Warren, I sense you really want something explicit. Try this: Let w be a complex parameter, and let X be all solutions x of Beta[x,1/5,1/2] == w. Now map the elements x of X by x --> Beta[x,2/5,1/2] and call the resulting set Y. Y is an analytic analog of the Penrose tiling vertex set. The elements of Y are analytic functions of the parameter w, and share all the symmetry and “phason” transformation properties (behavior with respect to w) of the standard Penrose tiling vertex set. -Veit On May 21, 2014, at 11:47 AM, Warren D Smith <warren.wds@gmail.com> wrote:

OK, given any elliptic function, i.e. doubly-periodic meromorphic function W(z) of complex z, e.g. a Weierstrass function if you want concreteness, the function of a real variable x W((a+b*i)*x) will in general be quasiperiodic.

Next, we can construct quadruply-periodic analytic functions of two complex variables. More interestingly, there is a 100+ year-old theory of quadruply periodic functions of 2 complex variables, whose output also is a 2-tuple of complex numbers, and each output is meromorphic in each input (or any fixed complex linear combination of outputs is meromorphic in any fixed complex linear combination of inputs). There also are generalizations to any dimension p -- the input is a complex p-tuple and so it the output, and the function has 2p period vectors -- these are called "Abelian functions." I will call them "Abelian functions of order p." The case p=1 is elliptic functions.

The case p=2 is the quadruply-periodic things in 2 complex dimensions. It was used by Jacobi 1834, Georg Rosenhain 1851 and Adolph G"opel 1847 to solve the inversion problem for "hyperelliptic integrals of the first kind" (integrands have denominator which is square root of polynomial of degree 5 or 6). B.Riemann and K.Weierstrass and Felix Klein then redid and superseded this work.

Henry Frederick Baker: An Introduction to the Theory of Multiply Periodic Functions, Cambridge Univ Press 1907 now available as google e-book http://books.google.com/books?id=0EQLAAAAYAAJ Multiply periodic functions, Cambridge Univ Press 2007 [I assume is a re-issue of the same book, perhaps with updates?]

Henry Frederick Baker: Abelian Functions: Abel's Theorem and the Allied Theory, Including the Theory of the Theta Functions. Cambridge, England: Cambridge University Press, 1995 e-book: https://archive.org/details/abeltheoralltheor00bakerich

A more recent book is David Mumford: Curves and their Jacobians, Uni Michigan Press 1975.

A promising-sounding modern re-do of all this is here (online book): http://arxiv.org/abs/1208.0990

But the only source I actually read is this encyc. of maths article: http://www.encyclopediaofmath.org/index.php/Abelian_function

So anyway, if you take any abelian function of order 2 i.e. F1(z1,z2) and F2(z1,z2) and take a generic 1-complex dimensional "cross section" of it, i.e. F1(a*z+b, c*z+d), F2(a*z+b, c*z+d) with a,b,c,d, generic complex constants, then you are going to get a pair of meromorphic functions which are both "quasicrystalline" in nature in the complex z-plane, like the Penrose tiling. For example the locations of the zeros, or poles, both should be sets of points forming a "quasicrystal." If you plotted contourish plots of them, pretty pictures should result. If the zeros were the locations of positive ions and the poles of negative ions presumably this would be a fake quasicrystalline chemical substance.

(CRAZY QUESTION: do these functions have anything to do with the "Painleve transcendents"?)

-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)