Navigating the Dark Heart of Chaos

I've complemented the Riemann zeta viewer with a suite of Mac XCode applications to visualize a full spectrum of complex functions, from the familiar 'dark heart' of the Mandelbrot set, through polynomials, and rational functions to a variety of transcendental functions.

The dark heart viewer enables exploration of functions with multiple critical points and shows where the higher dimensional variants of the parameter plane (Mandelbrot set) cardioid appear in subtle places in functions like the 'cubic' Cos(z) +c (compare Coz(z)+c below with the cubic z^3-z+c in the above icon) and the 'quartic' c(Cos(z)+Cos(2z))

The most up-to-date downloadable releases of the major XCode applications, which have been tested for both Tiger and Snow Leopard are as follows:

1. Riemann Zeta Viewer: Application - Source - RZ Flight Manual
2. Dark Heart Viewer: Application - Source - DH Flight Manual
3. Wave Function Method Viewer: Application - Source - WF Flight Manual
4. Modified Inverse Iteration Viewer: Application - Source
5. Herman Ring 4D parameter Viewer: Application - Source
6. Eight Critical Point csin(z)/z Viewer: Application - Source
7. Collatz 3n+1 Complex Map Viewer: Application - Source

DHViewer and RZViewer are the twin primary applications covering most of the research examples.

RZViewer deals with all the weird analogues of the Riemann zeta function, while the twin application DHViewer examines a wide variety of rational and transcendental functions. This means that the widest variety of complex functions are explored including some of the most difficult ones to model.

The Wave Function method is a method I invented, which performs effective inverse iteration by forward mapping the domain and colouring by a wave function of the eventual iterated range. As far as I know it is the only method which enables inverse iteration of functions like zeta or compex functions whose inverses cannot be solved explicitly. For comparison see the modified inverse application which works only for the standard map f(z)=z²+c.

The Herman viewer examines the Herman ring map h(z)=cz²(z-r)/(rz-1), which has two complex parameters. Following each parmeter in sequence to give an effective 4D parameter exploration involving whole classes of Mandelbrot sets for the second parameter c.

The remaining XCode viewers and source code cover specific techniques with the standard quadratic function, have also een tested on Tiger and Snow Leopard and can be accessed through links in the papers.

Research Papers

• Exploding the Dark Heart of Chaos: March-December 2009 An exploration of the universality of the cardioid at the centre of the Mandelbrot set extended to the diversity of complex analytic functions with Mac XCode applications and Quicktime movies illustrating each example.
• Experimental Observations on the Riemann Hypothesis and the Collatz Conjecture 22 May 2009 This paper seeks to explore whether the Riemann hypothesis falls into a class of putatively unprovable mathematical conjectures, which arise as a result of unpredictable irregularity and to provide an experimental basis to discover some of the mathematical enigmas surrounding these conjectures.
• Exploring Quantum and Classical Chaos in the Stadium Billiard 9 July 2009 Abstract: This paper explores quantum and classical chaos in the stadium billiard using Matlab simulations to investigate the behavior of wave functions in the stadium and the corresponding classical orbits believed to underlie wave function scarring.
• The Ising Model of Spin Interactions as an Oracle of Self-Organized Criticality, Fractal Mode-Locking and Power Law Statistics in Neurodynamics Aug 2009.

Exceptionally Simple? E8, Golden Ratio and Ising spins

240-vertex graph of the E8 lattice

The exceptional simple group E8 (lattice, lie group) appears to not only be at the centre of most cosmological theories, from Super-string theory to Garrett Lisi's "Exceptionally Simple Theory of Everything". But now it appears that it is also manifest in simple linear magnetism and in a form which is revealed by the golden ratio.

Garrett Lisi's predictions for missing particles in E8 from his 2008 TED seminar.

Having done a simple Matlab simulation looking for the fractal devil's staircase in a model of Isin spins, I was fascinated that a group of physicists have found evidence for E8 symmetries in the Ising chain of a crystal made of cobalt and niobium at 0.04 °C above absolute zero by detecting the Golden ratio.

