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The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds
324 pages, Hardcover
First published August 8, 1985
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January 20, 2022
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 :)
---
- 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: https://www.brainscape.com/p/212G3-LH...
- 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).
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 :)
---
- 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: https://www.brainscape.com/p/212G3-LH...
- 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).
October 22, 2018
It is amazing. Starts with a description of classic "mathematical" novel- Flatland, where we get to feel superior by explaining ordinary objects, like a sphere, to a 2D being; Then it takes us humblingly through dimensions we ourselves don't understand, whilst also showing us, how best to try to comprehend them. It is filled with interesting topological exercises( with solutions).
I would highly recommend it.
I would highly recommend it.
January 24, 2017
The author presents two most popular methods to determine the shape of our universe. Prior to explaining the principles of those methods, there is an extensive introduction to topology of space, and the possible theoretical shapes of the universe.
My Notes:
As of 2002, two research projects are underway to measure the shape of space:
• The method of Cosmic Crystallography looks for patterns in the arrangement of the galaxies.
if the Universe is finite and small enough, we should be able to see "all around" it because the photons might have crossed it once or more times. In such a case, any observer might recognize multiple images of the same light source, although distributed in different directions of the sky and at various redshifts. The main limitation of cosmic crystallography is that the presently available catalogues of observed sources at high redshift are not complete enough to perform convincing tests for topology. But the large and deep surveys (up to redshift z=6), such as the LSST (Large Synoptic Survey Telescope) planned in the next decade, should make such methods applicable.
• The Circles in the Sky method uses the 2D cosmic microwave background (CMB) maps. The last scattering surface from which the CMB is released represents the most distant source of photons in the Universe, and hence the largest scales with which we can probe the topology of the universe. One of the methods is the circles-in-the-sky test. It uses pairs of circles with the same temperature fluctuation pattern. So far, no positive results have been found.
An example of three-dimensional manifold is three-dimensional torus (also called three-torus) is a cube (room), where ceiling is glued to the floor, and front wall to the back wall, whilst the left wall is glued to the right one. In this three-torus if we walk towards the back wall and step into it, we will re-enter our room by the front wall, and so on.
Topology vs. Geometry: the aspect of a surface’s nature that is unaffected by deformation is called the topology of the surface. For example, an eggshell and a ping-pong ball have the same topology. A surface’s geometry consists of those properties that do change when the surface is deformed. An example here is curvature of surface.
Intrinsic vs. Extrinsic Properties: two surfaces have the same intrinsic topology, if the creatures living on the surface cannot (topologically) tell one from the other. Two surfaces have the same extrinsic topology if one can be deformed within three-dimensional space to look like the other. For example, a rubber band and a twisted rubber band (Mobius strip) have the same intrinsic properties. Another example is a sheet of paper and its bent version to make up a half-cylinder – a creature living on the sheet of pare could not detect whether the paper was bent or not
Local vs. Global Properties: local properties are those observables within a small region of the manifold whereas global properties require consideration of the manifold as a whole.
Homogeneous manifold is one whose local geometry is the same at all points. A sphere is a homogenous surface; a doughnut is non-homogenous. A flat torus (a piece of paper that enables to get from the top end to the bottom end, and from left to right, etc.) is homogenous.
A Klein bottle is made up of a square, where bottom left is joined to top right corner, bottom left side with the top right side, etc.
Manifolds that bring traveller back mirror-reversed are called nonoreintable manifolds. The sphere and torus are orientable surfaces. A Klein bottle is a nonorientable surface. A nonorientable three-manifold is different from its orientable version by gluing the front wall to the back wall with side-to-side flip – similar to the Klein bottle.
Spherical triangle – is made up of three great circles. Each great circle divides the sphere in two equal hemispheres. The spherical geometry belongs to elliptic geometry and has positive curvature.
There is an example of how to calculate the area of spherical triangle.
The hyperbolic geometry has negative curvature. An example here is a hyperbolic plane.
My Notes:
As of 2002, two research projects are underway to measure the shape of space:
• The method of Cosmic Crystallography looks for patterns in the arrangement of the galaxies.
if the Universe is finite and small enough, we should be able to see "all around" it because the photons might have crossed it once or more times. In such a case, any observer might recognize multiple images of the same light source, although distributed in different directions of the sky and at various redshifts. The main limitation of cosmic crystallography is that the presently available catalogues of observed sources at high redshift are not complete enough to perform convincing tests for topology. But the large and deep surveys (up to redshift z=6), such as the LSST (Large Synoptic Survey Telescope) planned in the next decade, should make such methods applicable.
• The Circles in the Sky method uses the 2D cosmic microwave background (CMB) maps. The last scattering surface from which the CMB is released represents the most distant source of photons in the Universe, and hence the largest scales with which we can probe the topology of the universe. One of the methods is the circles-in-the-sky test. It uses pairs of circles with the same temperature fluctuation pattern. So far, no positive results have been found.
An example of three-dimensional manifold is three-dimensional torus (also called three-torus) is a cube (room), where ceiling is glued to the floor, and front wall to the back wall, whilst the left wall is glued to the right one. In this three-torus if we walk towards the back wall and step into it, we will re-enter our room by the front wall, and so on.
