Fractal Gallery
This is a selection of finished, singleedition prints.
All prints are on highquality archival photo paper.
These prints are oneofakind and are for sale. To preserve their
value, I do not provide the parameters, and will never reprint.
Please inquire through the contact button, above.
Pricing
Pricing is calculated linearly based upon print size, taking into account average render time, printing
cost, and the creative effort required for the size. For photo prints, I use basic frames without mat
(just a spacer), so the image goes to the edge of the frame. If a piece is to be shipped, I strongly suggest
it without frame, then you will be able to choose your own framing and mat options to match your decor.
 Prices are as listed for picking up artwork without shipping.
 Add 20% to cover cost of shipping framed photo prints (heavy).
 Add 10% for shipping canvases or unframed photo prints (light).
Stage II  Trigonometric Fractals
After spending some time exploring and developing my fractal application and my style,
I feel that these works have achieved a new artistic level above my initial exploratory works.


 Carousel
 11/7/2020
 20" x 20"
 Photo Print
 Single
 $445

Reminiscent of the lights on a ride around the merrygoround, this is a deep zoom into
negative exponents. Pitting tangent against arctangent under frequencycounting produces
a complicated interaction of lights and bursts.


 Connections
 3/23/2020
 20" x 20"
 TBD
 Single
 $445

Here is the Mandelbrot rendered with a fractional exponent of 1.5 with logarithm applied to both real
and imaginary values. The trigonometric function takes over, creating a pattern just keeps repeating
and connecting the deeper you look.


 Currents
 6/1/2021
 20" x 20"
 Canvas Print
 Single
 $445

It looks like
Currents
in a pool of water. This is a light magnification into a Mandelbrot fractal that was
significantly altered by taking the arctangent of the square of both axes. I added a green tint for a watery feel.


 Curtain Call
 1/17/2020>
 20" x 20"
 Photo Print
 Single
 $445

The logarithm function creates patterns all on its own, so here it is with the exponent set
to near unity, 1.1. Swoops and folds look like curtains draping on a stage in infinite patterns.


 Flight
 6/16/2019
 20" x 20"
 Photo Print
 Single
 $445

There are multiple methods to depict a fractal rendering. This one uses angular cycle counting
to measure how rapidly each point changes. The result was unexpected. This is a fairly deep zoom into
the Negabrot (exponent of 2) using this technique. There is a lot of energy in this image.


 Frost
 1/12/2021
 20" x 20"
 Canvas Print
 Single
 $445

This is a cubed Mandelbrot with a logarithm on imaginary and tangent on real.
These functions interact to create a complex pattern of repeating curves.
Just a touch of tint added to warm it up.


 Magnetic Sands
 12/18/2020
 20" x 20"
 Canvas Print
 Single
 $445

A sandy texture and bubbles of energy create a texture reminiscent of a beach with washedup, dried
seaweed or crustaceans somehow arranged in swoops and swirls. Frequencymapped at a 5 exponent, the
fractal produces quite a lot of detail. This one took a week to render in the background.


 Microcircuitry
 11/11/2020
 20" x 20"
 Canvas Print
 Single
 $445

Creating a Mandelbrot variation with an exponent of one (or unity) seems silly, until you modify it with trigonometry.
The base formula, z=z+c, is fast, allowing the calculation to concentrate on the trig functions. In this image
the logarithm function adds mechanical curves and straight lines, giving the impression of microcircuitry.


 Misty Landscape
 3/6/2020
 20" x 20"
 Photo Print
 Single
 $445

The cosine function tends to generate rippleandwind effects. This is a mild magnification
of a vertical slice just under the exponent of 2, with cosine applied and a linear render method.
Mistcovered mountains await in the background.


 Misty Mountains
 3/1/2021
 20" x 20"
 Photo Print
 Single
 $445

I found this mysterious landscape inside the set of a vertical Multibrot with cosine applied. The
function produces a lot of loops that blend together into cloudy formations and sweeping textures.


