question: The equation that is iterated to create the Mandelbrot set.
comment: The Mandelbrot set is a collection of complex numbers, and the picture you commonly see drawn to represent the Mandelbrot set is created by coloring the points on the complex plane corresponding to numbers \*in\* the set black and numbers \*not\* in the set white, or with other colors (see below). I'm gonna assume you're comfortable with complex numbers and the complex plane. Whether or not a number is in the set depends on what happens when you apply a certain mathematical rule based on the number over and over and over again. Does the output from each application of the rule stay near zero? Or does it blast off towards infinity? If it stays near zero then the point is in the Mandelbrot set. The mathematical rule has two parts "start with zero" and "at each step, square whatever number you have and then add the number you're testing for Mandlebrot membership". If we test the number -2, we see it passes: First application of the rule: 0^(2)\+(-2) = -2 Second application of the rule: (-2)^(2)\+(-2) = 2 Third application of the rule (2)^(2)\+(-2)=2 And now we're very clearly in a loop that will repeat forever. The output will keep being 2, staying near zero and not blasting off towards infinity. -2 is in the Mandlebrot set. In the picture, it's the black point at the very far left. If we test the number -1, we see it passes: First application of the rule: 0^(2)\+(-1) = -1 Second application of the rule: (-1)^(2)\+(-1) = 0 Third application of the rule (0)^(2)\+(-1)=1 And now we're very clearly in a loop that will repeat forever. The output will keep oscillating between 0 and -1, staying near zero and not blasting off towards infinity. -1 is in the Mandlebrot set. In the picture, it's in the middle of the circular region left of center. If we test the number 1, we see it fails: First application of the rule: 0^(2)\+(1) = 1 Second application of the rule: (1)^(2)\+(1) = 2 Third application of the rule (2)^(2)\+(1)=5 And now we're very clearly getting away from 0. The next output will be 26, then 677. It will speed off towards infinity. 1 is not in the Madlebrot set. In the standard picture its a white color. Many pictures of the set don't just use black and white, they use a whole spectrum of colors for points not in the set. In these cases, the colors are chosen based on how "quickly" the rule speeds off towards infinity. So you might color "hot" points - points that create rules which grow away from 0 very quickly - with a red color and "cold" points - points which create rules which you have to apply dozens or even hundreds of times before they grow away from 0 - with a blue color and have a whole rainbow in between.
passages: Section::::Other properties.
  
  Section::::Other properties.:Main cardioid and period bulbs.
  
  Upon looking at a picture of the Mandelbrot set, one immediately notices the large cardioid-shaped region in the center. This "main cardioid"
  
  is the region of parameters formula_1 for which formula_36 has an attracting fixed point. It consists of all parameters of the form
  
  for some formula_54 in the open unit disk.
   The simplest algorithm for generating a representation of the Mandelbrot set is known as the "escape time" algorithm. A repeating calculation is performed for each "x", "y" point in the plot area and based on the behavior of that calculation, a color is chosen for that pixel.
   In pseudocode, this algorithm would look as follows. The algorithm does not use complex numbers and manually simulates complex-number operations using two real numbers, for those who do not have a complex data type. The program may be simplified if the programming language includes complex-data-type operations.
  
  Here, relating the pseudocode to formula_1, formula_12 and formula_36:
  
  BULLET::::- formula_100
  
  BULLET::::- formula_101
  
  BULLET::::- formula_102
  
  and so, as can be seen in the pseudocode in the computation of "x" and "y":
  
  BULLET::::- formula_103 and formula_104
   Section::::Generalizations.
  
  Section::::Generalizations.:Multibrot sets.
  
  Multibrot sets are bounded sets found in the complex plane for members of the general monic univariate polynomial family of recursions
   Finally, the computed value is used, eg. as an index to a color palette.
  
  If one has a continuous color palette ranging from 0 to 1 for each color component, then the normalized color of each pixel can be computed as follows, using the variables from above.
  
  This method may be combined with the smooth coloring method below for more aesthetically pleasing images.
  
  Section::::Computer drawings.:Escape time algorithm.:Continuous (smooth) coloring.
   The Mandelbrot set has its place in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. This fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups. On 1 March 1980, at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York, Benoit Mandelbrot first saw a visualization of the set.
   The color of each point represents how quickly the values reached the escape point. Often black is used to show values that fail to escape before the iteration limit, and gradually brighter colors are used for points that escape. This gives a visual representation of how many cycles were required before reaching the escape condition.
   remains bounded. That is, a complex number "c" is part of the Mandelbrot set if, when starting with "z" = 0 and applying the iteration repeatedly, the absolute value of "z" remains bounded however large "n" gets. This can also be represented as
   Section::::Description.:Sections.:"Hamatum".
   The recursive formula for multiplication contains arithmetic expressions involving the operands and their left and right sets, such as the expression formula_7 that appears in the left set of the product of "x" and "y". This is to be understood as the set of surreal numbers resulting from choosing one number from each set that appears in the expression and evaluating the expression on these numbers. (In each individual evaluation of the expression, only one number is chosen from each set, and is substituted in each place where that set appears in the expression.)
   The seahorse "body" is composed by 25 "spokes" consisting of two groups of 12 "spokes" each and one "spoke" connecting to the main cardioid. These two groups can be attributed by some kind of metamorphosis to the two "fingers" of the "upper hand" of the Mandelbrot set; therefore, the number of "spokes" increases from one "seahorse" to the next by 2; the "hub" is a so-called Misiurewicz point. Between the "upper part of the body" and the "tail" a distorted small copy of the Mandelbrot set called satellite may be recognized.
   Section::::Geometry.:Image gallery of a zoom sequence.
  
