The term fractal now commonly used to define this family of non-differentiable functions that are infinite in length was introduced in the mid 1970s by Benoit Mandelbrot. The term fractal is derived from the Latin adjective FRACTUS, whose corresponding verb FRANGERE means “to break”; a description which well suits the look of these curves.
There are various forms repeated by many including use of the julia sets. The Julia set is named after the French mathematician Gaston Julia who investigated their properties circa 1915 and culminated in his famous paper in 1918.
Since then many programs have been written which have used these forms in conjunction with others and a set of colors to generate images. My first use of fractals in my employment as a digital artist came in 1989 using what is now called the plasma form to generate a height field. The people from the stone soup group at the time also made the color map which used the color registers (numbers) to color the surface based on it’s distance from z=0 which was blue for water at sea level. That program was in beta at the time and later would be calld Fractint. Over the years I used this method a few times but stopped when it started being incorporated into 3-D programs like 3D Studio, Alias-Wavefront, Lightwave, Softimage, Prisms, NeoVisuals and others like Maya and Houdini all of which I’ve used.
The form used for flowers, currently, goes back to at least 2002 and includes at the minimum, the following sequence of Transforms and Variations: Transform 1 – some sort of Blur, often Gaussian Blur – the amount of value varies from say 0.3 to 0.6 generally. The transform weight of transform 1 is often around 0.5-0.7 but that varies also. Transform 2 – a linear and a spherical. Linear higher than the spherical but that is not hard coded or set in stone either. Often Linear is around 0.9 and spherical around 0.15 for instance. A cross variation is added to this transform at a value of 0.01-0.04 once again very subjective. A weight of 15-30 is set for transform 2. This transform is then moved (I for instance used the northeast quadrant close to center) rotated more than 90 degrees and scaled slightly up.
The final transform is often a member of the julia set mentioned above often with values of opposing signs like a minus 2 power and dist of 1. Julia3D is often used for 3D blooms here.
So an overview of this simple base is as follows:
Transform one – a blur
Transform two – linear spherical cross plus more
Final Transform – a member of the Julia set
From there one can add the many various and sundry variations to transform 2 to enhance their particular flower flame. Wedge, butterfly, crackle, fan2, pdj, log, bubble, droste(in apophysis) the various wave variations and others have been used by many since 2002 or from whenever they were written as some didin’t exist then.
Also a third transform can be added to provide highlights for instance a very small amount of Gaussian Blur with a very minimal weight.
My research is from those who graciously shared their knowledge with everyone.It includes but is not limited to Stan Ragets, Ian Anderson, Silvia Codeddo, Susan Wallace and Peter Sbodnov, all of whom have shown this form to us all.
Our great programmers, many in this group, make this all possible with the programs they write like Jwildfire and Apophysis. The flower form itself is universal, meaning that many, many have done things this way. Some have shared their methods and we thank those who have shared with us for doing so. I want to leave this on a positive note so I’ll just say that no one person OWNS this flower fractal form. It doesn’t belong to anyone. That’s like saying someone OWNS 2+2=4 – NOT HAPPENING OKAY – I reference the tutorials and scripts I have used for this and other individuals, as those individuals have shared their knowledge.
The flower shown below is from Stan Ragets tutorial His blurb is on that page at the bottom.
This is just a flower base – much can be added to the second transform – have fun. Also highlights can be added with a third transform like a Gaussian Blur in a small amount with a light transform weight, not added here in this one, however, but possible. Please note this image is not intended as a great work of art but merely a basis for this conversation and starting point for your images.
Sample flame
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