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Sophie Amundsen was on her way home from school. She had walked the first part of the way with Joanna. They had been discussing robots. Joanna thought the hum From time to time there would be a few letters from people called Moller, and quite a few called Knag. But there was nobody in the entire directory called Moller Knag. She examined the mysterious card again. It certainly seemed genuine enough; it had a stamp and a postmark. Why would a father send a birthday card to Sophie"s address when it was quite obviously intended to go somewhere else? What kind of father would cheat his own daughter of a birthday card by purposely sending it astray? How could it be "the easiest way"? And above all, how was she supposed to trace this Hilde person? So now Sophie had another problem to worry about. She tried to get her thoughts in order: This afternoon, in the space of two short hours, she had been presented with three problems. The first problem was who had put the two white envelopes in her mailbox. The second was the difficult questions these letters contained. The third problem was who Hilde Moller Knag could be, and why Sophie had been sent her birthday card. She was sure that the three problems were interconnected in some way. They had to be, because until today she had lived a perfectly ordinary life.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]

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，瓶去探索宇宙的秘密，从而推进了天文学和地理学的发展。1492年，意大利著名的航海家哥伦布发现新大陆，麦哲伦和他的同伴绕地球一周，证明地the girl in the mirror winked with both eyes It was only a quarter past seven. There was no need to hurry home. Sophie"s mother always took it easy on Sundays, so she would probably sleep for another two hours. Should she go a bit farther into the woods and try to find Alberto Knox? And why had the dog snarled at her so viciously? Sophie got up and began to walk down the path Hermes had taken. She had the brown envelope with the pages on Plato in her hand. Wherever the path diverged she took the wider one. Birds were I.D. card. It showed a picture of a girl with fair hair. Under the picture was the girl"s name: Hilde Moller Knag ... Sophie shivered. Again she heard the dog bark. She had to get out, at once! As she hurried past the table she noticed a white envelope between all the books and the pile of paper. It had one word written on it: SOPHIE. Before she had time to realize what she was doing, she grabbed the envelope and stuffed it into the brown envelope with the Plato pages. Then she rushed out of the door and slammed it behind her. The barking was getting closer. But worst of all was that the boat was gone. After a second or two she saw it, adrift halfway across the lake. One of the oars was floating beside it. All because she hadn"t been able to pull it completely up on land. She heard the dog barking quite nearby now and saw movements between the trees on the other side of the lake. Sophie didn"t hesitate any longer. With the big envelope in her hand, she plunged into the bushes behind the cabin. Soon she was having to wade through marshy ground, sinking in several times to well above her ankles. But she had to keep going. She had to get home. Presently she stumbled onto a path. Was it the path she had taken earlier? She stopped to wring out her dress. And then she began to cry. How could she have been so stupid? The worst of all was the boat. She couldn"t forget the sight of the row-boat with the one oar drifting helplessly on the lake. It was all so embarrassing, so shameful. . . The philosophy teacher had probably reached the lake by now. He would need the boat to get home. Sophie felt almost like a criminal. But she hadn"t done it on purpose. The envelope! That was probably even worse. Why had she taken it? Because her name was on it, of course, so in a way it was hers. But even so, she felt like a thief. And what"s more, she had provided the evidence that it was she who had been there. Sophie drew the note out of the envelope. It said: What came first--the chicken or the "idea" chicken ? Are we born with innate "ideas"? What is the difference between a plant, an animal, and a human? Why does it rain? What does it take to live a good life? Sophie couldn"t possibly think about these questions right now, but she assumed they had something to do with the next philosopher. Wasn"t he called Aristotle? When she finally saw the hedge after running so far through the woods it was like swimming ashore after a shipwreck. The hedge looked funny