Neat answer; although, the bullet points come off a bit like mathematical horoscopes (perhaps because of the use of the second-person, "you"). I often wonder if mathematics would be more popular if the beauty of works by it's masters were more accessibly appreciable by those without the mastery, as it is, e.g., in the arts or music. Anyone can look at a painting, see a movie, or listen to Bach, and at least have an opinion - not true for any random person flipping through, say, a functional analysis book.
We must remember the frame of reference for these works. The audience.
Most art, and certainly most paintings, movies, and most of Bach's work, is written for a general audience. Works of advanced, pure math however are written for a very specialized audience, purposefully.
This does not mean, though, that math cannot be written for a general audience and be understandable or even "beautiful". As an example, Cantor's diagnolization is understandable by a 10 year old. The profoundness of the question it asks is also understandable, as is its initially seemingly mind-bending inscrutability. What's more the solution and the logic of the proof are perfectly understandable and some of the easiest to understand explanations literally use a child's method of counting (mapping fingers to numbers). This is a beautiful work of math. And much math could be written for broader consumption. It's not done frequently only because that isn't the task most mathematicians set themselves to.
Moreover, even outside the actual proof, many math results can be described in understandable and even beautiful ways. And the derivative results of mathematical discoveries are often easy to describe, understand, and appreciate as impactful and beautiful.
So, I think this is a function of choice and focus. Perhaps math could indeed more popular if greater efforts were made here, but mathematicians as a group on the whole haven't set out to popularize it.
Now, no matter how much effort is applied, math will probably never quite be as enjoyable to most people as music or movies. But with such efforts math may be more appreciable to a wider audience, as say paintings or poetry are.
Cantor's diagonalization is a great example; I'm definitely not a "math person", but I was awestruck the first time I learned about it. People had explained the idea of uncountability to me before, but I never really "got" it until then. I can't think of a better word to describe its effect on me other than "beautiful".
You can do Cantor's diagonal for positive integers. It's still true but seems ugly to most people. Higher math is filled with just as many ugly as butiful ideas, but they are simply less talked about.
Here is a scifi short story in which a machine is built that allows anyone to directly perceive mathematics similarly to music. The mathematician inventors hoped that this would increase the popularity of math but the opposite was true: http://www.rudyrucker.com/transrealbooks/completestories/#_T...
You are right that a lot of math is pretty inaccessible to the mainstream, but so is a lot of music. Sure, an average person can nod their head at the beauty of a Bach piece just as they might marvel at an (to them) unintelligible equation, but just as they do not understand the higher math they may also lack an understanding of counterpoint, canon in inversion, fugue, etc.
Likewise for avant garde cinema, modern art, poetry, and more!
It's qualitatively different. The nature of math work is analytic, that of music is more expansive. Someone can appreciate the beauty of Bach intuitively, without consciously breaking it down or even being aware of a construction, depedends on their intuitive senses. In math the information is encoded and mostly can't even start to be processed by the layman.
