Geometry and the Imagination 大卫·希尔伯特：几何与想象（19世纪末和20世纪前期极具影响力的数学家之一）

基本信息

Format: Paperback| 357 pages  

Dimensions: 152.4x 228.6x 19.05mm| 333g  

Publication date: 17 Mar 2021 

Publisher: American Mathematical Society 

ISBN: 9781470463021

页面参数仅供参考，具体以实物为准

内容简介

这本了不起的书作为数学论述的真正杰作而经久不衰。很少有数学书在半个多世纪过去后仍被广泛阅读，并继续提供如此多的内容！这本书充满了数学思想，并始终以清晰、优雅的方式进行解释。书中洋溢着数学思想，这些思想总是被解释得很清楚，更重要的是，具有深刻的洞察力。无论是对初学者还是有经验的数学家来说，它都是一种阅读的乐趣。

"希尔伯特和科恩·沃森 "充满了有趣的事实，其中许多你希望你以前就知道。也很可能你以前就听说过这些事实，但肯定想知道在哪里可以找到它们。这本书从简单的曲线和曲面的例子开始，包括某些四边形和其他曲面的螺纹构造。有规律的点系统一章引出了晶体学群和R3中的有规律多面体。在这一章中，他们还讨论了平面格子。通过考虑单位格子，并在必要时加入少量的数论，他们毫不费力地推导出莱布尼茨的数列：? /4=1?1/3+1/5?1/7+?.... 在关于三维和更多维度的格子的章节中，作者考虑了球体包装问题，包括著名的开普勒问题。

极其引人注目的一章是 "投影构型"。在一个简短的介绍性章节中，希尔伯特和科恩-沃森给出了也许是Z简洁明了的描述，说明为什么一个普通的几何学家会关心投影几何，以及为什么这样一个表面上平淡无奇的设置确实具有丰富的结构和思想。在这里，我们从一个不同的角度再次看到了正多面体。本章的高潮之一是对Schlafli双六的讨论，这导致了对一般光滑立方体表面的27条线的描述。如同全书一样，这一章中华丽的图画对读者的帮助是不可估量的。

This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer-even after more than half a century has passed! The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians.

"Hilbert and Cohn-Vossen" is full of interesting facts, many of which you wish you had known before. It's also likely that you have heard those facts before, but surely wondered where they could be found. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in R3. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: ?/4=1?1/3+1/5?1/7+?.... In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem.

One of the most remarkable chapters is "Projective Configurations". In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader.

作者简介

大卫·希尔伯特（1862年1月23日-1943年2月14日）是一位德国数学家，是19世纪和20世纪初极具影响力的数学家之一。希尔伯特在许多领域发现并发展了广泛的基本思想，包括不变量理论、变化微积分、换元代数、代数数论、几何学基础、算子谱理论及其在积分方程中的应用、数学物理学以及数学基础（特别是证明理论）。

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory).