Math is a beautiful thing. People just don’t tend to understand it well. Mathematical relationships can be found everywhere, especially in nature. One of my favorite examples of math in nature is fractals. Fractals are fragmented geometric shapes that can be split into parts that are a reduced copy of the whole. There is also what is known as a fractal dimension, or something that is in between 2 dimensions. For example, a fractal dimension can be something that is located between the first and second dimension, and is neither an area nor a line, but is what is in between these 2 dimensions. An example of a fractal found in nature is snowflakes.
Another example of a mathematical concept that many people use while describing something considered to be beautiful is the concept of symmetry, or correspondence of size and form on the opposite sides of a plane. Many people think that a perfectly symmetrical face would be beautiful. However, I have found this to be false. Like the snowflake, there are always small differences in each person’s face that make them unique. These differences do not necessarily, diminish the person’s beauty. Therefore, whether or not a person’s face is beautiful, is not dependent upon its symmetry or even necessarily the “flaws”. The spatial relations between and how each part of the face interacts with another is what makes a face beautiful. For example, I was born with 2 different colored eyes (Heterochromia iridium http://en.wikipedia.org/wiki/Heterochromia_iridum) ; one of them is hazel and changes from a golden color to bright green while the other one is a darker brown.
My eye color makes my eyes asymmetrical. However, most people do not notice them upon meeting me unless my hazel eye happens to be bright green at the time and, even then, they still often do not notice. When they do notice, they may be taken aback. This is not a big deal because the colors and shape of my eyes, in my opinion, go well with the rest of my facial features. Therefore, overall, balance should be emphasized over symmetry as far as beauty is concerned. It would be strange to see someone with a perfectly symmetrical face, but the face would not necessarily beautiful. Beauty may be a more dangerous and active thing, than simple symmetry. As Debussy puts it, an effort to create a thing of beauty is always regarded by some people as a personal attack.
Finally, one of my favorite mathematical concepts is harmonics. Harmonics are not often fully understood by musicians even if they use it a lot, but having a good understanding of them would be helpful to any musician. In physics, harmonics are defined as waves at proportional frequencies and inversely proportional amplitudes.
In music, harmonic waves are what produce many different feelings to a song. For example, the same song can be sung in a way that can make it sound pleasant, dark, bright, bluesy or tragic all because of harmonics. To produce harmonics, the singer causes his or her vocal chords to oscillate in manner appropriate to the feeling that he or she is expressing. When someone is singing off key they are creating harmonics with their vocal chords for notes that do not exist.