and was delivered into the hands of the maid in the lobby, before she could think of an effective resistance. Indeed they were seated at a snowy little board brightened by an electric candle, before she really got her breath. At Jack's elbow stood a post-graduate waiter with a deferential bend in his back, and at just the right distance an orchestra was discussing the Meditation from Tha?s. A sigh escaped Kate, for after all she was a perfectly human girl. "Oh, this is heavenly!" Jack's eyes sparkled. "Good! I was wondering when you'd begin to let yourself go." He leaned forward. "You should worry! You're the prettiest girl here—and the best dressed!" Which was true—on both counts. There was no doubt about her prettiness; Heaven had attended to that. Eyes of the deepest blue with a glance steady and deep; an adorable little nose, and a mouth at once firm and most kissable. As for her clothes, it may be they were of cheap materials, but the taste that had chosen redeemed them. The hat, most important item, was of Kate's own manufacture, being copied from the window of a milliner whose name is a household word. "Don't be silly," said the wearer severely. "The waiter is waiting." "That's what he's here for! Oh, dear! I wish we could stay all afternoon!" This was put forth really as a proposal rather than a wish. But Kate was relentless. "We'll have to hurry," she said firmly. "Well, we've time for a cup of green turtle, a lobster paté and a coupe St. Jacques," said Jack. A whispered order was added, and one of the yellow backs changed hands. The waiter departed. "One would think you had been coming here all your life," said Kate demurely. This was delicious flattery. "I've planned it in dreams," he said. Presently the waiter returned, smiling from ear to ear, and bearing a bunch of violets almost as big as a cart wheel. Their delicious fragrance filled all the air. With a flourish he placed them before Kate. She gasped. "Oh! How wonderful! For me!" "Who do you think?" said Jack. "But—but what shall I do with them?" "Put them on. Any woman can wear violets without hurting." "But what will they think when I get back to the office." "The worst!" said Jack solemnly. "Oh, Mr. Norman!" "Why go back to the office?" asked Jack very offhand. "Oh, Mr. Norman!" she said again, with a scandalized air. "My name is Jack," he said unabashed. She made believe not to hear. "I can't bear to think of you working even for a day longer in that stuffy hole! Why, my first thought when I heard the news was I can take her out of that! What fun will it be for me to fluff around town spending money when you are still jailed there, punishing the alphabet." "What do you mean?" she said, trying to look indignant. "You know what I mean. Or if you don't, look at me and you'll see!" She did not avail herself of the invitation. "You don't seem to have thought much of me. What I might like. Am I nothing to you, but a sort of little follower, a hanger-on to help you spend money!" "Oh, Katy, that's unjust. Look at me! Katy darling, I love you. Will you marry me?" "Somebody will hear you," she murmured glancing nervously around. "That's no answer." "Why—I scarcely know you!" "Time will fix that." "You're not in earnest." "I am! Look at me! I know you well! For months I have thought of you night and day. Oh, I tried to cut you out at first; I thought I was only storing up trouble for myself. Poor devil of a stool-warmer like me. What chance did I have? But I couldn't help myself! Every time I saw your face at the window I forgot my hard-headed resolutions. You see you had me at a disadvantage. I had an ideal of what a lady was, that I got from my mother—but knocking round in cheap boarding houses, well you don't meet that kind. It was just plumb luck my meeting you. First time I heard your voice you just knocked me out. That was what I had wanted—all my life. Look at me! Don't you think I'm in earnest now?" "Please, not here!" she murmured. He suddenly realized that a girl is entitled to a certain degree of privacy in receiving a proposal. "Oh! I clean forgot where we were!" he said contritely. "I'm sorry. The two things are so mixed up in my mind, I felt I couldn't tell you quick enough." A silence fell between them. He studied her face wistfully, but could read nothing in the closed lips and downcast eyes. "Katy, dear, can't you give me one word to go on?" She shook her head. "Nothing definite, Katy—but just a hint I can't stand the suspense." She murmured softly: "My answer is no." "Oh, Katy!" he said brokenly. "Sometimes I thought you looked at me as if—my mistake, I suppose. Don't you like me, Katy?" "One doesn't marry on liking. I used to like you as a poor boy; But money changes people's characters. I'll have to wait and see." 