The Bodaciously Excellent Blog of Doctorwhen

Mandelbrot & Julia Sets

The Mandelbrot Set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the quadratic map

{\displaystyle z_{n+1}=z_{n}^{2}+c}

remains bounded. That is, a complex number c is part of the Mandelbrot Set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets.

Images of the Mandelbrot Set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The “style” of this repeating detail depends on the region of the set being examined. The set’s boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

The Mandelbrot Set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization and mathematical beauty.

Each point in the Mandelbrot Set defines a corresponding Julia Set.

The images in this gallery were created by Dr. Ronald Joe Record who holds a Ph.D. in Mathematics from the University of California. Dr. Record’s research focused on applications of Dynamical Systems Theory, an area of Mathematics popularly known as Chaos Theory.

All photography and digital photo manipulation by Ronald Joe Record, Copyright 2019, all rights reserved.

MandelbrotLyapunovIteratedFractalsPhotos

Iterated Systems

These images represent Iterated endomorphisms of the plane and iterated functions systems.. The color represents the frequency with which the trajectory of iterates landed in that pixel’s space.

The images in this gallery were created by Dr. Ronald Joe Record who holds a Ph.D. in Mathematics from the University of California. Dr. Record’s research focused on applications of Dynamical Systems Theory, an area of Mathematics popularly known as Chaos Theory.

All photography and digital photo manipulation by Ronald Joe Record, Copyright 2019, all rights reserved.

MandelbrotLyapunovIteratedFractalsPhotos

Lyapunov Exponents

These images represent the Lyapunov exponents of a forced oscillator modeled by parameterizing the Logistic map. The Lyapunov exponents of a dynamical system can be used to numerically characterize the qualitative behavior of the system.

The images in this gallery were created by Dr. Ronald Joe Record who holds a Ph.D. in Mathematics from the University of California. Dr. Record’s research focused on applications of Dynamical Systems Theory, an area of Mathematics popularly known as Chaos Theory.

All photography and digital photo manipulation by Ronald Joe Record, Copyright 2019, all rights reserved.

MandelbrotLyapunovIteratedFractalsPhotos

Fractal Art

In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension. Here however, we attempt to provide a visual introduction to fractals that does not require any mathematics.

Fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractional dimension, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found innaturetechnologyart, architecture and law. Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals.

The fractals in this gallery were created by Dr. Ronald Joe Record who holds a Ph.D. in Mathematics from the University of California. Dr. Record’s research focused on applications of Dynamical Systems Theory, an area of Mathematics popularly known as Chaos Theory.

All photography and digital photo manipulation by Ronald Joe Record, Copyright 2019, all rights reserved.

MandelbrotLyapunovIteratedFractalsPhotos