## A Course in Modern GeometriesA Course in Modern Geometries is designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. Chapter 1 presents several finite geometries in an axiomatic framework. Chapter 2 continues the synthetic approach as it introduces Euclid's geometry and ideas of non-Euclidean geometry. In Chapter 3, a new introduction to symmetry and hands-on explorations of isometries precedes the extensive analytic treatment of isometries, similarities and affinities. A new concluding section explores isometries of space. Chapter 4 presents plane projective geometry both synthetically and analytically. The extensive use of matrix representations of groups of transformations in Chapters 3-4 reinforces ideas from linear algebra and serves as excellent preparation for a course in abstract algebra. The new Chapter 5 uses a descriptive and exploratory approach to introduce chaos theory and fractal geometry, stressing the self-similarity of fractals and their generation by transformations from Chapter 3. Each chapter includes a list of suggested resources for applications or related topics in areas such as art and history. The second edition also includes pointers to the web location of author-developed guides for dynamic software explorations of the Poincaré model, isometries, projectivities, conics and fractals. Parallel versions of these explorations are available for "Cabri Geometry" and "Geometer's Sketchpad". Judith N. Cederberg is an associate professor of mathematics at St. Olaf College in Minnesota. |

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### Contents

Axiomatic Systems and Finite Geometries | 1 |

12 Axiomatic Systems | 2 |

13 Finite Projective Planes | 9 |

14 An Application to ErrorCorrecting Codes | 18 |

15 Desargues Configurations | 25 |

16 Suggestions for Further Reading | 30 |

NonEuclidean Geometry | 33 |

22 Euclids Geometry | 34 |

Projective Geometry | 213 |

42 The Axiomatic System and Duality | 214 |

43 Perspective Triangles | 221 |

44 Harmonic Sets | 223 |

45 Perspectivities and Projectivities | 229 |

46 Conies in the Projective Plane | 240 |

47 An Analytic Model for the Projective Plane | 250 |

48 The Analytic Form of Projectivities | 258 |

23 NonEuclidean Geometry | 47 |

24 Hyperbolic GeometrySensed Parallels | 51 |

25 Hyperbolic GeometryAsymptotic Triangles | 61 |

26 Hyperbolic GeometrySaccheri Quadrilaterals | 68 |

27 Hyperbolic GeometryArea of Triangles | 74 |

28 Hyperbolic GeometryUltraparallels | 80 |

29 Elliptic Geometry | 84 |

210 Significance of the Discovery of NonEuclidean Geometries | 93 |

Geometric Transformations of the Euclidean Plane | 99 |

32 Exploring Line and Point Reflections | 103 |

33 Exploring Rotations and Finite Symmetry Groups | 108 |

34 Exploring Translations and Frieze Pattern Symmetries | 116 |

35 An Analytic Model of the Euclidean Plane | 121 |

36 Transformations of the Euclidean Plane | 129 |

37 Isometrics | 136 |

38 Direct Isometries | 144 |

39 Indirect Isometries | 154 |

310 Frieze and Wallpaper Patterns | 165 |

311 Exploring Plane Tilings | 173 |

312 Similarity Transformations | 183 |

313 Affine Transformations | 190 |

314 Exploring 3D Isometries | 198 |

315 Suggestions for Further Reading | 207 |

49 Cross Ratios | 264 |

410 Collineations | 270 |

411 Correlations and Polarities | 283 |

412 Subgeometries of Projective Geometry | 298 |

413 Suggestions for Further Reading | 311 |

Chaos to Symmetry An Introduction to Fractal Geometry | 315 |

51 A Chaotic Background | 316 |

52 Need for a New Geometric Language | 334 |

53 Fractal Dimension | 347 |

54 Iterated Function Systems | 360 |

55 Finally What Is a Fractal? | 377 |

56 Applications of Fractal Geometry | 380 |

57 Suggestions for Further Reading | 382 |

Euclids Definitions Postulates and the First 30 Propositions of Elements Book I | 389 |

Hilberts Axioms for Plane Geometry | 395 |

Birkhoffs Postulates for Euclidean Plane Geometry | 399 |

The SMSG Postulates for Euclidean Geometry | 401 |

Some SMSG Definitions for Euclidean Geometry | 405 |

The ASA Theorem | 409 |

413 | |

427 | |

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