For more random interesting theorems with outrageously clever and beautiful proofs, the "Proofs from THE BOOK" [1] book is a fantastic collection. As a curiosity, "THE BOOK" in the title comes from Erdos, who often referred to the book in which God keeps nice proofs of math theorems :)

[1]: https://www.amazon.com/Proofs-BOOK-Martin-Aigner/dp/36420085...

If you can find a way to get a hold of it, Saunders MacLane's Mathematics: Form and Function is a tremendous book for understanding how math arises from the world and reflects it. What's written here is a lot more specific than that, of course, but if this illustration interests you then it might be a good book to try digging into a little bit.

On the other hand, while I am always cognizant and amazed by the POV of math as a human construction, there's something a little otherworldly about it from time to time. In the same way that you sometimes hit a code design which feels so damn good, Math, especially older math, is just a huge collection of these. Somehow, despite this all coming, apparently, from our minds, we hit on these design decisions that are just so sweet that they last millennia. This is what inspired things like Voyager---maybe it's hubris, but it just has to be the case that aliens speak mathematics.

So, I encourage anyone excited by this: math isn't "hard", it's just big and wonderful. You'll never finish learning it, but the journey will be incredible.

I'll leave this linking two more great resources (edit: to be clear, really the first one is the great resource... the second is just me talking, not really great at all). First, Paul Erdös, a famous mathematician who perhaps specialized in combinatorics, loved this idea I espouse above. In his mind, God had a small number of "proofs" in his mind when he designed mathematics. These are so wonderful that their beauty is completely self-evident. After Erdös "stopped doing math" (passed away) people compiled some of his Proofs from The Book along with others they imagine he would have so regarded into a great text book called, unsurprisingly, Proofs from The Book [0].

Finally, I'll self plug a little essay I wrote, actually in another HN comment, a while ago about learning mathematics.

http://jspha.com/posts/there_is_no_royal_road_to_mathematics...

[0] http://www.amazon.com/Proofs-THE-BOOK-Martin-Aigner/dp/36420...

A young George Polya, the Hungarian mathematician (and Stanford faculty) was in England once, and asked Hardy to see one of Ramanujan's notebooks. The next day, he returned the notebook in a panic, saying "there is so much in there, that I'll spend my entire life working on these proofs and never be able to do any original work". (paraphrased).

Ramanujan's accounts of how he arrived at the proofs ("the Goddess revealed it to me"), reminds me of Paul Erdos's "theory", that "God" has a big book of all possible proofs, called "The Book"; and if you're a nice mathematician, once in a while He will open the book and reveal one to you. Taking up this thread, a couple of mathematicians compiled some of the most beautiful proofs and published them as "Proofs from The Book". http://www.amazon.com/Proofs-THE-BOOK-Martin-Aigner/dp/36420...

The wonderful book, Proofs from THE BOOK,

http://www.amazon.com/Proofs-BOOK-Martin-Aigner/dp/364200855...

discusses the art gallery problem, among many interesting mathematical problems, and then exemplifies the playful spirit of mathematical proof by including in that discussion a photograph of what is quite possibly the ugliest art museum ever built--on the campus of my alma mater university.

http://www1.umn.edu/twincities/maps/WeismanArt/photo.jpg

(The photo shown in the book is different from this official photo, but you get the idea.)

For the rest of my life, I will always smile when I think of the art gallery problem, because it will remind me of what the authors of Proofs from THE BOOK asked their readers about the Weisman Art Museum (which I refer to as the "Ugly Art Museum"). Thanks for making me smile by submitting this link to the Wikipedia article.