Birkhoff, Gombrich and Lorand purport different, and at times antithetical, views on art and beauty. The three coincide, however, in taking for granted a basic dichotomy in its analysis.

On the one hand, the formal aspects of an artwork (which roughly coincide with the formal associations in Birkhoff, the sense of order in Gombrich and the discursive order in Lorand) and, on the other hand, the semantic aspects (connotational associations, sense of meaning and aesthetic order respectively). Not that these are watertight compartments. On the contrary, the interplay between them is a defining feature of many artforms (poetry being a notable example).

Unlike Lorand, but in agreement with Birkhoff and Gombrich, I see symmetry – in the various forms we have seen in Chapter 3 and in other forms we will meet – as a formal aspect, and a notable one for that matter. Furthermore, I see discursive and aesthetic orders as possible raw materials for each other (see the end of Section 5.1). In particular, I see symmetry as a raw material in the interplay mentioned above. This view is the theme of the present chapter and a starting point to develop it is the realization that, as a raw material, symmetry does not need to be exact.