The notion of beauty has historically led many mathematicians to fruitful work. Yet, I have yet to find a mathematical text which has attempted to elucidate what exactly makes certain geometric figures aesthetically pleasing and others less so. Naturally, some would mention the properties of elegance, symmetry and surprise but I think these constitute basic ideas and not a well-developed thesis.
In this spirit, I would like to know whether there are any references to mathematicians who have developed a mathematical theory of aesthetics as well as algorithms(if possible) for discovering aesthetically pleasing mathematical structures.
To give precise examples of mathematical objects which are generally considered aesthetic, I would include:
- Mandelbrot set
- Golden ratio
- Short proofs of seemingly-complex statements(ex. Proofs from the Book)
I think the last example is particularly useful as Jürgen Schmidhuber, a famous computer scientist and AI researcher, has attempted to derive a measure of beauty using Kolmogorov Complexity in his series of articles titled 'Low Complexity Art'. Meanwhile, I find the following research directions initiated by computer scientists particularly fruitful:
- Bayesian Surprise attracts Human Attention
- Curiosity and Fine Arts
- Low Complexity Art
- Novelty Search and the Problem with Objectives
Note: From a scientific perspective, researchers on linguistic and cultural evolution such as Pierre Oudeyer have identified phenomena which are both diverse and universal. Diversity is what makes our cultures different and universality enables geographically-isolated cultures to understand one another. In particular, many aesthetics have emerged independently in geographically isolated cultures especially in cultures which developed in similar environments. Basically, I believe that if we take into account what scientists have learned from the fields of cultural and linguistic evolution, embodied cognition, and natural selection I think we could find an accurate mathematical basis for aesthetics which would also be scientifically relevant.