Symmetries

Symmetries by D.L. Johnson.

" ... many eminent scholars, endowed with great geometric talent, make a point of never disclosing the simple and direct ideas that guided them, subordinating their elegant results to abstract general theories which often have no application outside the particular case in question. Geometry was...

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Bibliographic Details
Main Author: Johnson, D.L (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: London : Springer London : Imprint: Springer, 2001.
Edition:1st ed. 2001.
Series:Springer Undergraduate Mathematics Series,
Springer eBook Collection.
Subjects:
Geometry.
Group theory.
Algebra.
Ordered algebraic structures.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 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 Schläfli’s Theorem
  • Solutions
  • Guide to the Literature
  • Index of Notation.

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