Topology vs. Geometry: the aspect of a surface’s nature that is unaffected by deformation is called the topology of the surface. For example, an eggshell and a ping-pong ball have the same topology. A surface’s geometry consists of those properties that do change when the surface is deformed. An example here is curvature of surface.
Intrinsic vs. Extrinsic Properties: two surfaces have the same intrinsic topology, if the creatures living on the surface cannot (topologically) tell one from the other. Two surfaces have the same extrinsic topology if one can be deformed within three-dimensional space to look like the other. For example, a rubber band and a twisted rubber band (Mobius strip) have the same intrinsic properties. Another example is a sheet of paper and its bent version to make up a half-cylinder – a creature living on the sheet of pare could not detect whether the paper was bent or not
Local vs. Global Properties: local properties are those observables within a small region of the manifold whereas global properties require consideration of the manifold as a whole.
Homogeneous manifold is one whose local geometry is the same at all points. A sphere is a homogenous surface; a doughnut is non-homogenous. A flat torus (a piece of paper that enables to get from the top end to the bottom end, and from left to right, etc.) is homogenous.
A Klein bottle is made up of a square, where bottom left is joined to top right corner, bottom left side with the top right side, etc.
Manifolds that bring traveller back mirror-reversed are called nonoreintable manifolds. The sphere and torus are orientable surfaces. A Klein bottle is a nonorientable surface. A nonorientable three-manifold is different from its orientable version by gluing the front wall to the back wall with side-to-side flip – similar to the Klein bottle.
Spherical triangle – is made up of three great circles. Each great circle divides the sphere in two equal hemispheres. The spherical geometry belongs to elliptic geometry and has positive curvature.
There is an example of how to calculate the area of spherical triangle.
The hyperbolic geometry has negative curvature. An example here is a hyperbolic plane.
September 7, 2015
I found this (text)book after a traveling down a rabbit hole that begin with Jorge Borges's The Library of Babel and went through a great book titled The Unimaginable Mathematics of Borges Library of Babel. The chapter on topology caught me... and pointed me here.
A fun book, that requires some work. It is in fact a textbook but it doesn't entirely read like one. It's a firm but gentle introduction to the exciting topic of Topology. I'm probably going to continue on and try and find some related Coursers courses.
Great topic covered by great book.
A fun book, that requires some work. It is in fact a textbook but it doesn't entirely read like one. It's a firm but gentle introduction to the exciting topic of Topology. I'm probably going to continue on and try and find some related Coursers courses.
Great topic covered by great book.
November 27, 2016
This is a semi-expository book about the geometry and topology of surfaces and 3-manifolds with some cosmological flavor.
July 26, 2015
An explanation of how things can be twisted in space
Any author attempting to explain and visualize dimensions higher than three and/or the elliptic and hyperbolic geometries is engaged in a significant undertaking. In this book, Weeks does succeed in doing both but the reader is presented with a difficult task.
With 141 exercises and plenty of illustrations packed into 324 pages, it is short on explanation and the reader is forced to learn by problem solving. This is not to say that the exercises are poorly developed. On the contrary, they and the illustrations are very well done. However, doing an exercise after every few paragraphs does make the book a slow read, and in many cases it is necessary to understand a problem before the next material can be comprehended. Fortunately, complete solutions to all problems are given at the end of the book, but even so, a great deal of thought must be given to some of them before they are understood. As the book progressed, I found myself reading only fifteen to twenty-five pages on any given day. This necessitated a great deal of back-pedaling to previous illustrations and exercises, but it was the limit that I seemed able to comprehend at any given setting.
Beginning with Flatland (by A Square-actually Edwin A. Abbott) and going through the creation of manifolds, the presentation of the basic concepts, like all of the text, is very well written. It is just unfortunate that there is not more of it. For example, in Chapter 9 (concerning spheres), there are seven exercises and five and one-half pages of illustrations packed in twelve pages. Chapter 17 (describing bundles), has thirteen problems and seven and one-half pages of diagrams in a total of fifteen pages. Illustrations are valuable, but in this case they describe abstract phenomena not easily followed, and more words than usual are needed to explain precisely what is occurring.
And so, in conclusion, this book is highly recommended for those who wish to learn about the properties of manifolds and surfaces and are highly motivated to do so. But lacking that, the chances are very good that you will not make it beyond the midpoint.
Published in Journal of Recreational Mathematics, reprinted with permission. This review also appears on Amazon.
Any author attempting to explain and visualize dimensions higher than three and/or the elliptic and hyperbolic geometries is engaged in a significant undertaking. In this book, Weeks does succeed in doing both but the reader is presented with a difficult task.
With 141 exercises and plenty of illustrations packed into 324 pages, it is short on explanation and the reader is forced to learn by problem solving. This is not to say that the exercises are poorly developed. On the contrary, they and the illustrations are very well done. However, doing an exercise after every few paragraphs does make the book a slow read, and in many cases it is necessary to understand a problem before the next material can be comprehended. Fortunately, complete solutions to all problems are given at the end of the book, but even so, a great deal of thought must be given to some of them before they are understood. As the book progressed, I found myself reading only fifteen to twenty-five pages on any given day. This necessitated a great deal of back-pedaling to previous illustrations and exercises, but it was the limit that I seemed able to comprehend at any given setting.
