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Context and mathematical maturity: I have knowledge of the usual engineering math courses, meaning differential+integral+vector calculus, linear algebra, probability and statistics, etc. and some pure math courses like analysis. I am a pure mathematics enthusiast, so I have been working through some texts in pure maths like Abbott's analysis and Pinter's Abstract algebra.

Request: I would like a recommendation for a geometry class. I know there are many different areas of Geometry, and I have no idea of where to start. Since I don't know much about geometry I would like a relatively introductory book, but much preferably if it is not at a high school level. Hopefully the book would also have some exercises so I can also practice what I read.

I have seen some people recommending Euclid's elements, but to be honest I don't find it to be super easy to read (I guess it is mostly about the language), and I have also seen visual differential geometry by Needham, which seems interesting, but I am not sure whether that is a good place to start learning about Geometry.

Goal: My dream would be to read Hilbert's Geometry and Imagination and not be completely clueless of what is happening on the book.

Thanks all for your recommendations!

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    $\begingroup$ The different areas of geometry have less to do with one another on average than two randomly chosen subjects in mathematics, so the answer will depend heavily on what you really want. The last text that tried to be an introduction to all of them is probably by Coxeter rads.stackoverflow.com/amzn/click/0471504580 . $\endgroup$
    – darij grinberg
    Oct 14, 2021 at 15:29
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    $\begingroup$ Would it be considered bad style to self-promote? We put some effort into that book... Reid-Sz: Geometry and Topology, CUP 2005, written for the student with the background that you are describing. cambridge.org/core/books/geometry-and-topology/… $\endgroup$
    – Balazs
    Oct 14, 2021 at 15:35

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Your requirements: "much preferably if it is not at a high school level" and "My dream would be to read Hilbert's Geometry and Imagination" are contradictive. The book of Hilbert and Cohn-Vossen, Geometry and imagination IS on the high school level.

Other highly recommended geometry books on the same level are Marcel Berger, Geometry, (in 2 volumes), and Robin Hartshorne, Geometry: Euclid and beyond, and "Foundations of Projective geometry" by the same author. All these books require no prerequisites.

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  • $\begingroup$ Thank you for your answer. Your comment on Hilbert's book is interesting. Do you suggest to try and tackle that as my first approach to geometry? It seemed to me, from the index, that it includes some advanced topics such as differential geometry and topology. I am now embarrassed to say it, but I didn't take any of those in high school. Maybe it is a light treatment of those? $\endgroup$
    – Sergio
    Oct 14, 2021 at 19:18
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    $\begingroup$ It explains all those advanced concepts from scratch. I read this book while in high school, and I think it really requires no prerequisites. And this is one of the best books I ever read:-) One defect is that it has no exercises. But other books that I recommended do have them. $\endgroup$
    – Alexandre Eremenko
    Oct 14, 2021 at 19:26
  • $\begingroup$ I will wait a couple more days to check other recommendations. If not, I will accept your answer as the accepted one. I checked Hartshorne "Geometry" and it seems quite interesting (a bit long, but I guess that is what I asked when I wasn't more specific about the area of geometry I am interested in). I couldn't find any TOC of Berger's books, though. $\endgroup$
    – Sergio
    Oct 18, 2021 at 18:02

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