 Order
 4/20/2019
 20" x 20"
 Canvas Print
 Single
 $445

An orderly progression of chromed surfaces resulting from a mild zoom into the multibrot set at a fractional
exponent below 2 with an Exponential function applied produces a fairly imposing image. It is somewhat...
Orwellian... I attempt to title my pieces with words that reflect not only what appears to be
there, but how they feel.


 Peacock
 6/1/2021
 20" x 20"
 Canvas Print
 Single
 $445

A touch of color on the blackandwhite pattern appears similar to Peacock feathers. This is the Multibrot
set rendered at a fractional exponent with a logarithm function applied to both real and imaginary axes. The
function tends to “stir” the fractal, giving it a rippled or feathered appearance.


 Song
 6/1/2021
 20" x 20"
 Photo Print
 Single
 $445

Using the same algorithm as Flight, Song is a deeper zoom into the Negabrot (2 exponent) using
angular cyclecounting. It's music to the eyes.


 Splash!
 6/1/2021
 20" x 20"
 Photo Print
 Single
 $445

This is fun. Splash! is a Mandelbrot with sine and cosine applied, along with a bit of color.
These functions are difficult to control, as the ripples that are created do complicate quickly when zooming in.


 Synergy
 4/10/2019
 20" x 20"
 Photo Print
 Single
 $445

Inserting trigonometric functions into the periodic negative exponent
Multibrot space results in smooth, curvilinear lines in complex patterns.
Rendered in greyscale, it looks like the curves are holding in the energy
of this image.


 Turbulence
 9/17/2020
 20" x 20"
 TBD
 Single
 $445

A mild zoom into vertical Multibrot space with cosine variation. This is on the border of the set, so
the overlap of the trigonometric function produces a liquid or glassy texture.
Stage I  Traditional Fractals
These are works from what I would consider an exploratory stage, where I
was building my fractal application and exploring what is possible. This is not to say that exploration
has ceased, just that I feel that my work is starting to mature beyond this stage.


 Amphibian
 10" x 10"
 Photo Print
 Single
 $225

This is a vertical “slice” rotated an odd 70degree angle through Multibrot space,
positioned just below the traditional Mandelbrot set. Is it a school of fish, or a lounge
of lizards? I had to look up what a group of lizards was called. It reminds me of
Escher's “Reptile” tiled etchings, and his choice of subject has multiple levels,
rep tile. This fractal's complexity renders countless creatures.


 Aquifer
 10" x 10"
 Photo Print
 Single
 $225

This is a vertical slice of the multibrot set, just below an
exponent of 1. This area tends to be very fragmented, and
looks like water embedded in rock.


 Arachnoid
 01/06/2019
 12" x 12"
 Photo Print
 Single
 $265

I found this spidery image in a “vertical slice” through multibrot space
at an exponent of around 119. Things get very squished up here, so this had to be
stretched 4,000% horizontally to see the image. The fractal is rendered with a
complex color scheme with both escapetime and periodic rendering through some
trig functions to enhance its leggy details (whew).


 Bamboo Forest
 10/21/2017
 10" x 10"
 Photo Print
 Single
 Sold

This leafy image is a moderate zoom in a vertical slice of the negative space
under the multibrot set, with an exponent between 1 and 2. It is rendered
with periodic rendering, complicated by the extra iterations required for the
smoothing algorithm. The leafy, twiggy result reminds me of bamboo.


 Bluebird
 2/20/2017
 12" x 8"
 Photo Print
 Single
 Sold

A flutter of wings and feathers appears just underneath the
traditional Mandelbrot set in the Multibrot space. The
exponent is rendered vertically in this “Multibrot Slice”
at a 70degree angle from the real axis. “Bluebird” won
“Special Recognition” in the Light, Space, and Time Gallery
2017 “Abstracts” exhibition.


