Dhushara

Intrepid Tour Exploding the Dark Heart of Chaos Fractal Geography of the Zeta Function Ising Model and Criticality Taming Riemann's Tiger The Core Concept & the Math Mind Quantum Chaos and the Stadium Billiard Experiments of the Riemann Hypothesis A Dynamical Key to RH Graphical Site Map - click on thumbnails to visit pages.

Fractal and Chaos Research

Software . . . Research

An Intrepid Tour of the Complex Fractal World using
Dark Heart 2.0 Complex Viewer Package for Mac: Download

Exploring chaos through a new integrated 2.0 package depicting the fractal dynamics of virtually every complex function
from simple quadratics,
through polynomials, rational and transcendental functions to Riemann zeta and cutting edge zeta and L-functions.

CA2D - 2D Cellular Automata Application for Mac


Mandelbrot-maps extended function viewer for Android

Fractal Research

  • An Intrepid Tour of the Complex Fractal World using Dark Heart 2.0 Complex Viewer Package
  • The Physics and Computational Exploration of Zeta and L-functions This article presents a spectrum of 4-D global portraits of a diversity of zeta and L-functions, using currently devised numerical methods and explores the implications of these functions in enriching the understanding of diverse areas in physics, from thermodynamics, and phase transitions, through quantum chaos to cosmology. The Riemann hypothesis is explored from both sides of the divide, comparing cases where the hypothesis remains unproven, such as the Riemann zeta function, with cases where it has been proven true, such as Selberg zeta functions.
  • Taming Riemann's Tiger: Has a young Indian i-Phone Programmer Solved the Most Challenging Enigma in Mathematics?
  • A Dynamical Key to the Riemann Hypothesis May 2011 This note sets out a dynamical basis for the non-trivial zeros of the Riemann zeta function being on the critical line x = 1⁄2. It does not prove the Riemann Hypothesis (RH), but it does give a dynamical explanation for why zeta and the Dirichlet L-functions do have their non-trivial zeros on the critical line and why other closely related functions do not. It suggests RH is an additional unprovable postulate of the number system, similar to the axiom of choice, associated with the limiting behavior of the primes as .
  • Fractal Geography of the Zeta Function Mar 2011 http://arxiv.org/abs/1103.5274. The quadratic Mandelbrot set has been referred to as the most complex and beautiful object in mathematics and the Riemann Zeta function takes the prize for the most complicated and enigmatic function.  Here we elucidate the spectrum of Mandelbrot and Julia sets of Zeta, to unearth the geography of its chaotic and fractal diversities, combining these two extremes into one intrepid journey into the deepest abyss of complex function space.
  • Exploding the Dark Heart of Chaos: March 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 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.
  • The 'Core' Concept and the Mathematical Mind 4th Feb 2007 PDF This paper examines a variety of evidence from brain studies of mathematical cognition, from mathematics in early child development, from studies of the gatherer-hunter mind, from a variety of puzzles, games and other human activities, from theories emerging from physical cosmology, and from burgeoning mathematical resources on the internet that suggest that mathematics is more akin to a maze than a focally-based hierarchy, that topology, geometry and dynamics are fundamental to the human mathematical mind.