A collection of computergenerated images, including fractals, chaotic attractors and other mathematicallyderived graphics. Most were created with my own software.


A "breather" surface masquerading as a black hole with its associated quasar. 

A Julia set made using the Collatz conjecture formula, with superimposed fragments from a formula using the secant function. 

An image of a "complex rational map" from Boston University's fractal graphics pioneer Robert Devaney. 

This image is formed completely from chaotic attractors, except for the moth's proboscis, which is a logarithmic spiral, its antennae, which are IFS ferns, and the border, which is a combination of Euler spirals. 

A "flame" attractor, a concept created by Scott Draves, using the "popcorn" formula popularized by Fractint. 

A Julia set inspired by Figure 15 in the classic text The Beauty of Fractals, by HeinzOtto Peitgen and his colleagues at the University of Bremen. 

A Julia set generated by mapping iterated points based on their relation with a superimposed conic section curve  in this instance, a circle. The concept was developed by contemporary fractal artist Kerry Mitchell. 

An inverted Julia set generated from the conic section formula for an ellipse. 

A Julia set generated from the conic section formula for a hyperbola. 

A Julia set using Kerry Mitchell's "Fibonacci" formula. The series used is not actually a Fibonacci series, but rather the product of a sequence of previously iterated values. 

A Julia set with outside coloring using a triangle inequality averaging algorithm. 

A similar Julia set, with inside coloring provided by samples of paintings by Monet. 

An inverted Julia set. 

A raytraced Julia set. 

A rendering of Kuen's surface, which is noteworthy for having constant negative curvature. It was inspired by an exhibit in the Phillips Collection in Washington, D.C., in 2015. It featured mathematical models from the Institut Henri Poincaré in Paris, photographs of the models taken by Man Ray in the 1930's, a series of his paintings based on the models entitled Shakespearean Equations and sculptures by Hiroshi Sugimoto. 

This is a photograph of Sugimoto's sculpture of Kuen's surface. 

A 4dimensional representation of a Lambert series in complex variables using domain coloring. 

A wellknown map of a Lyapunov exponent known as "Zircon City".. 

A zoom into the Mandelbrot set inspired by Map 36 in The Beauty of Fractals. Outside coloring uses a triangle inequality averaging algorithm. 

A zoom into the Mandelbrot set generated using an optimization known as the distance estimation method. 

The Mandelbrot set, showing a group of external rays, which follow the shapes of the triangle inequality averaging. 

The Mandelbrot set, colored by a "fusion" algorithm obtained from Jeff Field involving a normalized "size" value and various trigonometric manipluations. 

A zoom into a modified Mandelbrot set, using the "lambda" formula  a tweak of the standard formula that produces graceful spirals. 

A ray tracing of the Mandelbrot set. 

A zoom into the Mandelbrot set, rendered in Python with a shading algorithm from the "PyPlot" package. 

The famous "nebulabrot" variation of the "Buddhabrot" formula. These fractals are calculated by reiterating only the points that are not in the Mandelbrot set, and assigning incremental color values to those points each time they are hit. 

A fractal generated using Newton's method for finding complex roots. A "relaxation factor" is added to the formula to cause the "twisting" of the image. 

A Newton's method fractal using a more complex polynomial. 

A zoom into the Mandelbrot set, mapped onto a ray traced sphere as a Christmas ornament. 

A polynomial in complex variables using domain coloring. 

A chaotic attractor using a formula from prolific fractalist and author Cliff Pickover. 

Homage to the psychedelic concert posters of the 1960's advertising events at the Fillmore West (San Francisco) and the Fillmore East (NYC). 

Quaternion Julia satellite and moon orbiting a Mandelbrot planet, with a fractal flame nebula in the background. 

A representative Star Wars image mapped to various functions in complex variables using a domain coloring algorithm. 

One of my first ray tracings: a Mandelbrot landscape with the Eta Carinae star complex in the background. 

Inspired by the opening credits of Season 9 of the "modern" Doctor Who series. 

A zoom into the "Tree of Life" Julia set generated with a variation of the standard formula. 

An attractor generated using the VolterraLotka predatorprey differential equations. 