I've been reading about quaternions and octonions and something that immediately stood out to me was how they all fit 2n in the amount of "axis" that exist in said number system. Is there some reason for this and where can I read more about thr others possible systems for example 3 or 9? Do they not hold any special properties like the ones that fit 2n? Is 16 something that has been explored? Thanks.
pi being 3.14159....is from one of Euler's textbooks where he calculated the ratio of a SEMICIRCLES length to its radius. so first off, we already got pi originally in terms of the radius, and not the diameter, and also, he was using pi to just say P (a greek letter translation) , much like we would describe an angle by using the variable theta. the symbol pi was just a variable until we for some reason decided to use pi to annotate the circle constant we know today. i bet that somewhere in his writings he used pi to mean "the ratio of the circumference to the radius" (if he has and u know about it, please tell me that would be cool to learn). So i think that tau is better.
I'm interested in learning more about calabi yau manifolds but before I get there I need to learn about k3 surfaces. there dont seem to be many online resources that explain it without already assuming that one has a prerequisite understanding of it.
In the context of research, I just try to change some things up a bit such as tackling a different portion of the study. If it's a book, I try my best to get as many exercises as possible and ignore the "unimportant" bits of text.
I’m legitimately curious, because it’s some of the most frustrating, mind-boggling stuff I’ve ever seen in my life. Maybe it’s because I’m currently a community college kid and I have a project due in a week where I can’t even get past the first question, but it’s legitimately sort of insane to me.