Navigating Curved Spaces

Hands-On Explorations in Hyperbolic and Elliptic Geometry

by Michael E. McDaniel, PhD

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View First 25 Pages: (free download)

Synopsis

This book invites readers into the strange, beautiful worlds of hyperbolic and elliptic geometry, where triangles don't behave as Euclid promised and space itself bends in unexpected ways. Using compass and straightedge, graph paper, Temari spheres, viral capsids, and even collage, the book puts non-Euclidean geometry directly into the reader's hands--something to build, test, and explore rather than simply read about.

Drawing on twenty-five years of teaching college geometry, Dr. McDaniel writes in a chatty, personal style built around examples readers can make or hold themselves. The book moves from short, playful explorations toward a longer mathematical saga--including the six-year research story behind the elliptic Wallace-Simson line, which began with a high school student pulling a key idea out of a wastebasket. Along the way, readers are invited to pause, weigh the facts, and predict the next big conjecture for themselves--a chance to stand at the edge of the unknown.

On the practical side, construction tricks for orthogonal circles and point inversions generate hyperbolic and elliptic objects with genuinely oddball properties: a hyperbolic rectangle that splits into three equal-area triangles with no Euclidean counterpart, a viral capsid modeled through a pentagon construction traced to Ptolemy, and an elliptic construction of Desargues's ten-point geometry that gives every point and line a double meaning.

Written for college geometry students, engineers, artists, philosophers, biochemists, and anyone mathematically curious, this is a book for readers who want to see the world with both Euclidean and non-Euclidean eyes--on the page and in real life. The major results here have been refereed and published; some of the construction tricks have never appeared anywhere else.

About the Author

Dr. McDaniel earned his Ph.D. in 1997 at The George Washington University and has taught college geometry at Aquinas College since 1998. Over the years, he has become a guide to both hyperbolic and elliptic geometry, encouraging students to explore mathematical landscapes both familiar and unexpected. Support from the Mohler-Thompson Research Fund allowed him to lead students beyond introductory exploration into full-time work in curved spaces. Fifteen of his students have published refereed mathematics articles, forming the foundation for his two books. He retired in 2025.