MA532 Foundations of Geometry Presentation 19 Friday, July 17, 2015 Reminder: Major Exercise and Projects I is due today (Friday, July 17). Presentation 19 Objectives: After completing the Presentation 19 reading and associated assignment, you will be able to state the properties of several models of hyperbolic geometry that were developed to \"visualize\" hyperbolic geometry and to discuss the philosophical foundations and profound implications of the discovery of non-Euclidean geometry. Textbook Reading Assignment: A. B. C. D. Chapter 7: Chapter 7: Chapter 7: Chapter 8: pp. 289 - 311 (stop at \"A Model of the Hyperbolic Plane from Physics\") pp. 319 - 321 (Read the Definition and Important Remark) p. 346 (Conclusion) pp. 371 - 389 Today's presentation is largely a reading assignment about some of the models that were developed to visualize hyperbolic geometry and the profound philosophical implications of the discovery of non-Euclidean geometry. The Klein model actually appeared in one of our earlier assignments (Assignment 3, Problem #9 part c, p. 92 of the text). A fascinating aspect of the Poincar models is the definition of Poincar length as AB log AP BQ BP AQ Note that AP, BQ, BP, and AQ denote the Euclidean length of the Euclidean segments, which it is possible to measure since the circle in the Poincar model is in the Euclidean coordinate plane. In the text on pages 319 and 320, the definition uses the cross ratio, but I have incorporated the cross ratio into the equation above. Be sure to refer to Figure 7.27 on p. 321 for the corresponding diagram. Assignment 19: Due by 11:55 pm on Tuesday, July 28, 2015 A. In Figure 7.27, fix points A, P and Q and imagine point B moving continuously along the P-line from A to P. Calculate AA . As B approaches P, what is the value of AB approaching? B. Chapter 9 Review Exercise (p. 389): For those statements that are false, make the appropriate correction. 1 of 1 \f