Beginning with Flatland (by A Square-actually Edwin A. Abbott) and going through the creation of manifolds, the presentation of the basic concepts, like all of the text, is very well written. It is just unfortunate that there is not more of it. For example, in Chapter 9 (concerning spheres), there are seven exercises and five and one-half pages of illustrations packed in twelve pages. Chapter 17 (describing bundles), has thirteen problems and seven and one-half pages of diagrams in a total of fifteen pages. Illustrations are valuable, but in this case they describe abstract phenomena not easily followed, and more words than usual are needed to explain precisely what is occurring.
And so, in conclusion, this book is highly recommended for those who wish to learn about the properties of manifolds and surfaces and are highly motivated to do so. But lacking that, the chances are very good that you will not make it beyond the midpoint.
Published in Journal of Recreational Mathematics, reprinted with permission. This review also appears on Amazon.
January 12, 2012
very intuitive explanations of topological topics, though definitely limited in depth due to difficulty level of the subject matter. For those very interested in topology, this will give VERY GOOD rough idea of what it is all about and what it takes to at least grasp topology at its basics. Some concepts, though without esoteric notations, are still accurately explained.
April 18, 2016
This is a nice introduction to 3-dimensional geometry aimed at people who don't know the definition of a topological space, with some fun references to Flatland, lots of good exercises that ask you to imagine and draw various interesting spaces, and a non-technical discussion of how this material applies to cosmology. I had fun.
March 17, 2009
This is a great introduction to topology and non-Euclidean geometry. It provides the reader a different perspective on what shape is--or what it can be.
January 2, 2018
One of the few books in the library about topology that isn’t just flatland, or abstract math. Want to know what periodic boundary conditions look like? Fun! Good introductory text, or a relaxing read if you are already familiar with the subject matter. I read this textbook for entertainment and enjoyed it for that purpose.
June 23, 2018
I liked this, but wow, it is not for everyone. How much you appreciate this book will depend on how much you enjoy math for its own sake. I loved the early parts, but felt a large portion of the middle was too abstract for my taste (my aim was to get ideas for writing fictional hyperdimensional spaces), but I enjoyed the conclusion about the universe again. Your mileage will definitely vary.
February 28, 2019
This was great. To those of you, like me, with no real knowledge of topology, it will transform your understanding of what “space” is. It is an induction into the possible weirdness of reality that I found enchanting. Some of the exercises were a bit too hard for me, or at least overcame my patience. :( Someday maybe I’ll have a better maths brain.
May 7, 2017
I've been looking for a book like this for years now. The author explains surfaces and 3-manifolds in the most visual and intuitive way possible. If you think you like topology but never really read about it (as was my case), this is your book.
August 8, 2021
An informative read that provides an introduction to topology for people of all mathematical backgrounds. You do not need to have studied any advanced math to understand its contents. I enjoyed reading about topology along with its applications to cosmology and the universe!
November 11, 2020
Inspiring but heavily lacking in definitions. I got lost at chapter 5.
August 7, 2011
This was a very interesting book. It's a textbook, but it's not written like most textbooks. You don't have to understand everything to find it enjoyable - but basically it's an "exploration" (courtesy, Jeff) of topology, and dimensions above the third. If you like reading about physics, and Einstein, and string theory, then chances are you'll like this book. There are many references to another great book called Flat Land, which is a novel (not a textbook) so I would recommend you read that first and if you like that then you will like this book as well.
October 5, 2009
Jeffrey Weeks is the kind of writer who can introduce complex ideas in an engaging way. A lay person can follow his explanations and work through the problems without much outside help. The best part is that most problems have the solutions in the back of the book. This helps clarify the ideas presented in the text. The only reason I did not finish this book is because it was due back at the library. I'll be checking it out again!
June 7, 2014
This book does a great job of explaining geometry, topology and physics at an accessible level without dumbing it down too much or focusing on irrelevant conceptually easy topics; when I took a differential geometry class in college there were a few sections I already knew because of this book. But be prepared to do or at least really think about the problems suggested in the book, because you get a lot more out of it that way.
May 18, 2013
A remarkably concise explanation of somewhat difficult to master topics. Weeks uses Flatland as a metaphor to explain to his readers multiple ways that space in the fourth dimension (three dimensional manifolds) could be constructed, and takes care to pepper the pages with concrete diagrams to make it simple. Some topics could be explained in a bit more detail, but there are plenty of exercises to make sure that you're following along and understanding the material presented.
January 4, 2016
This book is full of joy but it left me wanting a more rigorous exploration of the ideas of this topic. It's also feels quite dated in ways that even older texts in the discipline of Topology don't. For that reason I will get Munkres textbook and do the applications in it. This is a very good text, and is even approachable to high or junior high students if they are dedicated enough but it is an introductory text.
Displaying 1 - 20 of 20 reviews