References and Further Reading

[Abbott-1884]: Edwin A. Abbott. Flatland: A Romance of Many Dimensions. 1884. Reprint. New York: Barnes & Noble, 1983. [CADE-93]: Joel Castellanos, Joe Dan Austin, Ervan Darnell, and Maria Estrada. "An Empirical Exploration of the Poincaré Model for Hyperbolic Geometry", Mathematics and Computer Education, pp. 51-68, (Winter 1993), Volume 27, number 1.

[Health-56]: Sir T. Health. The Elements (Euclid), Dover, New York, NY (1956).

This book is a translation from Euclid's original text written in Greek over 2000 years ago.  It is remarkably similar to modern high school geometry text books.  The editor has added extensive, and useful commentary.

[Kedder-85]: R. M. Kedder. "How High-schooler Discovered New Math Theorem", The Christian Science Monitor, pp. 19-20, (April, 1985).

[Moise-74]: E.E. Moise. Elementary Geometry from an Advanced Standpoint, Addison-Wesley, Reading, MA (1974).

[NCTM-89]: The National Council of Teachers of Mathematics, Curriculum and Evaluation Standards, pp. 157, (1989).

[Rucker-84]: Rudy Rucker. The Fourth Dimension: Toward a Geometry of Higher Reality. Boston: Houghton Mifflin Company, 1984.

[Tello-92]: H. G. Tello, and Y. O. Yang. Formative Evaluation of NonEuclid, Unpublished manuscript, The Ohio State University, Educational Theory and Practice Department (1992).

[Polking-98]: John Polking. The Geometry of the Sphere, Web site of the Department of Mathematics of Rice University,

[Ramsay&Richtmyer-95]: Arlan Ramsay, Robert D. Richtmyer. Introduction To Hyperbolic Geometry,  Springer-Verlag, New York, Berlin, Heidelberg (1995).

This text for advanced undergraduates emphasizes the logical connections of the subject.  The derivations of formulas from the axioms do not make use of models of the hyperbolic plane until the axioms are shown to be categorical; the differential geometry of surfaces is developed far enough to establish its connections to the hyperbolic plane; and the axioms and proofs use the properties of the real number system.  Topics include: Axioms for Plane Geometry, some Neutral Theorems of Plane Geometry, Qualitative Description of the Hyperbolic Plane, Differential Geometry of Surfaces, Quantitative Considerations, Consistency and Categoricalness of the Hyperbolic Axioms, the Upper Half-Plane Model, Matrix Representation of the Isometry Group, Tilings, Differential and Hyperbolic Geometry in More Dimensions, and Applications to Special Relativity.  Elementary techniques from complex analysis, matrix theory, and group theory are prerequisite to this text.

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