# Geometry by Construction

## Object Creation and Problem-solving in Euclidean and Non-Euclidean Geometries

### by Michael McDaniel

#### Synopsis

College geometry students, professors interested in undergraduate research and secondary geometry teachers will find three rich environments in this textbook. The first chapter contains many of the standards of Euclidean college geometry. The second and third chapters introduce non-Euclidean models where some Euclidean rules hold and others do not. With emphases on constructions and proofs, the reader is encouraged to create the objects under investigation and verify the results with reasoning. Since both models of “bent” spaces exist in Euclidean geometry, the reader gains facility with Euclidean moves through the whole book, even while exploring non-Euclidean spaces. The book itself is meant to be unpacked, expanded and taken further, just like the problems it contains.

*Geometry by Construction* challenges its readers to participate in the creation of mathematics. The questions span the spectrum from easy to newly-published research and so are appropriate for a variety of students and teachers. From differentiation in a high school course through college classes and into summer research, any interested geometer will find compelling material.

Teachers and professors might especially appreciate the way constructions provide open-ended questions which resist internet searches for solutions. College students should find the five refereed results from undergraduates like themselves encouraging. The active reader joins the mathematical tradition of a laboratory being a notebook plus a compass and ruler (or a dynamic geometry program on a computer.) New ideas await exploration and here are examples!

#### About the Author

Michael McDaniel has won teaching awards at his schools in the USVI, New Jersey and Michigan. He earned his Ph.D. in 1997 at the George Washington University and has been at Aquinas College since 1998. During his first sabbatical, he picked up some non-Euclidean construction skills and creating the objects of his college geometry course has become his favorite pastime. He has become a hyperbolic and elliptic tour guide, pushing students to explore landscapes both familiar and unusual. The Mohler-Thompson Research Fund provided the opportunity to take some students beyond tourism into full-time residency in bent spaces.