from the other side. She didn"t look at her watch until she had crawled into the den. It was ten-thirty. She put the big envelope into the biscuit tin with the other papers and stuffed the note with the new questions down her tights. Her mother was on the telephone when she came in. When she saw Sophie she hung up quickly. "Where on earth have you been?" "I... went for a walk ... in the woods," she stammered. "So I see." Sophie stood silently, watching the water dripping from her dress. "I called Joanna..." "Joanna?" Her mother brought her some dry clothes. Sophie only just managed to hide the philosopher"s n "What do you do when you"re together, Sophie? Why are you so wet?" Sophie sat staring gravely at the table. But deep down inside she was laughing. Poor Mom, now she had that to worry about. She shook her head again. Then more questions came raining down on her. "Now I want the truth. Were you out all night? Why did you go to bed with your clothes on? Did you sneak out as soon as I had gone to bed? You"re only fourteen, Sophie. I demand to know who you are seeing!" Sophie started to cry. Then she talked. She was still frightened, and when you are frightened you usually talk. She explained that she had woken up very early and had gone for a walk in the woods. She told her mother about the cabin and the boat, and about the mysterious mirror. But she mentioned nothing about the secret correspondence course. Neither did she mention the green wallet. She didn"t quite know why, but she had to keep Hilde for herself. Her mother put her arms around Sophie, and Sophie knew that her mother believed her now. "I don"t have a boyfriend," Sophie sniffed. "It was just something I said because you were so upset about the white rabbit." "And you really went all the way to the major"s cabin ..." said her mother thoughtfully. "The major"s cabin?" Sophie stared at her mother. "The little woodland cabin is called the major"s cabin because some years ago an army major lived there for a time. He was rather eccentric, a little crazy, I think. But never mind that. Since then the cabin has been unoccupied." "But it isn"t! There"s a philosopher living there now." "Oh stop, don"t start fantasizing again!" Sophie stayed in her room, thinking about what had happened. Her head felt like a roaring circus full of lumbering elephants, silly clowns, daring trapeze flyers, and trained monkeys. But one image recurred unceasingly-- a small rowboat with one oar drifting in a lake deep in the woods--and someone needing the boat to get home. She felt sure that the philosophy teacher didn"t wish her any harm, and would certainly forgive her if he knew she had been to his cabin. But she had broken an agreement. That was all the thanks he got for taking on her philosophic education. How could she make up for it? Sophie took out her pink notepaper and began to write: Dear Philosopher, It was me who was in your cabin early Sunday morning. I wanted so much to meet you and discuss some of the philosophic problems. For the moment I am a Plato fan, but I am not so sure he was right about ideas or pattern pictures existing in another reality. Of course they exist in our souls, but I think--for the moment anyway-- that this is a different thing. I have to admit too that I am not altogether convinced of the immortality of the soul. Personally, I have no recollections from my former lives. If you could convince me that my deceased grandmother"s soul is happy in the world of ideas, I would be most grateful. Actually, it was Are we born with innate "ideas"? Most unlikely, thought Sophie. She could hardly imagine a newborn baby being especially well equipped with ideas. One could obviously never be sure, because the fact that the baby had no language did not necessarily mean that it had no ideas in its head. But surely we have to see things in the world before we can know anything about them. And Joanna, of course. And Jeremy, perhaps. But that"s for you to decide. I remember my own fifteenth birthday so clearly, you know. It doesn"t seem all that long ago. I felt I was already quite grown up. Isn"t it odd, Sophie! I don"t feel I have changed at all since then." "You haven"t. Nothing changes. You have just developed, gotten older..." 球是圆形的，使人们开始真正认识地球。 [4] 对他国的影响 在教会严密控制下的中世纪，也发生过轰轰烈烈的宗教革命。因为天主教的很多教义不符合圣经的教诲，而加入了太多教皇的个人意志以及各类神学家的自身成果，所以很多信徒开始质疑天主教的教义和组织，发起回归圣经的行动来。 捷克的爱国主义者、布拉格大学校长扬·胡斯（1369～1415年）在君士坦丁堡的宗教会议上公开谴责德意志封建主与天主教会对捷克的压迫和剥削。他虽然被反动教会处以火刑，但他的革命活动在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。新教的教义得到许家的支持，波兰也深受影