Mandelbulb3D/Reference/Formulas/Iterations

Iterations
The power of fractal formulas is unlocked through Iterations: repeated calculations of the formulas. The bottom part of the Formulas window provides controls to set the number of iterations per formula and globally. In DECombinate mode, the maximum number of iterations can be set independently for two fractal sequences, known as Hybrid Part 1 and 2.

Iterating an escape time formula produces an orbit. The initial point, P0, is passed through the formula, resulting in a new point P1. This new point is then passed through the formula, resulting in point P2. This process is repeated, resulting in the orbit for P0: P1, P2, P3, ... Pn. For each orbit point, its distance R from the origin is calculated. If P0 is part of the fractal object, all of the orbit points will also be within the fractal object and all of the distances will be relatively small. If not, the orbit points will get further and further from the object and the distances will get larger and larger. The orbit (especially its length) is used to estimate the distance of the initial point from the fractal object.

Iterating a dIFS formula is very different. It produces a sequence of distances, one for each iteration, the smallest of which is taken as the distance of the initial point from the dIFS shape.

Iterationcount
Iterationcount sets the number of iterations for the selected Formula tab, within the current formula sequence.

Mandelbulb3D calculates multiple formulas in series, a sequence determined by the order of formulas in the tabs, numbered 1 through 6. Each formula may be iterated more than once within the sequence. Iterationcount sets the number of times the current formula will iterate before the sequence continues to the next formula.

Formulas are iterated until R becomes greater than the R Bailout value, or Max Iterations is reached.

Min and Max Iterations
Min Iterations and Max Iterations set the number of global iterations, so control the length of the orbit. Min Iterations must be less than Max Iterations, and is the minimum orbit length; this many iterations will be done even if the bailout condition is reached sooner.

Max Iterations is a hard limit on the total number of iterations for all formulas. If Iterationcount is 100, and Max Iterations is 10, Mandelbulb3D will only calculate 10 iterations. For dIFS shapes, the formula will always be iterated this many times. For escape time fractals, if the orbit reaches this length without escaping, the point is considered an inside point.

For DEcombinate hybrids, Max Iterations is replaced by two values: Maxits hybrid part1 and Maxits hybrid part2, which specify the Max Iterations for part 1 and part 2 of the hybrid fractal respectively. (The Min Iterations and R Bailout values are the same for both parts.)

R Bailout
The R Bailout field determines when the orbit of some point "escapes", indicating that the point is not part of the object. If the distance R from an orbit point is less than the R Bailout value, it has not escaped and iteration continues. If it exceeds the R Bailout value, it escapes and iteration stops.

The value for R Bailout must be at least large enough that the orbits for inside points don't incorrectly escape; otherwise the results will be distorted. Making it larger than this minimum value will result in a larger range of orbit sizes for the fractal, giving more control over the coloring. But making it too large will just increase the render time, so set R Bailout large enough to provide the desired coloring, but not larger.

Repeat From Here
The Repeat From Here checkbox specifies a breakpoint in formula tab sequencing. All active formula tabs are processed for the first sequence of calculations. Unless the Max Iterations or R Bailout value have been reached, the sequence repeats. The Repeat From Here checkbox indicates that the current formula is the beginning of all subsequent sequences. On the second and later sequences, calculations are only performed for tabs including and after the tab with Repeat From Here enabled. Tabs before the one with Repeat From Here enabled are ignored.

Disable Analytical DE
The checkbox to Disable Analytical DE will increase the render time of analytical escapetime formulas. This checkbox may be of use if the analytical mode is producing visual artifacts such as noise, but generally the analytical mode is very reliable and should not be disabled.

Outside and Inside Render
The pulldown list at the bottom right of the Formulas window chooses three rendering modes: Outside, Inside, and In and Outside. Fractals such as the Mandelbulb can be viewed from the exterior or the interior. For exterior renderings, choose Outside. For interior renderings, choose Inside. Note that Inside mode will only work if the camera viewpoint is actually inside the fractal object.

The In and Outside mode is generally only applicable to animations in which the camera travels through the surface of the fractal object.