Michael McDaniel

This is the book referenced in the International Journal of Geometry in 2024, and Mathematics Magazine and the Rose-Hulman Undergraduate Math Journal in 2023. This is the only book with proofs for elliptic compass and straightedge AND squaring the circle constructions in non-Euclidean geometries. This book would be fun for the arm-chair geometer who wants to build stuff in bent space. Try that video with the circles if you want to see how to play with this book.

Now for the biography.

Raised in Muskegon Heights, Michigan, in the 1970's, I fell in love with geometry when I first taught math at Sts. Peter and Paul School on St. Thomas in the US Virgin Islands. Before going for my Ph.D., I taught in New Jersey at Mount Saint Mary Academy for five more magic years.

The George Washington University was a great place for graduate work. I was lucky to score a tenure-track job one year after graduation and I've been at Aquinas College ever since. Teaching college geometry rekindled my love for compass and straightedge. Exploring the non-Euclidean disk models quickly exceeded the chapters I could find in textbooks.

Every year, my geometry classes went deeper into constructing non-Euclidean objects. Once the Mohler-Thompson Summer Research Program got going, my students and I started finding publication-worthy results, enough for a book in 2014. Those are the students pictured in the photo, holding copies of the book with their published work in it.

The deep magic of working in Euclidean and non-Euclidean geometry at the same time continues giving new results, into 2019. We have constructed a hyperbolic rectangle which can be cut into three triangles of equal area (impossible in Euclidean) and we have solved a problem from 1797 in elliptic geometry.

The book, Geometry By Construction, opens the door to this magic, the product of half my life in math. This is the only book with instructions for squaring the circle in both hyperbolic and elliptic geometries; my students and I felt as if we were contesting with a living opponent as we probed and scraped for results and proofs. Such engagement can be yours for the price of the book, drawing tools and a notebook.

The first figure is from my summer 2020 research on elliptic triangles and viruses. The second figure is from summer 2019, on Wallace-Simson lines. Our articles use figures which have elliptic triangles and their polar triangles with special lines constructed correctly. All our constructions use the ideas in this book.

Are you a teacher or professor? Steal this idea. I give the students a construction to perform. I give them some support to get about half-way and then we look to see if they are going in a correct direction. Then they have to finish and write up their justification. The second photo is an example: construct a hyperbolic triangle with two 60 degree angles. (The shaded blue triangle has been constructed correctly. The justification takes three paragraphs.)

Now that people are buying and using the book, here are the typo-graphical errors. The hassles are in the figures. Page 59, the unlabeled point is point B. Page 60, the ray OB in the first paragraph is meant to be the ray OE. Page 79, not triangle ABC but EDB in circle F is the triply asymptotic one. Page 81, the top right hexagon has a little circle whose diameter is OC and F is the marked intersection of the two circles. Page 100 does not have the point Y and Y' drawn nor marked; undrawn Euclidean line OB contains them both. If you continue the circle through AB just a bit on the left, the Euclidean line OB will hit at Y'. Page 116, center point O is unmarked because the drawing was so crowded. Page 125 problem 32 should have OB equal to alpha.