Mundy Reimer

Book Review - The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds

14 March 2020

Created: 2019-03-14
Updated: 2020-12-31
Topics: Physics, Mathematics, Philosophy
Confidence: Certain
Status: N/A

TL;DR - My review of Jeffrey Week’s book exploring low-dimensional topology, non-Euclidean geometries, and connections with the cosmological shape of the universe. 5/5

Cross-posted from my Goodreads reviews found here.

Excellent book! The Shape of Space is a comfortable and enjoyable read for beginners with little to no experience in alternative geometries and topology. As context, I would consider myself having been trained in the sciences with some graduate-level exposure to applied maths using calculus, linear algebra, statistics, etc (and mostly a recreational / hobby-level enthusiasm for math!), but I would consider that WAY overkill and you definitely don’t need to have experience in any of that to appreciate this book. No further knowledge is needed other than basic algebra 1 and maybe some familiarity with geometry terms one usually is exposed to when younger. My motivation for reading it is just because I really love geometry :)

If I hadn’t misplaced it in the middle of moving to another city, I think it would have taken just 3-4 reading sessions over the course of a few days to consume! I mention this because the book is really conversational in tone and quite easy to work your way through and get lost in (and depending on your habits, maybe even something you can relaxingly read before bed).

The author does a great job jumping back-and-forth between the actions of the characters in the fictional Flatland and how their scenarios apply to the current mathematical topic, and he spends a lot of time teaching you how to see things before moving on, which I personally wish more authors would do. I especially enjoyed the exercises (answers provided in the back) as they reinforced the concepts quite well at an appropriate level without doing what some math textbooks do and solely relying on exercises to teach. It might be trying to some of the more mathematically adept, but this book definitely holds your hand and brings you through concepts gently (if you are impatient, just skip over those parts!)

Content-wise, most of this book is composed of geometry & topology, with about the last ⅙ portion of the book finally getting into physics concepts like homogeneity, isotropism, expansion of the universe, the relationship between density-energy-curvature, cosmic crystallography, microwave background radiation, etc. I particularly enjoyed this format with a greater emphasis on math content, but I just wanted to mention it to those who prefer more physics.

With regards to style, on the spectrum of “physics-hand-wavy” to “definition-lemma-proof math-speak” the author obviously leans towards the former (minus much of hand-wavy if you do the exercises). This book contains LOTS of pictures, which definitely aided my intuition (thank you Professor Weeks!!!), while also not shying away from equations. I heavily advise going through this book like a fun adventure game and doing the exercises yourself to hone your intuition.

I always read books like these with a pencil in hand and “converse” with the author by writing in the margins of the book my comments, questions to look up, speculative thoughts, etc, which I would also heavily recommend to others. After some time I noticed that I became rather comfortable adding projective planes to Klein bottles via connected sums, multiplying spheres by circles, giving tori (donut surfaces) with n>1 holes a hyperbolic geometry, applying the Gauss-Bonnet formula to find the curvature or areas of polygons on curved surfaces, etc, which I think are all pretty darn neat things to be able to do! Also, I think learning that “research by Bill Thurston suggests that three-dimensional hyperbolic geometry is by far the most common geometry for three-manifolds, just as two-dimensional hyperbolic geometry is the most common geometry for surfaces” (p. 249) was one of the best insights that have influenced my thinking so far, and will quite possibly play an important role going forward in my own biological complexity-theoretic interests.

Overall, a really fun book! I really wish this author wrote more :)

1) As a side note, besides actively engaging with the book with a pencil in hand and writing up this review, I also took the time to record some flashcards of the material I learned in this book. Over the years I’ve found that I actually forget quite a bit of material(!) and that doing the aforementioned while additionally using spaced-repetition systems all help curb this. Feel free to check out those flashcards here.

2) For a summary of Thurston’s evidence, this author recommends the article Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Bulletin of AMS 6(1982), pp. 357-381). An interesting corollary of Thurston’s ideas is that a randomly chosen three-manifold is unlikely to be a connected sum (see p. 255 in book).

3) Another notable review by the Mathematical Association of America (MAA) is given here.

Cover art created by Brian Weinstein at Fouriest Series.