 Braided
 3/20/2015
 12" x 18"
 Photo Print
 Single
 $335

The twisting threads, colors, and textures in this traditional
Mandelbrot magnification remind me of a braided rug. The
color ramps use prime numbers to limit repetition, increasing
the color depth of the resulting fractal.


 Chains
 11/29/2014
 18" x 12"
 Photo Print
 Single
 $335

This is my very first finished piece. It is a deep magnification
into the Mandelbrot set, with a large escape depth setting
and the beginnings of simplified Lyapunov periodic mapping to
produce the gentle, smooth gradients.


 Champagne
 11/29/2016
 1:3 to be printed
 Photo Print
 Single
 $535 (at 12x36)

A “Multbrot” is a variation of the Mandelbrot set
rendered with an exponent other than the traditional “2.”
This is a magnification of the set with a high exponent of 71.
Why 71? Why not? The swirling, bubbly texture lends to its name.
“Champagne” won “Special Merit” in the
Light, Space, and Time Gallery 2017 ”Abstracts“ exhibition.


 Clematis
 2/16/2017
 10" x 10"
 Photo Print
 Single
 $225

You're “not supposed to” render a negabrot with
escapetime mapping. It's “wrong.” That said,
this is a negabrot (a Mandelbrot set with an exponent of 2)
rendered with a combination of escapetime mapping and
simplified Lyapunov periodic mapping. In other words, it's a
“hybrid,” like a lot of plants you can find in a
garden. This image reminds me of its namesake, a climbing,
flowering vine.


 Coriolis
 5/24/2015
 3" x 3" cube
 Photo Print Cube
 Single
 Sold

Here is a deep zoom into the Mandelbrot set. The actual image was renderes
into six squares and mounted on the sides of a cube. The colorful swirls seem
to wrap around the edges.


 Dragon Scales
 9/1/2016
 8" x 12"
 Photo Print
 Single
 Sold

The scaly texture comes from a deep magnification of the
Mandelbrot set. I selected colors reminiscent of traditional
Chinese art. “Dragon Scales” won “Special Merit”
in the Light, Space, and Time Gallery 2016 “Open” exhibition.


 An Evening in the Garden
 06/30/2017
 10" x 10"
 Photo Print
 Single
 $225

This is a moderate zoom into a sinewave variation of the Mandelbrot set.
Traditional escapetime mapping renders it as “snow,” but
periodmapping reveals a whole different picture. Careful adjustment of
the color parameters produces an organic, yet otherworldly feel.


 Equinox
 12/20/2016
 3:1 to be printed
 Photo Print
 Single
 $535 (at 36x12)

Swirls of winter snow, crunchy autumn leaves, and hot summer
breezes are all reflected in this long slice of the Multibrot
set with the exponent rendered vertically just below 1.
Simplified Lyapunov periodic mapping (whew) reveals intricate
details in this area.


 Fern Garden
 06/15/2018
 10" x 10"
 Photo Print
 Single
 $225

This is a sinewave variation of the Mandelbrot set with both
escape and periodic rendering applied.
Overlaying these techniques brings out the periodic nature of
the sine function with the enhanced detail of the escape function.


 Fluidity
 10/24/2016
 3½" x 3½" cube
 Photo Print Cube
 Single
 $65

This deep magnification of the Mandelbrot set uses complex
color ramps to highlight oily, swirly forms. The limits of
the ramps form broad, curving edges, creating overlapping
fluid shapes. This is just one of six renders of this space,
each mounted in the side of a transparent cube.


 Fragments of Imagination
 06/15/2018
 10" x 10"
 Photo Print
 Single
 $225

What is it? It could be a broken stainedglass window, an
abstract bushy tree... Setting the exponent to a fractional
value, in this case around 5.6, produces fragmented patterns.


 Glass #2
 10/28/2017
 10" x 10"
 Photo Print
 Single
 $225

The trig functions normally tend to produce a random mess when applied to the
Mandelbrot formulas. However, since trigonometric functions are basically periodic in
nature, applying periodic mapping can draw out hidden patterns. Using an
arctangent variant of the Mandelbrot set with period mapping applied, a glassy,
transparent image emerges.


