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Request for Book Recommendations:

Background introduction

Disclaimers: If the tone below is a little arrogant I apologize beforehand, but I'm being very specific here because I want to make sure that I will be learning and accessing the right material, and also that which is more tailored to my interests

So this is the first time I'm using the forum here to ask a question, although I've visited this site a couple of times already. But I do notice similar sounding questions have been asked several times, so I want to be highly specific so as to separate this question from other similar questions, so that it does not get tagged as a duplicate or lacks specificity to the point it is vague. What I want is a personalized recommendation, (not a generalized recommendation), tailored to my level of knowledge and interest.

First off, I am an incoming freshman going to college, and am majoring in physics, however I am interested in eventually double majoring in mathematics as well, and one area of mathematics I would really like to choose as an area of study is geometry.

Specifically, I'm searching for a recommendation in Euclidean geometry/Non-Euclidean Geometry, whether it is a book, a pdf, or a website tutorial. I do not want an book with an axiomatic treatment style for right now. It would be highly helpful if the book were more problem oriented, teaching specific techniques in solving geometry problems That being said, the book should not just be a collection of problems without any exposition or explanations. Hopefully the book contains a complete treatise that connects concepts and techniques, while also managing workable problems

Preferred style and difficulty of the material

For example, some geometric techniques I want to learn about include homotheties, spiral similarity, inversion, projective transformations(I know this lies out the scope somewhat), complete quadrilaterals. There's much more, but in general I want a somewhat comprehensive, encyclopedic text on the different theorems and techniques(without being pedantic and overtly wordy, and having enough exercises)

For reference two books whose chapters catalog and encompass what I am looking for, but are inaccessible for my current level are Evan Chen's "Euclidean Geometry in Mathematical Olympiads", and Dan Pedoe's "A comprehensive course in geometry". In EGMO, the problems are clearly meant for those who go to IMO and math olympiads, which is not what I'm looking for, while in Dan Pedoe's book, the exercises are scant and few.

A note about the difficulty of the books: The geometry taught in high school is boring, SAT style-dry and does not vary in both concepts and problem type. Obviously I am not asking for a book introducing me to the extreme basics in Euclidean geometry. That being said, I'm still a novice in comparison to many of the seasoned Geometers out there(or those who have studied geometry in depth) I have read AOPS Introduction to geometry, which I think is closer to the level I want for getting to solve a variety of geometry problems while also using different techniques, the book's difficulty level is actually quite easy, but then again I don't want an International Mathematical Olympiad difficulty style type of book, but not a book where everything is spoon-fed to you either.

Please do not recommend the following: BOOKS NOT TO RECOMMEND

I recognize some of the texts included below are good texts in themselves, but for my present purpose, I do not want to get mired in the actual theory behind what a geometry/geometries is, as well as the axiomatic foundations of geometry. I want to learn math, not what the math behind really means at this point.

"Elementary Geometry from an Advanced Standpoint" Edwin Moise

"Advanced Euclidean Geometry" Robert Hartshorne

Euclid's elements(Seriously? Why would anyone recommend this book to a beginner? Just as no-one would recommend Newton's principia to someone who is first learning physics either)

EGMO Evan Chen (See above)

"Geometry-A comprehensive Course" Dan Pedoe

"An Introduction to Geometry" Richard Ruscyzyk (Already read the book, good for beginners but only focuses on some of the more basic techniques)

"Geometry revisited" Coxeter (Too few problems, there is little to be gained from reading it imo)

"Geometry Unbound" Kiran S. Kedlaya

"PROBLEMS IN PLANE AND SOLID GEOMETRY" Viktor Prasolov (This is almost all problems and no exposition at all)

Also-no Olympiad style books, unless the difficulty is manageable.

Further questions

One final question to those who have studied the aforementioned and more in geometry: I do want to eventually learn projective geometry and differential geometry, are there any specific course recommendations, or areas I have to learn first?(With regard to geometry, topology, and abstract algebra, since I have some knowledge of analysis) Thanks-

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    $\begingroup$ Have you seen Greenberg's book $\endgroup$ – Conifold Jul 27 '20 at 4:48
  • $\begingroup$ I upvoted the query; nice detail given in the query. $\endgroup$ – user2661923 Jul 27 '20 at 6:40
  • $\begingroup$ I'm surprised you feel there aren't enough problems in Pedoe or in Geometry Revisited. In any case, see if Geometric Transformations I-IV by Yaglom is close to what you're looking for. $\endgroup$ – Anonymous Jul 27 '20 at 13:03
  • $\begingroup$ With respect to your last question, how much knowledge of linear algebra and analysis do you have presently? $\endgroup$ – Anonymous Jul 27 '20 at 13:11
  • $\begingroup$ @Conifold Seems like an interesting book, I'll check it out later. Thanks for the suggestion. $\endgroup$ – Ren Jul 27 '20 at 16:04
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It looks like you've looked around for books and found a lot that are not to your taste, or are at the wrong level of difficulty.

