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书名 对称(Springer大学数学图书影印版)
分类 科学技术-自然科学-物理
作者 (美)约翰逊
出版社 清华大学出版社
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简介
编辑推荐

本书以通俗易懂的方式讲述几何与群的本质,以及两者问的联系(即对称),并且自然地延伸到一些高级的观点和材料(如有限和仿射Coxeter群,这是李群李代数以及Kac—Moody代数的基础;球面的分割,这是球面几何的内容;上半平面被群SL2(z)的作用,这是双曲几何与自守函数的基础)。阅读本书所需的几何与群的知识在书中均有通俗易懂的介绍(附有大量几何直观图形)。

本书是一本优秀的数学教材,适用于数学系本科生和其他专业对数学有兴趣的本科生用作数学参考书或课外读物。

目录

1. Metric Spaces and their Groups

1.1 Metric Spaces

1.2 Isometries

1.3 Isometries of the Real Line

1.4 Matters Arising

1.5 Symmetry Groups

2. Isometries of the Plane

2.1 Congruent Triangles

2.2 Isometries of Different Types

2.3 The Normal Form Theorem

2.4 Conjugation of Isometries

3. Some Basic Group Theory

3.1 Groups

3.2 Subgroups

3.3 Factor Groups

3.4 Semidirect Products

4. Products of Reflections

4.1 The Product of Two Reflections

4.2 Three Reflections

4.3 Four or More

5. Generators and Relations

5.1 Examples

5.2 Semidirect Products Again

5.3 Change of Presentation

5.4 Triangle Groups

5.5 Abelian Groups

6. Discrete Subgroups of the Euclidean Group

6.1 Leonardo's Theorem

6.2 A Trichotomy

6.3 Friezes and Their Groups

6.4 The Classification

7. Plane Crystallographic Groups: OP Case

7.1 The Crystallographic Restriction

7.2 The Parameter n

7.3 The Choice of b

7.4 Conclusion

8. Plane Crystallographic Groups: OR Case

8.1 A Useful Dichotomy

8.2 The Case n = 1

8.3 The Case n = 2

8.4 The Case n = 4

8.5 The Case n = 3

8.6 The Case n - 6

9. Tessellations of the Plane

9.1 Regular Tessellations

9.2 Descendants of (4, 4)

9.3 Bricks

9.4 Split Bricks

9.5 Descendants of (3, 6)

10. Tessellations of the Sphere

10.1 Spherical Geometry

10.2 The Spherical Excess

10.3 Tessellations of the Sphere

10.4 The Platonic Solids

10.5 Symmetry Groups

11. Triangle Groups

11.1 The Euclidean Case

11.2 The Elliptic Case

11.3 The Hyperbolic Case

11.4 Coxeter Groups

12. Regular Polytopes

12.1 The Standard Examples

12.2 The Exceptional Types in Dimension Four

12.3 Three Concepts and a Theorem

12.4 Schlafli's Theorem

Solutions

Guide to the Literature

Bibliography

Index of Notation

Index

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