Reddit mentions of Fearless Symmetry: Exposing the Hidden Patterns of Numbers - New Edition

At some point these "Pop" reading books get wholly unsatisfying and you need textbooks, but I think that's a story for a different semester. Theres a good set of books written by Avner Ash and Robert Gross (Boston College) that anyone with calculus 1 can easily get into:
Elliptic Tales:
https://www.amazon.com/Elliptic-Tales-Curves-Counting-Number/dp/0691151199
Fearless Symmetry:
https://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-Numbers/dp/0691138710/ref=pd_sbs_14_t_1?_encoding=UTF8&psc=1&refRID=JG1NQ2F2XS0WJJ5PBKVV

Well worth the read, entertaining, and great introductions to their respective subjects!

I have a few books I read at that age that were great. Most of them are quite difficult, and I certainly couldn't read them all to the end but they are mostly written for a non-professional. I'll talk a little more on this for each in turn. I also read these before my university interview, and they were a great help to be able to talk about the subject outside the scope of my education thus far and show my enthusiasm for Maths.

Fearless Symmetry - Ash and Gross. This is generally about Galois theory and Algebraic Number Theory, but it works up from the ground expecting near nothing from the reader. It explains groups, fields, equations and varieties, quadratic reciprocity, Galois theory and more.

Euler's Gem - Richeson This covers some basic topology and geometry. The titular "Gem" is V-E+F = 2 for the platonic solids, but goes on to explain the Euler characteristic and some other interesting topological ideas.

Elliptic Tales - Ash and Gross. This is about eliptic curves, and Algebraic number theory. It also expects a similar level of knowlege, so builds up everything it needs to explain the content, which does get to a very high level. It covers topics like projective geometry, algebraic curves, and gets on to explaining the Birch and Swinnerton-Dyer conjecture.

Abel's proof - Presic. Another about Galois theory, but more focusing on the life and work of Abel, a contemporary of Galois.

Gamma - Havil. About a lesser known constant, the limit of n to infinity of the harmonic series up to n minus the logarithm of n. Crops up in a lot of places.

The Irrationals - Havil. This takes a conversational style in an overview of the irrational numbers both abstractly and in a historical context.

An Imaginary Tale: The Story of i - Nahin. Another conversational styled book but this time about the square root of -1. It explains quite well their construction, and how they are as "real" as the real numbers.

Some of these are difficult, and when I was reading them at 17 I don't think I finished any of them. But I did learn a lot, and it definitely influenced my choice of courses during my degree. (Just today, I was in a two lectures on Algebraic Number Theory and one on Algebraic Curves, and last term I did a lecture course on Galois Theory, and another on Topology and Groups!)

I don't really have anything super manly on my lists, the only somewhat semi-manly I can think of, is all the mathematics books, dealng with gemoetry and symmetry and I guess that is somewhat manly in itself, considering that if you take higher level mathematics/science classes you're pretty much surrounded by men. yay mathematics

I'd recommend the following:

(1) Either of the Ash and Gross books Fearless Symmetry of Elliptic Tales

(2) Anything by Paul Nahin for instance Dr. Euler's Fabulous Formula

(3) Get yourself a Rubik's cube and copy of David Joyner's Adventures in Group Theory

(4) Prime Numbers and the Riemann Hypothesis

It's a bit complicated and I can't claim to be an expert but here's what I'm familiar with.

Group theory is the idea that certain systems can be represented as a closed group governed by some operator. If you apply the operator to some member of the group, you will always get another member of the group. G(Z, +), a group of integers governed by the + operator, is a group because you can add 1 (or subtract 1 -- the reverse of the operator) continuously to get every other integer.

Field theory governs fields, which are basically sets of numbers in which addition, subtraction, multiplication, and division are defined (so all of your favorite number systems -- real numbers, natural numbers, the whole family -- they're fields). Galois theory is dedicated to re-interpreting fields as groups to better understand them. The single operator of groups typically gives a better way of describing the relationship of numbers within a set than field theory does.

I wish I had a concise example to give but I really can't think of one. If you want to read more about it though I'd recommend Fearless Symmetry.