This is the Behind the Scenes page for Basic Fractal Concepts. Let’s start with information on the fractals. For parameters, click the pastebin link.
This fractal was made in JWildfire using a single transform with the pdj variation.
Made in JWildfire using three transforms, two with julian and one with bubble.
Made in Ultra Fractal using the Julia formula and Basic outside coloring.
Notice the banding that appears outside the fractal; this comes from the fact that the escape time is always an integer: the length of the orbit before the bailout is reached. It is possible to smooth this banding. There is a second layer in the parameters that uses Smooth (Mandelbrot) to do this; to see it just make that layer visible. Just how smoothing works and how to use it is a potential future blog topic. Here is a side-by-side comparison:
Made in Ultra Fractal with the Ducky Plus formula and Statistics inside coloring.
I started with a random Flowers3D (stunning) flame, and tweaked it a lot both to simplify it and make it more attractive. But the basic structure comes from the code that generates those flames.
Made with Mandelbulber, using the Riemann Sphere Msltoe formula.
I wrote a Python program to draw the Koch snowflake: https://pastebin.com/YAxk7bEv. This is a common example of programming recursion. The Koch snowflake is actually made from three Koch curves, one for each line of the triangle. To make the figure showing the development of a Koch snowflake, I interactively ran koch.snowflake with different orders, then combined the figures and added numbers.
Another common way to render a Koch curve is to use a Lindenmayer system or L-system, named for their developer Hungarian biologist and botanist Aristid Lindenmayer. There are several L-system coloring algorithms for Ultra Fractal in mtz.ucl and reb5.ucl that can be used to make Koch curves. It is also relatively easy to use an Iterated Function System (IFS), a type of flame fractal, to make a Koch curve. I’m not elaborating on L-systems and Iterated Function Systems here; those may become future blog posts. But here are links to parameters for those who want to try them out:
L-system Koch curve (Ultra Fractal): https://pastebin.com/ntxtX1da (pictured above)
IFS Koch curve (JWildfire): https://pastebin.com/tpZdffLu