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.

Visualizing the Fractal Dynamics of Riemann Zeta

Zeta, its Mandelbrot set, the root Julia set of c=0 and the Julia set of Newton's method, converging to the zeta zeros.

I have just develped a natty research application for Mac using XCode, which is a state of the art viewer and analytical tool for investigating the notorious Riemann zeta function and its Mandelbrot and Julia sets. You can download and explore this viewer and its source code and related research at:
http://www.dhushara.com/DarkHeart/

Thanks to Mike Lewinski (see comments) who noticed it was not working in Snow Leopard this application and the other three associated fractal applications, including the twin application DHViewer which looks at the dynamics of rational and transcendental complex functions now should all be working in all systems from Tiger to snow Leopard.

RZViewer contains all the bells and whistles in terms of interactive menus check boxes and input and output necessary to manipulate and investigate the Riemann zeta function and its fractal properties and those of a spectrum of related Dirichlet L-functions which share the same enigmatic zeros.


These are very strange functions, unlike the trigonometric, exponential, logarithmic, rational and polynomial functions we are used to dealing with in calculus. They require massive computation and are complicated to program as they are represented as infinite series and products of complex numbers, so it was somewhat of a victory to find that the coding could be done directly using the gcc C library complex.h.

The viewer exposes in ultra-fast code many features of these functions which have not to my knowledge been explored before. Many of the images are intriguingly different from the fractal sets of standard functions we are more used to, although they do share some universal fractal properties.