 Ice
 3/14/2015
 12" x 18"
 Photo Print
 Single
 Sold

I was challenged to remap “Magma” to create a
frozen, icy effect to contrast the deep, hot look of the
original work. This is the same fractal, with a completely
different effect.


 Incursion
 3/15/2015
 12" x 18"
 Photo Print
 Single
 $335

This is a magnification of the traditional Mandelbrot set with a
tightlyrepeating grayscale color map applied, resulting in a
highlyenergetic pattern.


 Magma
 3/20/2015
 10" x 20"
 Photo Print on Metal
 Single
 $335

This is a deep magnification into the Mandelbrot set. The
various cracks and veins inspired me to give it a hot, dark,
deepearth look. I added the “interference”
capability to my fractal application just for this particular
work to increase the available complexity of the color map.


 Melted
 06/15/2018
 12" x 12"
 Photo Print
 Single
 $265

This is from the sinewave variant of the set. I just love the
glassy, transparent textures that come out in this application.


 Outburst
 11/20/2018
 12" x 12"
 Photo Print
 Single
 Sold

This is a detail of the tower of powers fractal. I have
applied a complex gradient to enhance subtle changes that occur
within the “fronds” of this fractal. As you
zoom in, the interaction of these curves becomes more interesting.


 Periwinkle
 2/16/2017
 8" x 12"
 Photo Print
 Single
 $225

Here is a moderate zoom into the bottomright area of the
Mandelbrot set. I was experimenting with techniques to
render inside the set. The resulting curving
forms produce an iridescence and depth beneath the
complex, branching shapes.


 Ripples
 4/6/2017
 8" x 12"
 Photo Print
 Single
 $225

A variant of the Mandelbrot set takes the absolute value of
the formula on each iteration. The result is very different.
This is a deep magnification of the “Burning Ship”
fractal. It looks like you are looking upward through rippling
water at an old cityscape. This fractal tends to be noisy, so
rendering required 16X supersampling to smooth it out.


 Shoreline
 06/15/2018
 12" x 12"
 Photo Print
 Single
 $265

The burning ship variant of the Mandelbrot set uses the
absolute value (positive value) of the formula, and produces a
rather noisy, turbulent result. Applying periodic rendering to
the inside of the set reveals some interesting patterns. This
image is technically “inside” the set, but periodic
rendering brings out some normally invisible activity.


 Starlight
 5/24/2015
 3" x 3" cube / 14" x 11"
 Photo Print
 2/2
 Sold

Millions of stars glow in tight clusters in this deep
magnification of the Mandelbrot set. This fractal is split
into six separate squares and mounted into a cube frame.


 Storm Surge
 5/23/2015
 3" x 3"
 Photo Print Cube
 Single
 Sold

This is a more traditional rendering of the Mandelbrot set.
There is a lot of energy in the swirling, branching forms.
This is just one rendering of six in this space, mounted
into a cube frame.


 Striped Sky
 01/06/2019
 12" x 12"
 Photo Print
 Single
 265

This was found inside the “Burning Ship” fractal.
Periodic rendering reveals a lot of interesting turbulence in here. Applying
the Julia form in this area produces the repeating, reducing pattern.


 Sunrise
 3/13/2019
 12" x 12"
 Photo Print
 Single
 265

Studies find that the human brain prefers fractal repetition at a
factor of about 1.4. So I tried it. This sunny pattern is a Julia
variant at that exponent. Periodrendering brings out the warm
fire in the middle.


 Sunset on Ice
 12/17/2016
 14" x 11"
 Photo Print
 Single
 285

The fractured nature of this rendering comes from the very
turbulent space between 1 and 2 in the Multibrot set. The
exponent is rendered on the vertical axis, with simplified
Averagevector periodicity mapping to bring out the textures.
This pattern reminds me of ice crystals that form overnight
on the road, so I added a hint of morning sun.