Fractal Friday 2020.02.28

For today's post I'm showing off some the Apple Core Image effects capabilities built into MathPaint, with three Julia set fractals. By the way, MathPaint's release is now close enough for crowdfunding - please see our MathPaint IndieGoGo campaign for details, and a video of the application in action!

For each Julia set in this week's images I'm sharing the original color-gradient fractal, followed by three results applying similar sets of effects.

Here's a spiral Julia set with C = -0.63 + 0.4i, max iterations 500:

Adding vibrancy and posterize effects with MathPaint:

Here's the output of a Morphology Maximum blur, Edges, and Exposure Adjustment effects on the original:

The final example uses a Whitepoint Adjust effect to shift the original image colors (along with Brightness/Saturation/Contrast), then applies a Fourfold Rotated Tile effect:

MathPaint (Trailer #1)

Fractal Friday 2020.02.21

I started this week's Fractal Friday exploration by trying out higher-iteration-cutoff Mandelbrot sets.  Here are a few results with max iterations set to 1000, and higher breakout values than my usual setting of 10.0 - more detail definitely appears, especially in the boundary of the breakout value which becomes much more frond-like. The first one also includes a geometric texture backdrop layered in with MathPaint.





 I also tried the same with Julia sets, but at least for the parameters I was exploring had less luck there because Julia set thresholds seem to be a bit more abrupt. If values escape after 100 iterations, it's not easy to find higher breakout values that will permit higher iterative attempts to reveal more detail around the breakout boundary in the same way as Mandelbrot. Nonetheless I found some cool results, including this spiral that again has a MathPaint geometry-layer background:





I'm including a couple of different takes on the last one. First there's the base Julia Set image with some nice spiral figures. The image that follows uses the same parameters, xy-range and color mapping, but with a real-valued warp function added. The 'warp' is an inverse exponential function applied to the real part of z for each calculation, and the results are pretty cool. I include a couple of extra zooms into this warped fractal below.









I'm going to post some updates on the geometry renderer of MathPaint in the coming days - check back on the blog or follow on Facebook, Twitter, Instagram, or Pinterest!

Bring the noise! MathPaint gets more algorithms, plus templates

The forthcoming MathPaint beta now has a fairly complete noise-rendering feature set. There are several algorithmic noise-generation modes - Perlin, Billow, Ridged, and Voronoi - as well as MathPaint's own random-shape drawing mode.

In addition to the feature, I'm starting to build the collection of templates that will ship with the app as starting points - here's a screenshot of some Perlin noise applied as well as the templates window showing about a dozen noise templates that were added today:



While the templates are all grayscale, any color gradient can be applied to these noise patterns. In the case of the random shape patterns, instead of a gradient there's a pen color and a background color. As with all graphics layers in MathPaint, the layer opacity can be changed to make the noise a semi-transparent overlay of other generated content (or as image export, for use in other graphics programs).

Here are a few output examples from the noise templates, with colors changed:

Cloudy Perlin noise

Ridge-noise 'sandstone'

Gold billow noise

Grassy random segments (angle-constrained)

Chunky Voronoi noise in muted colors

MathPaint now has its own Facebook page you can Like/Follow for more pre-release screenshots and release updates.

Fractal Friday 2020.02.14

Here are some extended Julia Set (z^7) floral fractals for Fractal Friday, Valentine's Day edition - share them with your sweetheart!













This week's images are all zooms of the same set (unchanged function parameters and color mappings).

All images made with MathPaint, the flagship application for generative graphics on Mac OS X currently in development by Mathaesthetics.

You can follow Mathaesthetics on Instagram, Facebook, Twitter and Pinterest for more fractals and other math-generated art.

Fractal Friday 2020.02.07

This week I'm sharing the results of a cool new feature in MathPaint: fractals warped by real-valued functions. The application now supports defining a custom real-valued function that is applied during each iterative evaluation to the real part of the complex number 'z', before z is squared.

The possible effects are endless - from slight skewing to complete warping of the image, but generally the fractal retains its self-similarity and zoomability, as long as the real function doesn't drive it to a premature divergence or convergence.

A variety of approaches produced the images below, including use of a sine function with different powers of the input argument, a step function (a simple round() with some scaling applied), and a natural-log-base exponential function. All are warped Julia sets except for the next-to-last, which is based on Mandelbrot. A variety of MathPaint graphics features are also used such as edge drawing and different forms of color mapping.

More next week! You can follow Mathaesthetics on Instagram, Facebook, Twitter and Pinterest for more cool images and announcements about MathPaint, the app that makes them.