Artists worry a lot about aesthetics. How do you know if something is beautiful? We do know that one person will make efforts to pursue sensory experiences that another will find boring, unpleasant or distasteful. In spite of that people often show a good deal of agreement about what they consider beautiful.
Professional mathematicians also have a strong aesthetic sense, spend time talking about it, and sometimes let it guide their work. They frequently agree with each other on what is beautiful maths and what is not, and much respect and prestige goes to those who produce answers to important problems in an elegant way. They all agree that the feelings inside their heads when they encounter a beautiful mathematical idea are comparable to those experienced, for example, when listening to great music. (University maths departments are often hotbeds of high-level musical activity.)
One surprising but extremely useful bit of maths lets us construct closed curves of arbitrary shape using a very simple recipe. (See the figure on the right which show a square (-ish) curve being built from circular motions. Read on to find out how we do it.) See my Epicycles Gallery for some art-work which depends on this technique and my Cyclic Motions page for a detailed walk through of Processing code. It is, however, also used in constructing other images sets (e.g. Generative Circles) and as a step along the way in more complex methods (e.g. Trees in Snow ).
Unfortunately most of this experience is not easily accessible to anyone who has not studied maths for fifteen or twenty years. (Well, after all, great Chinese literature is a closed book to me - I do not know the language. Maths is a language that allows us to formulate and discuss concepts that are extremely difficult to express in natural languages.) The components of the experience are frequently similar, involving tensions between order and chaos, unexpected juxtapositions and connections, with surprising resolutions and revelations leaving a feeling of satisfaction.
There is no short cut to maths understanding, any more that there are short cuts to learning to play a musical instrument or to paint with skill (or for that matter learning Chinese). However, we can perhaps try to give some little insight, particularly when we can put together a few of the ideas that we have already encountered in a surprising way. This is not for the faint hearted - no one ever said that maths was easy - but even if you do not follow the detail you may get the general idea.