  The Mandelbrot set shows more intricate detail the closer one looks or magnifies the image, usually called "zooming in". The following example of an image sequence zooming to a selected "c" value gives an impression of the infinite richness of different geometrical structures and explains some of their typical rules.
   Section::::Other properties.:Local connectivity.
  
  It is conjectured that the Mandelbrot set is locally connected. This famous conjecture is known as "MLC" (for "Mandelbrot locally connected"). By the work of Adrien Douady and John H. Hubbard, this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the important "hyperbolicity conjecture" mentioned above.
   In the Blum-Shub-Smale model of real computation, the Mandelbrot set is not computable, but its complement is computably enumerable. However, many simple objects ("e.g.", the graph of exponentiation) are also not computable in the BSS model. At present, it is unknown whether the Mandelbrot set is computable in models of real computation based on computable analysis, which correspond more closely to the intuitive notion of "plotting the set by a computer". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.
  
  Section::::Other properties.:Relationship with Julia sets.
   Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. However, such a component "can" be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below).
   Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point formula_1, whether the sequence formula_7 goes to infinity (in practice — whether it leaves some predetermined bounded neighborhood of 0 after a predetermined number of iterations). Treating the real and imaginary parts of formula_1 as image coordinates on the complex plane, pixels may then be coloured according to how soon the sequence formula_9 crosses an arbitrarily chosen threshold, with a special color (usually black) used for the values of formula_1 for which the sequence has not crossed the threshold after the predetermined number of iterations (this is necessary to clearly distinguish the Mandelbrot set image from the image of its complement). If formula_1 is held constant and the initial value of formula_12—denoted by formula_13—is variable instead, one obtains the corresponding Julia set for each point formula_1 in the parameter space of the simple function.
   The third pass iterates through the NumIterationsPerPixel array and adds up all the stored values, saving them in "total". The array index represents the number of pixels that reached that iteration count before bailout. 
  
  After this, the fourth pass begins and all the values in the IterationCounts array are indexed, and, for each iteration count "i", associated with each pixel, the count is added to a global sum of all the iteration counts from 1 to "i" in the NumIterationsPerPixel array. . This value is then normalized by dividing the sum by the "total" value computed earlier.
   The tricorn (also sometimes called the "Mandelbar") was encountered by Milnor in his study of parameter slices of real cubic polynomials. It is "not" locally connected. This property is inherited by the connectedness locus of real cubic polynomials.
  
  Another non-analytic generalization is the Burning Ship fractal, which is obtained by iterating the following :
  
  Section::::Computer drawings.
  
  There are many programs used to generate the Mandelbrot set and other fractals, some of which are described in fractal-generating software. These programs use a variety of algorithms to determine the color of individual pixels and achieve efficient computation.
  
  Section::::Computer drawings.:Escape time algorithm.
   The nature of what happens in the mapping plane under repeated application of formula_1 depends on where formula_4 (or formula_7) is in the control plane. The "filled Julia set" consists of all points in the mapping plane whose images remain bounded under indefinitely repeated applications of formula_1. The "Mandelbrot set" consists of those points in the control plane such that the associated filled Julia set in the mapping plane is connected.
   However, this only works using discrete colors in the escape time algorithm. It will not work for smooth/continuous coloring.
  
  Section::::Computer drawings.:Optimizations.:Symmetry utilization.
  
  The horizontal symmetry of the Mandelbrot set allows for portions of the rendering process to be skipped upon the presence of the real axis in the final image. However, regardless of the portion that gets mirrored, the same number of points will be rendered.
  
  Section::::Computer drawings.:Optimizations.:Advanced bailout method.
   From a mathematician's point of view, this formula only works in limit where "n" goes to infinity, but very reasonable estimates can be found with just a few additional iterations after the main loop exits.
  
  Once "b" is found, by the Koebe 1/4-theorem, we know that there is no point of the Mandelbrot set with distance from "c" smaller than "b/4".
  
  The distance estimation can be used for drawing of the boundary of the Mandelbrot set, see the article Julia set.
  
  Section::::Computer drawings.:Distance estimates.:Interior distance estimation.
   Very highly magnified images require more than the standard 64–128 or so bits of precision that most hardware floating-point units provide, requiring renderers to use slow "BigNum" or "arbitrary-precision" math libraries to calculate. However, this can be sped up by the exploitation of perturbation theory. Given
  
  as the iteration, and a small epsilon and delta, it is the case that
  
  or
  
  so if one defines
   To get colorful images of the set, the assignment of a color to each value of the number of executed iterations can be made using one of a variety of functions (linear, exponential, etc.). One practical way, without slowing down calculations, is to use the number of executed iterations as an entry to a look-up color palette table initialized at startup. If the color table has, for instance, 500 entries, then the color selection is "n" mod 500, where "n" is the number of iterations.
  
  Section::::Computer drawings.:Escape time algorithm.:Histogram coloring.
   To the left of the main cardioid, attached to it at the point formula_55, a circular-shaped bulb is visible. This bulb consists of those parameters formula_1 for which formula_36 has an attracting cycle of period 2. This set of parameters is an actual circle, namely that of radius 1/4 around −1.
  
  There are infinitely many other bulbs tangent to the main cardioid: for every rational number formula_58, with "p" and "q" coprime, there is such a bulb that is tangent at the parameter
   A period-"q" limb will have "q" − 1 "antennae" at the top of its limb. We can thus determine the period of a given bulb by counting these antennas.
  
  Section::::Geometry.:Pi in the Mandelbrot set.
  