Have you browsed through archive.org? If you type the search term "Geometry" into the Internet Archive search box you get hundreds, maybe thousands of books on geometry. Browse away until you find the right level and mix of exercises.

Geometry as a subject was a much bigger part of the math curriculum during the 19th century than it was in the 20th century. Archive.org has a wealth of 19th century texts, and I've found that many of them are full of exercises. So I'd recommend that you look through some of those. The great thing about math is that it doesn't become obsolete, and everything you're likely to learn about Euclidean geometry at this stage was well known in the 19th century.

You can of course narrow down your search terms to focus on euclidean, solid, or projective geometry.

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  • $\begingroup$ Ahhhh- I know this website, although I used it in the past to search for literature instead of math. I haven't visited internet archives for awhile, but I'm wondering why I never thought of browsing archive for geometry textbooks. I'll search more thoroughly on archive. $\endgroup$ – Ren Jul 27 '20 at 16:10
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Not only do I agree with brainjam's suggestion re archive.org, but that suggestion dovetail's into mine. I think that is often challenging to find "just the right book" for you re a specific math topic (e.g. Geometry).

I offer generic suggestions:

  1. Look for a book that has a lot of exercises. In every math topic that I have explored with a book (e.g. calculus, number theory, complex analysis) I have always learned much more from the exercises. As far as I am concerned, the more exercises the better.

This is especially true given the existence of the MathSE forum. For an exercise that you deem valuable that you just can not solve, all you have to do is provide context (i.e. the theorems leading up to the problem) and show your work. Then you will generally get very helpful responses from this site. Given such a crutch, this adds value to a book that has a lot of exercises.

  1. Review queries on this site that are tagged [book-recommendation] and [geometry].

  2. Try to find impartial reviews of a book that you are considering. Personally, I have found Amazon book reviews to be very reliable.

  3. Try to obtain a free copy of the book, either through archive.org (as brainjam suggested) or by googling for a specific title, including the "pdf" suffix. For example, you might google : coxeter geometry pdf. Surprisingly, I have often found (and downloaded) totally free pdf copies of math books that were very highly recommended.

The point here is not to save the cost of one book. Instead, the point is that you may have to try 10 different books before you find the one that best fits your needs. After you find such a book, you might then consider buying the book (even if you already have a free pdf copy). I have found that having a physical copy of the book (either hardback or paperback) is often helpful.

  1. Ironically, you will have to take the recommendations of others (e.g. from this site or from Amazon reviews) with a large grain of salt. A math book needs to be customized to your intelligence, math-background, math-goals (re what you are trying to accomplish via the book) and level of committment. It isn't reasonable to expect a book customized for someone else to necessarily be the right book for you.
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  • $\begingroup$ I have bought a lot of books(using my parents money) in the past for physics on Amazon, but I have a budgetary limitation now, because I have reached the age of 18. The only problem with Amazon I sometimes have, is that there is no book preview, and that makes it hard to decide if a book is the right fit. I can probably purchase 3 books under the price of 15$, but I want to be careful about the purchase. $\endgroup$ – Ren Jul 27 '20 at 16:16
  • $\begingroup$ @Ren If you try hard enough, you should be able to find 10 different books on geometry for free (either from a library, a pdf archive that generally allows free downloads, or simply googling for the pdf of a title : author). Simply keep digging until you find the right book (for free). Then your only issue will be whether the book is worth buying. I wouldn't spend a nickel until I knew that the book is right for me. An approach that I like to take is to get recommendations from mathSE or Amazon, and then try to get a free copy of a recommended book. $\endgroup$ – user2661923 Jul 28 '20 at 0:30
  • $\begingroup$ That's true-with regard to your suggestions- although it seems like google's algorithms manages to always push irrelevant(and sometimes frankly annoying) things like "high school geometry tutoring" or "geometry for k-12 kids". That being said, maybe I can apply some type of filter for the search results so it's less diluted. $\endgroup$ – Ren Jul 28 '20 at 2:42
  • $\begingroup$ @Ren I've had good experience googling a string formatted like <subject> <author> <pdf>. For example, google : geometry coxeter pdf. $\endgroup$ – user2661923 Aug 10 '20 at 18:20

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