John Stillwell

This book is unique in that it looks at geometry from 4 different viewpoints - Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants Approach makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic Abundantly supplemented with figures and exercises

The Four Pillars of Geometry

Mundy Otto Reimer

As usual, this is another excellent book by Professor John Stillwell, filling in the rare niche between formal textbooks with little to no narrative exposition and popular math books with little to no rigor. 

No prerequisites are assumed besides basic algebra 2 skills. The level of mathematical maturity expected is that of an advanced high school student to undergraduate. As context, I'm approaching this book as a dedicated recreational mathematical enthusiast / amateur. 

Topic-wise, this author brings you through four different perspectives on geometry, first through the classical Euclidean geometry you learn in school, starting with straightedge + compass constructions and introducing the axiomatic-styled proof approach. The author follows this up with the second perspective taking a linear algebra approach, starting with an explanation centered around coordinates, followed by vector spaces and inner products. Then for those that have only some exposure to applied math or have a quantitative-bent science background, the author introduces two new perspectives on geometry, that of projective geometry (think perspective drawings) and that of transformation groups, symmetry, and basic group theory, all of which lead nicely into a discussion of fascinating non-Euclidean hyperbolic geometry concepts. I really enjoyed this latter half of the book. 

I thought it would be good to mention that this book provides a nice rigorous treatment to alternative perspectives of geometries that complements the superbly excellent and intuition-laden The Shape of Space by Jeffrey Weeks (review here: https://www.goodreads.com/review/show...) as well as the more comfortable historical read from The Poincare Conjecture by Donal O'Shea. I read all of these in parallel and I'd recommend the same for those who want various levels of geometric explanation that mutually reinforce each other.

With respect to the author's writing style, I really enjoyed the conversational exposition and historical details that accompanied each section. Although there were some points in the book that came across as stilted, and the ending was rather terse and abrupt in nature, I don't want this to detract from the excellent effort this author gave in trying to weave mathematical ideas into a more conversationally flowing pedagogical style, as it is rather rare to find authors who do this while not giving up on the rigorous mathematical treatment expected of a great textbook. It's definitely not the dry definition-lemma-proof style that you find in most traditional advanced math topic books, but neither does it discard proofs in their entirety, bringing the reader gently through each argument. The author mentions that this could be used as a textbook and I am sure he uses this in his own class, but I see this as more of filling in that rare niche of "self-study textbooks" that talk you through the ideas side-by-side with the equations. Overall, I think the author struck a nice balance between deductive-style proofs and casually talking to the reader.

My main complaint is that there are no answers in the back for the exercises (Gah, I hate it when authors do this, especially for us self-study people!) I often go through these books with a pencil in hand, working my answers in the margins or giving my thoughts and "conversing" with the author. I would have really appreciated it if I could easily flip to the back of the book and check my answers in real-time on the spot. I think this might have been the result of this book coming out of a series of notes the author wrote and eventually wrapped up into a nice textbook-like package that he could use for his own class (which is somewhat common), so that is understandable, but it is my major reason for subtracting the one star. 

My main take-away that changed how I now think (and for this I can't be more grateful) is the author's explanation of Felix Klein's perspective of "geometry as the study of *invariants* of groups of transformations" (pg. 143). This is probably a rather simple concept for working mathematicians, but this perspective really struck me as significant (especially with the potential physical interpretations and parallels I see with conservation laws and Emily Noether's work) and has fascinated me such that I might take some time to study some Group theory after I satiate some of my geometry addiction. If anybody reading this has some good book recommendations on Klein's Erlangen Program, I'd really appreciate you letting me know! 

(Also, another simple yet insightful perspective that will stick with me is that the group of rotations is also another geometric object!)

Overall, this is a nicely written introductory book for those who would like to self-study geometry from different perspectives. This book makes me really wish I got the chance to take a class with this professor :) 

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As a side note, in case anybody would like to review some of the concepts in this book, I'm trying to get into the habit of using spaced-repetition more since I read so much and tend to forget a lot, so here's a link to some Brainscape flashcards I made out of the important concepts and definitions found in this book: https://www.brainscape.com/p/212G3-LH...

Hope it helps :)

The Four Pillars of Geometry

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