几何的四大支柱（True PDF）
Many people think there is only one “right” way to teach geometry. For two millennia, the “right” way was Euclid’s way, and it is still good in many respects. But in the 1950s the cry “Down with triangles!” was heard in France and new geometry books appeared, packed with linear algebra but with no diagrams. Was this the new “right” way, or was the “right” way something else again, perhaps transformation groups? In this book, I wish to show that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Euclid-style construction and axiomatics seem the best way to start, but linear algebra smooths the later stages by replacing some tortuous arguments by simple calculations. And how can one avoid projective geometry? It not only explains why objects look the way they do; it also explains why geometry is entangled with algebra. Finally, one needs to know that there is not one geometry, but many, and transformation groups are the best way to distinguish between them. Two chapters are devoted to each approach: The ?rst is concrete and introductory, whereas the second is more abstract. Thus, the ?rst chapter on Euclid is about straightedge and compass constructions; the second is about axioms and theorems. The ?rst chapter on linear algebra is about coordinates; the second is about vector spaces and the inner product.
英文|PDF（True）|240页|ISBN:038725303|4.2 MB许多人认为只有一种“正确”的几何教学方法。两千年来，“正确”的方式是欧几里得的方式，在许多方面仍然很好。但在20世纪50年代，法国听到了“打倒三角形！”的呼声，出现了新的几何书籍，里面塞满了线性代数，但没有图表。这是新的“正确”方式，还是“正确”的方式又是其他方式，也许是转型团体？在这本书中，我想说明几何学可以用四种根本不同的方式发展，如果要充分展示这门学科的辉煌，就应该使用所有的方法。欧几里得式构造和公理似乎是最好的开始方式，但线性代数通过简单的计算取代了一些曲折的论证，使后期阶段变得平滑。如何避免射影几何？它不仅解释了为什么物体看起来是这样的；这也解释了为什么几何与代数纠缠在一起。最后，人们需要知道，没有一个几何体，而是有许多几何体，而变换组是区分它们的最佳方式。每种方法都有两章：？第一部分是具体的和介绍性的，而第二部分则更抽象。因此？欧几里得的第一章是关于直尺和圆规的构造；第二个是关于公理和定理。这个？线性代数的第一章是关于坐标的；第二个是关于向量空间和内积。本站不对文件进行储存，仅提供文件链接，请自行下载，本站不对文件内容负责，请自行判断文件是否安全，如发现文件有侵权行为，请联系管理员删除。
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