Doesn't the fugue imply the composer's submission to the rules? And is it not within those strictures that he finds the full flowering of his freedom as a creator? […] My freedom will be so much greater and more meaningful the more narrowly I limit my field of action.
A sweeping look at the preceding chapters would probably reveal the role of mathematics in rule-driven artistic creation as the main theme in this book. We initiated our discussion with the idea, stated in Section A.4, of visiting the crossing paths of art and mathematics. As this visit proceeded, though, we saw diverse conceptions of geometry (either as the assumption of some characteristics of space or as an idea of shape set via a group of transformations on this space) taking a prescriptive capacity with regard to the artist's work. The laws of geometry – of whichever geometry may underlie the artwork – acted both as a catalogue of what is possible (and we mentioned the role of mathematicians as catalogue-makers in this context) and as constraints precluding the impossible. We may talk about rule-driven (or constraint-based) artistic creation.
The framework of rule-driven artistic creation, however, is broader than the contents of our exposition: indeed, art practice can be driven by rules which do not have a mathematical character, and, actually, this is often the case.
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