Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Conventional view of Symmetry-Breaking. Symmetry-breaking is a reduction of symmetry group.

Thus the transition from a square to a parallelogram is conventionally given by the following reduction in symmetry group:

D4 ---> Z2.

That is, the eight operations in $D 4$ are reduced to the two operations in $Z 2$ . However, according to our view, this is inherently weak because it means a loss of algebraic structure. In contrast, our approach to symmetry-breaking can be illustrated with the example given in section 8 of the transition from a square to a parallelogram: This transition was modelled by adding, to the symmetry group of the square, the general linear linear group, via a wreath product. Thus, in our approach, symmetry-breaking actually preserves the original group. The breaking of a symmetry group $G 1$ is carried out by extending $G 1$ by another symmetry group $G 2$ via a wreath product thus: $G wOG 1 2$ . The original symmetry is given by the fiber copy of $G 1$ which corresponds to the identity element in the control group $G 2$ . Non-identity elements in $G 2$ break the symmetry of the fiber group. Wreath products of this kind will be called symmetry-breaking wreath products.

New View of Symmetry-Breaking. Symmetry-breaking is extension via a wreath product. The extending group is the symmetry group of the asymmetrizing action.

Most crucially, in our view, symmetry-breaking corresponds to an increase in symmetry group! More deeply, this undermines the standard notion that groups represent symmetry. Rather, we argue that they are maximally compact descriptions of asymmetry.

16 New Foundations to Geometry

Essentially, the recoverability of generative operations from the data set means that the shape acts as a memory store for the operations. More strongly, we argue in the book that all memory storage takes place via geometry. In fact, a fundamental proposal of our theory is this:

Geometry $=_$ Memory Storage.

This theory of geometry is fundamentally opposite to that of Klein’s in which geometric objects are defined as invariant under actions. If an object is invariant under actions, the actions are not recoverable from the object. Therefore Klein’s theory of geometry concerns memorylessness, and ours concerns memory retention.