![]() For e -| z| smoothing, however, no changes are necessary it works well for almost any divergent series.Revamping ground-breaking ideas from fractal geometry, we propose an alternative micro-supercapacitor configuration realized by laser-induced graphene (LIG) foams produced via laser pyrolysis of inexpensive commercial polymers. For other fractal equations, more radical reworking may be required. For similar equations where only the exponent is changed, replacing the 2 with the correct exponent is often sufficient. It should be noted that the formulas given above generally apply to the classic z 2+ c formula. Like normalized iteration counts, as long as the bailout radius is sufficiently large, the precise value bears little impact on the resulting image. This simply sums e -| z| over all iterations as | z| increases, e -| z| approaches zero and further iterations change the sum very little. The formula for calculating the normalized iteration count isĪn interesting quirk of this formula is that it does not depend on the size of the bailout circle used for matching up normalized iteration counts with the simpler discrete iteration counts, it is more convenient to use the formula :Īs a final distance estimator, we introduce e -| z| smoothing. This direct adaptation can remove the banding from older escape-time images while preserving the rest of the fractal shape. The primary advantage of this algorithm is that it produces isocontours which are the same as the escape-time algorithm, if given the correct parameters, yet it produces continuous values rather than discrete ones. Of particular interest to fractal artists who have many images created with the escape-time algorithm is the normalized iteration count algorithm. Note the banding effect in 2a compared with the continuous coloring of 2b. Discrete (Escape Time) and continuous (Distance Estimation) color algorithms. Most fractal artists do not care about electrostatic potential, but they are very interested in continuous values.įigs. ![]() Provided enough iterations ( n) are done that | z n| is large. If the fractal shape in the complex plane were extended infinitely above and below the plane as a fractal-shaped "cylinder", and treated as a wire that produces an electrical field, then the electrostatic potential for any point z 0 is approximated by Its isocontours are shaped differently than the visible bands from the escape-time algorithm.Īnother variation is the continuous potential algorithm. This value is closer to the Euclidean distance. It represents the distance of every point to the fractal set, calculated with the formula The first historical approach to continuous color values was the distance estimation algorithm. None of the algorithms here pretend to give accurate Euclidean distances, but generally they provide acceptable continuous values. Clearly, the goal has been to develop continuous functions for this distance measurement. Creative use of gradients can actually take advantage of this effect (so-called "tiger striping") but a large number of artists have explored algorithms to hide this effect. Its use of a discrete value (the number of iterations, always an integer) produces a banding effect similar to the contour lines of topographic survey maps. The escape-time algorithm can be considered as a measurement of the (non-Euclidean) distance from any point z 0 to the border of the set.
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