From an undergraduate perspective, coming from the rigid proofs and concrete constructions of middle- or high-school courses, the broad discipline of geometry can be at once intimately familiar and menacingly exotic. For most of its history, and perhaps for many of the same reasons, geometers struggled to come to terms with the unsolved problems, unstated assumptions, and untapped generalizability contained in the "bible of mathematics", Euclid's Elements
In their recent text, Geometry: The Line and the Circle
(MAA Press, 2018), Maureen T. Carroll
and Elyn Rykken
have produced a unified survey of Euclidean and many significant non-Euclidean geometries, one that draws from the patterns of historical development to immerse students into progressively new territory. Their book is organized around the Elements but soon (and often) detours into spherical, finite, and other geometries that bring the limitations of the classic text—and the contributions of subsequent geometers—to the fore. Throughout, they examine the shifting roles and behaviors of two fundamental geometric concepts, the line and the circle—a narrative hook that might deserve more play in mathematics texts! In addition to their historical vignettes, Carroll and Rykken include rich selections of exercises and incorporate a variety of tactile and online tools, and their treatment is held together in an accessible and absorbing writing style. The book is tailored to an upper-level undergraduate course but could also support a history of mathematics or introduction to proofs course.
Suggested companion works:
Edwin A. Abbott, Flatland: A Romance of Many Dimensions
(+ sequels & film adaptations)
Norton Juster, The Dot and the Line
(+ film adaptation)
Cory Brunson (he/him) is a Postdoctoral Fellow at the Center for Quantitative Medicine at UConn Health.