As a mathematician, what may you say are the beauties that you see in the Mandelbrot set??
Before anyone can answer, you have to define your terms - and beware of philosophy.
Is "beauty" a mathematically defined concept?
It would help to have a context for the question.
The general answer, though, would be "the same sort of beauty as one finds in Nature."
Ok, can i rephrase that.. From "the beauty" to "the functionality (with respect to nature and the human sense experience ) and the mathematical significance of its discovery...
Im sorry Im still coming to terms with the fact that philosophy (or some degree of it at least) is not allowed in a physics forum, i find the lines between the two a bit blurred and unconsciously most of the time i find myself leaning toward the philosophical side of physics..i feel thats where discussion is most fruitful..
For an implied definition of "fruitful" which is far from clear in this context ... it is a banned topic exactly because the discussions tend to be the opposite of fruitful in the sense of actually getting anywhere. One reason it tends not to go anywhere is the way people who prefer philosophical discussion keep missing out vital definitions like that and everyone ends up talking at cross purposes and then people get upset etc...
However, it is not possible to discuss science without taking some philosophical position. The standard accepted position is, loosely, empirical realism. Broadly, that there exists a real Reality "out there" that we can make sense of through our sense data via careful experimentation.[*]
A sensation of "being in the presence of beauty" (etc) forms part of our sense data which must be telling us something about the World just like a sensation of "the color yellow" does. Beauty, as with much of our immediate sense data, is notoriously observer dependent - but by careful examination of our sense data we can come up with relationships which do not depend on the observer. When we find one of those, we say we have found something "fundamental". A core goal of physics is to find these fundamental relationships in Nature.
What you are asking for here is basically the matter covered in a college course in fractals or chaos theory. That's a little big for this forum.
You can find out about the significance of the mandelbrot set, and fractals, the impact their investigation has had on maths and physics, simply by looking them up online. If you have trouble understanding that information, then we can help you.
But you can get a glimpse of the significance of the Mandelbrot set by considering: a characteristic of fundamental relationships is that they are informationally compact - they encode a lot of direct-experience Nature in a small space. So how fundamental is the Mandelbrot set?
See also: http://www.math.binghamton.edu/topics/mandel/mandel_why.html
Sorry - but it's that or RSI.
[*] More precisely - that there are statements about Reality whose truth may only be investigated this way.
Topics: Empiricism vs Rationalism and the problem of induction ... not to be discussed in these forums.
Significant authors: Carl Popper and Thomas Kuhn
Separate names with a comma.