Juan G. Escudero
A possible way to remove the gap between the worlds of sciences and humanities, is the search for interconnections between mathematics and physics with the sound and visual arts. There are several prominent examples in the 20th century in the domain of the sound arts like the greek architect, engineer and composer Iannis Xenakis, who used tools ranging from statistical mechanics to group theory, and the spanish composer Francisco Guerrero, who felt a fascination with mathematics and physics which is reflected on his high quality music. In the visual arts there are also well known artists inspired by mathematics, but perhaps there is a lack of perspective yet to analyze their significance.
A pentadecic surface with 1162 real nodes and some of its variations are the basis of this work. Mathematica and Surfer computing and geometric visualization tools have been used. The last part of the title refers both to the graphic work of the italian architect Giovanni Battista Piranesi, and the homonymous series by the english composer Brian Ferneyhough.
This work is based on a family of algebraic surfaces with many nodal singularities. They have been introduced recently, by means of a kind of duality in the basic geometric constructions corresponding to the generation of substitution tilings ("A construction of algebraic surfaces with many real nodes". http://arxiv.org/abs/1107.3401). Here the surface is a nonic with 220 real nodes. In general, the surfaces have degrees divisible by three and cyclic symmetry. They appear as mirror pairs not necessarily topologically inequivalent (see the sextic with 59 real nodes in arXiv:1107.3401).