白尼是正确的。[3] 阿里斯塔克斯提倡 阿里斯塔克斯（Aristarchus, 约公元前 310年- 约公元前230年），是人类历史上有记载的首位提倡日心说的天文学者，是古希腊时期、也是人类历史上有记载的最伟大的天文学家，数学家。他生于古希腊萨摩斯岛。他将太阳而不是地球放置在整个已知宇宙的中心，他是人类歴史上有记载的最早期的日心说的提倡者之一。但是在当时的古希腊、他的宇宙观和杰出的智慧并未能被当时的人们所理解，并被亚里士多德和托勒密的要求希腊人控告阿里斯塔克斯的渎神之罪。之后阿里斯塔克斯的思想学说就像珍贵的戒指被扔入大海般消失无踪。直到哥白尼的出现。伽利略的论证 伽利略是通过Café Savarin," said Jack to the chauffeur. The cab started with a jerk, throwing them back on the cushions. "Let me out!" she said—but not very strongly. He affected not to hear. There was a delicious satisfaction in seeing the self-possessed little lady overcome with confusion, if only for a moment. "To-morrow I'll come for you in my own car," he said, nonchalantly. "Are you crazy?" she murmured, really alarmed. He laughed. "Can't I have a car as well as anybody?" "But I thought—that is—you always said——" "That I was as poor as Job's turkey, eh? Only a stall. I just worked for Fisher for the sociological experience. I don't have to work really." She looked at him with troubled eyes. He couldn't resist the temptation to tease her a little. "My old man's a multi-millionaire," he rattled on. "Of course I get sick of that life sometimes, and scout about a bit." Her eyes became so reproachful his heart smote him. "Oh, that's only a joke," he said quickly. "Lord knows the poverty was real enough—but it's over for good!" "For both of us," he would have liked to add, but did not quite dare. "Look!" he cried, drawing his hand out of his pocket with the great roll of yellow-backed bills. "My income for half an hour!" "Where did you get it?" she said aghast He laughed again. "Honest, I didn't steal it." "Please!" He told her at last. The story sounded strange in his own ears. When he came to the end he saw to his astonishment that there were tears in her eyes. "Why—why, what's the matter?" he cried. "I don't know," she said smiling through the rain. "Am I not silly? But I suppose it means change. And I hate changes!" "A change for the better, only. If you knew how I hated poverty!" Her eyes dropped. "I, too," that meant, but she did not care to tell him so, audibly. "If you knew how mean I felt every day when we went to that beanery together, and you had to pay for your own lunch!" "But what was the difference? We both work for our living." "A man feels differently. Why I never would ask you if I could come to see you in the evenings, because I couldn't take you out anywhere. I was afraid I couldn't keep my end up with your gang." "I haven't any gang," she murmured. "Well all that's ended now! Now there's no limit but the sky! And here we are. The lawyer guy told me this was the swellest place down-town." A fresh panic seized her. "I can't eat in a place like this! I'm not fit to be seen!" "Nonsense! You always look like a lady!" Circumstances were too strong for her. She found herself being wafted across the sidewalk, and was delivered into the hands of the maid in the lobby, before she could think of an effective resistance. Indeed they were seated at a snowy little board brightened by an electric candle, before she really got her breath. At Jack's elbow stood a post-graduate waiter with a deferential bend in his back, and at just the right distance an orchestra was discussing the Meditation from Tha?s. A sigh escaped Kate, for after all she was a perfectly human girl. "Oh, this is heavenly!" Jack's eyes sparkled. "Good! I was wondering when you'd begin to let yourself go." He leaned forward. "You should worry! You're the prettiest girl here—and the best dressed!" Which was true—on both counts. There was no doubt about her prettiness; Heaven had attended to that. Eyes of the deepest blue with a glance steady and deep; an adorable little nose, and a mouth at once firm and most kissable. As for her clothes, it may be they were of cheap materials, but the taste that had chosen redeemed them. The hat, most important item, was of Kate's own manufacture, being