To me, math is beautiful. I have heard some say that takes the beauty away from the natural world around us. I disagree with that sentiment and agree with Dr. Feynman who said that understanding only makes the world more beautiful.
Math was invented as a method to describe what goes on around us. Ok, first it was probably invented to represent the counting of something like sheep or eggs. But are those not things around us? That was simple addition and subtraction, which when down repeatadly leads to multiplication. Then to division. That leads to fractions, rationals and then to irrationals, etc.
I am always a little taken-a-back when I hear someone say something like "I am not good at math." That is a load of... something I shouldn't say. I figure that if I am not good at something, it is because I did not try hard or long enough. Sure there are cases where I probably can not achieve some certain goal, but those are extremely few in comparison to those that I will not achieve because of self-imposed limitations. I wonder if that person had a teacher or parent that said the same thing and then when addition was a little difficult for them they emulated what they were already told, instead of repeatedly trying.
To continue, I find several specific math functions to be very beautiful. First, the Fourier Series and subsequent transform. Fourier developed the series when trying to understand heat transfer in a metal plate. Math was used to describe how the heat moved but originally there were only solutions to simple cases of the equations. Fourier found that by combining a bunch of cases he could solve more complex ones. Cool. His discovery has led to many amazing others. For example, coherent light passing through a transparent object will produce a fourier transform of the object in the focal plane. I often carry with me a clear card that looks like there are a bunch squares and dots on it. When a laser pointer is shined through it there are many different images, like one of Einstein.
Second, Euler's identity. This simple identity combines natural logarithms, pi, complex numbers (unfortunately often called imaginary numbers), and trigonometric functions (sin, cos) into one simple statement. I had a professor years ago, Dr. Moon, who called it the "death bed identity" saying that any good electrical engineer will quote this to family and friends as he or she lay dying. This was somewhat tongue-in-cheek but emphasing how important it is to electrical engineers and science in general.