Reddit mentions: The best non-euclidean geometry books

You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.

Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.

• The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.
  
  
• Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.
  
  
• Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.
  
  
• An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.
  
  
• Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.
  
  
• Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.
  
  
• A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.
  
  
• Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.
  
  
• Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.
  
  
• I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.
  
  
• Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.
  
  Basically, don't limit yourself to the track you see before you. Explore and enjoy.

I’m not sure precisely what you mean by “contemporary” or “geometric algebra” or “basic number elements and algebra”. What did you feel was missing from Lang’s book? (I’m not familiar with its contents.)

If you want something in line with the standard high school curriculum, but maybe a bit more rigorous than most, this book by Kiselev was the standard Russian school text for generations (review)

Or you could try the Art of Problem Solving geometry book (site).

There’s a lot of good stuff in Coxeter and Greitzer’s book Geometry Revisited, but I’d say it probably assumes a standard high school geometry course as a prerequisite.

Not really limited to plane geometry, but I really like Hilbert and Cohn-Vossen’s book Geometry and the Imagination (review). I’d recommend getting a used copy of the original printing; the recent ones are printed on demand and not as nice.

Also let me recommend Apostol and Mamikon’s lovely book New Horizons in Geometry (review), though it’s more about calculus than algebra per se.

If you want to study plane curves from a complex number perspective, you could try Zwikker’s 1963 The advanced geometry of plane curves and their applications

If by geometric algebra you mean Grassmann/Clifford/Hestenes style algebra, check out the stuff Jim Smith has been doing, or you could take a look at this thing (I haven’t read it), or try these papers.

They probably aren’t what you’re looking for, but I think Farouki’s Pythagorean Hodograph Curves are pretty neat (that book also has a lot of other interesting material in it). Also neat for formalistic theorizing about algebras for spline curves is Ramshaw’s monograph On Multiplying Points: The Paired Algebras of Forms and Sites (probably a bit abstract for what you want here).

What are your goals? Do you want to design lenses and mirrors for cameras? Model classical mechanics systems? Construct arbitrary shapes out of polynomial curves so you can draw fonts or animate characters on a computer screen? Design cut paths for CNC machines? Approximate transcendental functions by some type of function that you can more easily compute with? Find the prettiest proofs of thousand-year-old theorems about circles? Prepare yourself to study differential geometry or algebraic topology? ...

I didn't say the twins are comoving. The accelerating twin uses a comoving inertial proxy to avoid a measurement error. The observer that comoves with the accelerating twin measures the other twin's velocity. The MCIF isn't needed when the accelerating twin measures the other twin's velocity quick enough to avoid a significant measurement error. Regardless, the 2 measurements of the other twin's velocity are the same in any given moment.

> That calculation just doesn't have anything to do with what you write.

How so? Empty claims aren't convincing.

> It's unintuitive, cumbersome and more complicated.

To you. To others who don't understand "the Minkowski-space formulation based on invariants between different reference frames" the book's explanation might be better. As long as both models return expected results it's a matter only of personal preference.

I say you shouldn't so quickly dismiss such books because I see that you're lacking intuitive skills, leading you to fundamental mistakes, and such books could help you fix that. It doesn't have to be Relativity Visualized. Another good one is Understanding Einstein's Theories of Relativity. My favorite is Exploring Black Holes by Taylor and Wheeler. It's free & online. On page 1-20 guess what they recommend reading? Relativity Visualized! How do you explain that Taylor and Wheeler recommend the book that you think is crap?

A decent introductory book on SR will teach you that velocities are best measured using additional observers at rest relative to the main observers (in this case the twins), so that all velocities are measured when observers pass right by each other. Using the MCIF there are 2 inertial observers passing right by each other. It'd be real hard for you to explain how they could measure difference velocities for each other.

> You should question your approach based on an excel table if it gets wrong results especially if your demeanour is such that you go around saying "out of the way, Me street smartz, no education in relativity, but me knowzz best and most easiest explinationz"

But it gets correct results and I didn't say that. That "Here's the simple explanation you're looking for" is supported by the math and is indeed simpler than yours and the other one you linked to. It doesn't matter that I made a mistake somewhere else.

Books on Fractal Geometry tend to have pretty pictures:

Indra's Pearls: The Vision of Felix Klein by David Mumford et al.

Beauty of Fractals by Heinz-Otto Peitgen et al

Fractal Geometry of Nature by Benoit Mandelbrot

For what it's worth New Kind of Science by Stepeh Wolfram has tons of pretty pictures, even if the content is dubious.



you might also want to checkout the Non-Euclidean Geometry for babies and other similar titles.

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
  

Try What is Mathematics?, by Courant & Robbins. It's a good overview of mathematics beyond the elementary level you've completed. Another good book like that is Geometry and the Imagination, by Hilbert & Cohn-Vesson.

Only seems to be a massive problem in North America really. For my math undergrad masters in the UK (4 years), the stuff we were required to buy was really cheap and only for extremely specialised modules. For general calculus or whatever, they wrote their own notes with their own worksheets/coursework that could be accessed online.

The only textbooks I own are the four books that I was required to buy (for about £80 total over four years), textbooks I bought willingly after I decided to do my PhD, and textbooks I bought during my PhD. Looking at my bookshelf, I have eight books that were voluntary purchases/presents, and four required ones. The required textbooks were:

Hyperbolic Geometry (Jim Anderson) (fourth year Hyperbolic Geometry masters module);

Introducing Einstein's General Relativity (Ray D'Inverno) (third year GR module, also useful for advanced GR/gravitational waves in fourth year);

Complex Functions (Jones and Singerman) (fourth year Complex Function Theory masters module);

The Code Book by Simon Singh (first year Number Theory and Cryptography module).

The first three were all written by lecturers from my university and the code book is a fantastic cheap read regardless of course requirements. The GR book was the most expensive, but at the end of each year they would buy back the books from students who didn't want it after the course was over so that they could sell it second hand to next year's students for about half the price.

Edit: In this context, required means "required because you need to self study some stuff and we are generally following this book, so it would be really, REALLY fucking helpful". Some people never bought any.

You would probably like these two books:

• Geometry and the Imagination by David Hilbert and Stefan Cohn-Vossen.
  
  
• What is Mathematics? by Richard Courant.
  
  Neither of those are "popular math" books; the authors are famous mathematicians, and they explore various fields of mathematics without requiring too much advanced knowledge.

• The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel
  
• Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov et al.
  
• Visual Complex Analysis by Tristan Needham
  
• Geometry and the Imagination by David Hilbert and Stephan Cohn-Vossen
  

Relevant: http://www.amazon.com/How-To-Draw-Straight-Line/dp/1436877709

I think this is a book I used to teach an undergrad back when I was in grad school. I don't remember much, but it seemed fairly low in prerequisites.

This is actually a geometry book, but it did for me exactly what you're describing. It's a shame it's so expensive. https://www.amazon.com/dp/0716799480/ref=cm_sw_r_cp_awdb_t1_EcDLAb6NHVJSA