copied from the window of a milliner whose name is a household word. "Don't be silly," said the wearer severely. "The waiter is waiting." "That's what he's here for! Oh, dear! I wish we could stay all afternoon!" This was put forth really as a proposal rather than a wish. But Kate was relentless. "We'll have to hurry," she said firmly. "Well, we've time for a cup of green turtle, a lobster paté and a coupe St. Jacques," said Jack. A whispered order was added, and one of the yellow backs changed hands. The waiter departed. "One would think you had been coming here all your life," said Kate demurely. This was delicious flattery. "I've planned it in dreams," he said. Presently the waiter returned, smiling from ear to ear, and bearing a bunch of violets almost as big as a cart wheel. Their delicious fragrance filled all the air. With a flourish he placed them before Kate. She gasped. "Oh! How wonderful! For me!" "Who do you think?" said Jack. "But—but what shall I do with them?" "Put them on. Any woman can wear violets without hurting." "But what will they think when I get back to the office." "The worst!" said Jack solemnly. "Oh, Mr. Norman!" "Why go back to the office?" asked Jack very offhand. "Oh, Mr. Norman!" she said again, with a scandalized air. "My name is Jack," he said unabashed. She made believe not to hear. "I can't bear to think of you working even for a day longer in that stuffy hole! Why, my first thought when I heard the news was I can take her out of that! What fun will it be for me to fluff around town spending money when you are still jailed there, punishing the alphabet." "What do you mean?" she said, trying to look indignant. "You know what I mean. Or if you don't, look at me and you'll see!" She did not avail herself of the invitation. "You don't seem to have thought much of me. What I might like. Am I nothing to you, but a sort of little follower, a hanger-on to help you spend money!" "Oh, Katy, that's unjust. Look at me! Katy darling, I love you. Will you marry me?" "Somebody will hear you," she murmured glancing nervously around. "That's no answer." "Why—I scarcely know you!" "Time will fix that." "You're not in earnest." "I am! Look at me! I know you well! For months I have thought of you night and day. Oh, I tried to cut you out at first; I thought I was only storing up trouble for myself. Poor devil of a stool-warmer like me. What chance did I have? But I couldn't help myself! Every time I saw your face at the window I forgot my hard-headed resolutions. You see you had me at a disadvantage. I had an ideal of what a lady was, that I got from my mother—but knocking round in cheap boarding houses, well you don't meet that kind. It was just plumb luck my meeting you. First time I heard your voice you just knocked me out. That was what I had wanted—all my life. Look at me! Don't you think I'm in earnest now?" "Please, not here!" she murmured. He suddenly realized that a girl is entitled to a certain degree of privacy in receiving a proposal. "Oh! I clean forgot where we were!" he said contritely. "I'm sorry. The two things are so mixed up in my mind, I felt I couldn't tell you quick enough." A silence fell between them. He studied her face wistfully, but could read nothing in the closed lips and downcast eyes. "Katy, dear, can't you give me one word to go on?" She shook her head. "Nothing definite, Katy—but just a hint I can't stand the suspense." She murmured softly: "My answer is no." "Oh, Katy!" he said brokenly. "Sometimes I thought you looked at me as if—my mistake, I suppose. Don't you like me, Katy?" "One doesn't marry on liking. I used to like you as a poor boy; But money changes people's characters. I'll have to wait and see." 数学逻辑相信哥白尼。这一点与布鲁诺没有区别。同时，伽利略发明了天文望远镜，一定程度证明了哥白尼的正确。但是，在罗马宗教事务所组织的学术讨论中，伽利略没有战胜自己的对手，导致了最后的悲剧：当时“地球绕太阳”和“太阳绕地球”都有科学证据，而伽利略学说的破绽之一，是科学家探测不到“斗转星移”（Stellar Parallax）的现象。什么是斗转星移呢？这名堂十分吓人，其实意思很简单。如图一显示，假设星星 A 和星星 B 悬浮在太空中，我在地球表面之观察点 1 仰望星星 A 和星星 B 时，它们的距离好像十分接近，如果地球自转，即使我站在原地不动，我将会随着地球移动而去了观时，我们还应该注意到，天文学的应用范围不如物理学、化学和生物学那样广泛。从理论上来讲，人们即使对哥白尼学说的知识和应用一窍不通，也会造出电视机、汽车和现代化学厂之类的东西。但是不应用法拉第、麦克斯韦、拉瓦锡和牛顿的学说则是不可想象的。仅仅考虑哥白尼学说对技术的影响就社会上引起了强烈的反应。捷克农民在胡斯党人的旗帜下举行起义，这次运动也波及波兰。1517年，在德国，马丁·路德（1483～1546年）反对教会贩卖赎罪符，与罗马教皇公开决裂。1521年，路德又在沃尔姆国会上揭露罗马教廷的罪恶，并提出建立基督教新教的主张。新教的教义得到许多国家的支持，波兰也深受影响。
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