Thinking Geometrically: A Survey of Geometries

Why Geometry Still Matters

Geometry is one of the oldest branches of mathematics, but it is far from a closed subject. In Thinking Geometrically: A Survey of Geometries, Thomas Q. Sibley invites readers to see the discipline as an evolving landscape of ideas—from the familiar Euclidean plane to the unexpected worlds of hyperbolic and elliptic spaces.

A Wide-Ranging Mathematical Survey

The book moves systematically through six major areas: Euclidean geometry, axiomatic systems, analytic geometry, non-Euclidean geometries, transformation geometry, and symmetry. Each chapter builds on earlier ideas, offering historical context, formal development, and plenty of exercises.

Designed for Active Learners

Sibley’s approach encourages readers to work with definitions, diagrams, and proofs rather than simply memorize results. Exercises appear throughout, and end-of-chapter projects push deeper into topics such as computer-aided design, regular polytopes, and the geometry of the sphere.

From Classical Foundations to Modern Connections

While rooted in undergraduate mathematics, the text also highlights connections that extend beyond pure geometry—such as transformations used in CAD and iterated function systems. This makes the material relevant not only to mathematics students but also to anyone interested in the visual and computational applications of geometry.

What You Can Expect

• A careful introduction to axiomatic reasoning and mathematical models
• Clear treatment of hyperbolic and spherical geometries
• Hands-on exercises and suggested readings for further study
• A bridge from classical Greek geometry to modern transformational methods

For students, educators, and self-directed learners, this MAA textbook offers a coherent and engaging way to explore the many meanings of “geometry.”