Source research article: Coldea et. al. 2009 Science 327 177-180 (DOI: 10.1126/science.1180085).

Approaching the quantum critical state where special quantum critical symmetry
would be expected to apply, the two resonances have a ratio converging to the golden.

From the abstract: We realize this system experimentally by using strong transverse magnetic fields to tune the quasi–one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) through its critical point. Spin excitations are observed to change character from pairs of kinks in the ordered phase to spin-flips in the paramagnetic phase. Just below the critical field, the spin dynamics shows a fine structure with two sharp modes at low energies, in a ratio that approaches the golden mean predicted for the first two meson particles of the E8 spectrum.

The evidence for the E8 connection in the magnetism research paper hinges on resonances following the Golden ratio, but the original paper which predicted this by Alexander Zamolodchikov

Int. J. Mod. Phys. A 4, 4235 (1989).

appears to be unobtainable even through university electronic journal access, which does little credit to the International Journal of Modern Physics. Alex just confirmed this "I have the same problem with
accessing the IJMP papers. However I have scanned version (.ps format) from the KEK preprint library, which I attach."

I tried accessing it at KEK but found only some other of his papers, however from the ps he has sent, here is a brief summary of the Ising connection with the Golden ratio and E8.

Due to its basis in spin, Ising magnetic theory can be equated to the field theory of massive Majorana fermions. In the purely elastic scattering theory (PEST) of the field, it turns out that there are six integrals of motion Ps with [M,Ps]=sPs for s=1, 7, 11, 13, 17, 19 (all relatively prime to 30), but there can be no more than two non-trivial integrals of motion unless the first two fermions have masses in the Golden ratio, and then the minimal PEST has 8 stable particles with these two the lightest (hence E8!)

Golden ratio in human proportions. Golden angle 0.618 of a revolution
governs plant growth because it is the most irrational least mode-locked angle.

Now the Golden Ratio is something that appears not only in architecture, the geometry of figures like the pentagram and chaotic fractal processes, but throughout biology in the spiral angles of cacti, pineapples and the sunflower and the relative lengths of our digits, nose to mouth to eyes, navel-to-head and toes.

So what does the Golden Ratio have to do with E8 and how do they both relate to simple linear magnetism and what does it have to do with the centre of the cosmic cyclone? We have a sketch already to the Ising connection, so what about E8 and the Golden ratio?

To try to get this clear I came upon some exceptionally simple forthcoming explanations which I am listing here to try to help make this very arcane subject as clear as possible to others.

The first of these is John Baez's "My Favorite Numbers" 8 which is downloadable in pdf. This gives a beautiful straightforward exposition, starting from the reals and proceeding through complex and quaternions to the octonions, then looking at dense sphere packings in multi-dimensions it arrives naturally at the E8 lattice, while at the same time giving a rationale for super-symmetry in terms of equivalent dimensionalities of vectors and spinors (read here bosons and fermions when thinking of TOEs), mentioning in passing the Golden ratio numbers in terms of the binary icosahedral group and hence E8 in terms of icosians.

The octonions are the 8D basis for E8. Can you see the relationship
between these two representations of the octonian units?

In 2D, circular discs can be packed either in a square grid or a denser hexagonal honeycomb, governed by two transformation groups which preserve the centres. In 3D we also get a cubic and a denser tetrahedral packing by stacking hexagonal ones. As we increase dimension, at 4D the hypercubic grid can double up to form a second one filling the spaces between the first. This gives D4. When we hit 8D there is a second doubling up to give E8 in which each hypersphere meets 240 neighbours. How simple can you get?

However, somehow he leaves out all detail on what spinors are, which Wikipedia treats in an arcane unapproachable way, however "The Nature of Spinors" web page gives a beautiful clear complementary account.


Finally, I found a page that explains beautifully in terms of the Golden ratio construction of an isocahedron from an octahedron, how this leads on to the Golden Ratio construction of E8 based on D4.

Okay now you have a full glint of the phenomenon! ... ?