Ask HN: A math study program?

I took math courses through high school up to calculus in college and a course on discrete math, which i did well in. I just got John Stillwell's "Mathematics and It's History" and I'm dazzled by the way math is presented and the beauty inherent in it, unlike the way it was taught to me in school. However, I'm starting to struggle in some of the early geometry exercises like with regular polyhedra and conic sections, and later with exercises in projective geometry. Is there a course or series of courses I can take that can build my math skill level to solve such problems with ease? Stillwell mentions using this course to teach senior-level math undergrads.

I think H. S. M. Coxeter's books are good sources for training in geometry, leading to ideas in groups. I haven't looked at this book by Stillwell, so I am not aware of the difficulty of problems presented in this book.

There is an old Schaum's series book on Projective Geometry that was very useful to get the basic ideas quickly. It may not be an elegant presentation of the subject, but it was very quick, and to the point.

All said and done, synthetic geometry (not analytic geometry) is not very easy, and it might help to create models (wire-frame or 3d cardboard or plasticine/play-dough) to visualize things. They may not help to solve the problem rigorously, however.

By coincidence, I am slowly working through another of Stillwell's books, on Reverse Mathematics. You are right, his way of presenting things suddenly makes things you know snap into place.

The best resource I've found is Brilliant.org's courses:

https://brilliant.org/

There's no royal road to geometry. You just have to keep trying things and keep suffering. Sadly, I'm not familiar with that book or particularly with the topics you've mentioned, so I can't recommend specific books, but this is the basic recipe that I've used to teach myself some amount of mathematics:

Start with a book you want to read. If you get stuck, then buy another book (hopefully aimed at a lower level) on that topic and repeat the process with the new book.

Don't be afraid to read "easy" books. You should probably aim to start reading books where you look at the contents page and think you know 80-90% of the material already. I've wasted a lot of time trying to read books that were above my level. The path of least resistance is longer, but in my experience it pays off.

"Do the exercises" is good advice, but don't be too obsessive about it. Be more obsessive about regularly working on the topic, even if that means skipping exercises or jumping between books (on the same topic). You can often find the answer to an exercise in one book in a different book's presentation of the same topic, or on a website or in a paper. As long as you can integrate these discoveries into your conceptual framework of the subject, that's not cheating, it's success.

Writing things out in a lot of detail and working out examples in a lot of detail in a notebook can really help. This is like designing your own exercises and can be better than doing exercises in a book sometimes.

Khan Academy. I completed all their Math modules over the course of a Summer and, frankly, learnt more from their content than I had in 10+ years of mainstream Math education. I feel like if I had done it sooner I would have been an A* student, which haunts me a little.

Despite studying engineering up to postgraduate level, I tended to avoid any concept that required a deep understanding of the Mathematics behind it. Now, I love Math and feel like I could express most problems using it, as well as make sense of papers and text books that were closed off to me previously.