I was going through my fractal files and asked myself a question to which I don't know the answer: Exactly what am I seeing when I view a fractal? What is the relationship of the display to the <z> axis? Is it possible to rotate any fractal so that any axis becomes any other axis? Or is the computed display all there is? Is it possible to begin parsing with different axes? Basically, imagine a rectangle the ratios of which (x, y, z) are 2:4:1 (of any magnitude). Where on, or in, this rectangle is the position of the fractal? And if it were observed from the smallest axis (z), what would it look like? This has me baffled, and is mind game that is generating a lot of heat. Any answer would be appreciated, but especially one I could understand. Thx David M fisher
In article <D6368981A5754454B8FB422FB07A9ECC@charlie>, "David M Fisher" <sunfish8@verizon.net> writes:
I was going through my fractal files and asked myself a question to which I don't know the answer: Exactly what am I seeing when I view a fractal?
Depends on the fractal. With complex plane iterated formulas, what you are seeing is the visualization of stable orbits as the inside of the set and unstable orbits as the outside of the set. With IFS or L-system fractal types, you're seeing something else. Bifurcation diagrams and some of the other fractal types in fractint (ant, cellular) are different visualizations of chaotic systems.
What is the relationship of the display to the <z> axis?
Nothing standard, really. Some iterated equations in the plane are slices of higher dimensional structures and for those it makes sense to think of the Z axis being something relevant.
Is it possible to rotate any fractal so that any axis becomes any other axis? Or is the computed display all there is? Is it possible to begin parsing with different axes? Basically, imagine a rectangle the ratios of which (x, y, z) are 2:4:1 (of any magnitude). Where on, or in, this rectangle is the position of the fractal? And if it were observed from the smallest axis (z), what would it look like?
Well, most iterated plane equation fractals don't have a volume or a Z, so I'm not sure what you mean by this. You can artificially consider Z to be the number of iterations before behavior (stable or unstable) of the orbit is determined. However, this interpretation of "Z" isn't part of the fractal itself and is just a visualization trick. -- "The Direct3D Graphics Pipeline" -- DirectX 9 draft available for download <http://legalizeadulthood.wordpress.com/the-direct3d-graphics-pipeline/> Legalize Adulthood! <http://legalizeadulthood.wordpress.com>
----- Original Message ----- From: "Richard" <legalize@xmission.com> To: "FractInt Discussion" <fractint@mailman.xmission.com> Sent: Tuesday, 19 October, 2010 22 42 Subject: Re: [Fractint] What?
In article <D6368981A5754454B8FB422FB07A9ECC@charlie>, "David M Fisher" <sunfish8@verizon.net> writes:
I was going through my fractal files and asked myself a question to which I don't know the answer: Exactly what am I seeing when I view a fractal?
Depends on the fractal. With complex plane iterated formulas, what you are seeing is the visualization of stable orbits as the inside of the set and unstable orbits as the outside of the set.
With IFS or L-system fractal types, you're seeing something else. Bifurcation diagrams and some of the other fractal types in fractint (ant, cellular) are different visualizations of chaotic systems.
What is the relationship of the display to the <z> axis?
Nothing standard, really. Some iterated equations in the plane are slices of higher dimensional structures and for those it makes sense to think of the Z axis being something relevant.
Is it possible to rotate any fractal so that any axis becomes any other axis? Or is the computed display all there is? Is it possible to begin parsing with different axes? Basically, imagine a rectangle the ratios of which (x, y, z) are 2:4:1 (of any magnitude). Where on, or in, this rectangle is the position of the fractal? And if it were observed from the smallest axis (z), what would it look like?
Well, most iterated plane equation fractals don't have a volume or a Z, so I'm not sure what you mean by this. You can artificially consider Z to be the number of iterations before behavior (stable or unstable) of the orbit is determined. However, this interpretation of "Z" isn't part of the fractal itself and is just a visualization trick. --
As I understand what you are saying is that the orbits all remain on the x/y plane. There is no escape either out or in, but only up/down or right/left? I understand it is called a plane equation but is this due to it's characteristics or only because that is the manner in which it parsed. In other words, could there be <volume> that isn't represented, and therefore not displayed? And if there is how would you go about rotating the plane to see another angle? David M Fisher _______________________________________________
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In article <9A0023F6C8A24076831CCE1362BCC50F@charlie>, "David M Fisher" <sunfish8@verizon.net> writes:
As I understand what you are saying is that the orbits all remain on the x/y plane.
For iterated functions on the complex plane, yes.
There is no escape either out or in, but only up/down or right/left?
Correct.
I understand it is called a plane equation but is this due to it's characteristics or only because that is the manner in which it parsed.
Its due to its characteristics.
In other words, could there be <volume> that isn't represented, and therefore not displayed?
Some formulas are slices of higher dimensional quantities. For these formulas, you are viewing a slice of a larger volume. That volume may be 3D, or higher dimensionality.
And if there is how would you go about rotating the plane to see another angle?
For the formula types I mention above, the slices are picked by the use of parameters to the formula. Quaternions are 4-dimensional quantities that are analogous to the 2-dimensional complex number. Iterating formulas of quaternions yields fractals that are 4D shapes divided into regions of stable and unstable (diverging) orbits. There are ways to slice these as 3D volumes which are then projected into the 2D image plane of the rendering. -- "The Direct3D Graphics Pipeline" -- DirectX 9 draft available for download <http://legalizeadulthood.wordpress.com/the-direct3d-graphics-pipeline/> Legalize Adulthood! <http://legalizeadulthood.wordpress.com>
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