References and Further Reading

Flatland was written in 1884 by a Victorian schoolmaster named Edwin Abbott Abbott. It is a science fiction story about an imaginary two-dimensional world. The book works on three levels. Most obviously, it is a satire on Victorian society. "Irregulars" (cripples) are put to death, women have no rights at all, and when the main character, A Square, tries to teach his fellows about the third dimension, he is imprisoned. On the second level, Flatland is scientific. Flatland's relationship to the 3rd-dimension is analogous to our relationship to the 4th-dimension. Finally, at the deepest level, A Square's trip into higher dimensions is a metaphor for the mystic's experience of higher reality. It is a short book, and a fun read.

[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).

This book gives a clear, yet complete, axiomatic development of the Poincaré Disk Model, and other geometric systems.  The only prerequisite to this text is a solid understanding of typical high school geometry.

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

Rudy Rucker does an excellent job of helping the reader actually visualize the 4th-dimension. It is filled with great, cartoon drawings and thought provoking puzzles. Part III of the book is titled "How To Get There".  Execellent reading for the layperson and mathamatition alike.

[Polking-98]: John Polking. The Geometry of the Sphere, Web site of the Department of Mathematics of Rice University, http://math.rice.edu/~pcmi/sphere/.

[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.