Along the nπ/4 directions there are these interesting recurrent patterns, where as one looks further out they become more rococo. This does have a quasi-crystal sort of appearance. Off axis the pattern becomes more random appearing. Each one of these points defines the zero of the Riemann ς-function.

Phil’s observation about “24” is interesting. In fact the Dedekind η-function appears in this gemish as well. The η-function occurs in density of states calculations of strings. There are aspects of the paper I am not familiar with, so gaps appear pretty quickly in my understanding of this.

]]>Notice how the number 24 comes into the formulas and is related to modular forms. This tells us that these results are also related to physics.

]]>This gets into something I proposed some years ago. There is a curious pattern that emerges for prime numbers when written as

p_j = |x_j +/- y_j|^2

which appears as a complex series of repeating patterns with a rotational symmetry of π/2 and lines of reflection symmetry about π/4. These numbers go by the title of Gauss primes or some moniker. There is a repeating pattern of clustered points. I proposed that this was something like a fractal. However, I was corrected by saying the use of the term fractal was inappropriate, as fractals refer to structures which have infinitesimal structure.

]]>http://motls.blogspot.com/2011/01/bbc-horizon-what-is-reality.html ]]>

Here are the papers, http://www.aimath.org/news/partition/folsom-kent-ono.pdf

]]>