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 in the last century, but perhaps there is a lack of perspective yet to analyze their significance.
Hexagonal substitution tilings were introduced in (Int.J.Mod.Phys.B, Vol.18 (2004), p.1595) by means of simplicial arrangements of pseudolines. The analysis of the topological invariants of the associated space of tilings ("Topological invariants and CW complexes of cartesian product and hexagonal tiling spaces". Numerical Analysis and Applied Mathematics. AIP. Conf. Proc. Vol.1389 (2011), p.1702 ) leads to the introduction of a branched surface. This work can be interpreted as a metaphor of a nomad place, a space in constant change where local configurations of a very small number of shapes always reappear, but in different surroundings. This "ritornello" type property is preserved when we extend the pattern to infinity. The basic hexagonal symmetry is continuously broken and has to be perceived in a dynamical way, as would be the case if temporal phenomena were embedded.
Surfaces with many singularities of type Aj can be obtained also by using the polynomials introduced in arxiv:1107.3401. Explicit constructions are possible by adding classical Jacobi polynomials to the polynomials obtained as product of lines. We explore here deformations of a sextic with 59 real nodes and a dodecic surface with 37 nodes and 66 singularities of types A3 and A7, all of them also real. Mathematica and Surfer computing and geometric visualization tools are used.
The basic geometric constructions for the generation of
substitution tilings in the series of "Branched Surfaces" are
simplicial arrangments of lines or pseudolines. Simple
arrangements can be obtained either as subarrangements ("A
construction of algebraic surfaces with many real nodes".
http://arxiv.org/abs/1107.3401) or by rotations ("Substitutions with vanishing rotationally
invariant first cohomology" Discrete Dyn. Nat. Soc.,Vol.2012). One
of the two subfamilies of simple arrangements produces real
variants of Chmutov surfaces ("Real line arrangements and surfaces
with many real nodes". Geometric modeling and algebraic geometry,
Springer (2008)), and the other gives surfaces with a larger
number of real nodes as those shown in previous works like "Nueve
y 220-B" or "Quince y 1162-Carceri-B".
The polynomials obtained as product of lines in the simple arrangements can be used to construct hypersurfaces. This work is based on a projection in 3D of a degree-12 singular 3-fold.