Update

mandelbrot_set.md

@@ -65,7 +65,7 @@ Mandelbrot set (also M-set) is a famous two dimensional [fractal](fractal.md), a
 
 *Simple ASCII rendering of Mandelbrot set, below a zoomed-in detail.*
 
-**Definition**: we use [complex numbers](complex_number.md) to define the set. Consider the following series of complex numbers *z[n]*:
+**Definition**: we use [complex numbers](complex_number.md) to define the set. Take the following series of complex numbers *z[n]*:
 
 *z[0] = 0*, *z[n + 1] = z[n]^2 + p*
 
@@ -73,7 +73,7 @@ Mandelbrot set is the set of all points *p* for which the [absolute value](abs.m
 
 NOTE: here is the series equation rewritten to just work with *x* and *y* coordinates. Checking the point *[px,py]*, your series will be *x[n + 1] = x[n]^2 - y[n]^2 + px* and *y[n + 1] = 2 * x[n] * y[n] + py*.
 
-I.e. taking any point *p* in the complex plane (whose real and imaginary parts we see as the *x* and *y* coordinates), plugging it into the above equation and iterating the series infinitely many times, if the absolute value of *z[n]* stays bounded under some finite value (even very large, just not infinitely large), the number belongs to the set, otherwise not (if the absolute value diverges towards infinity). I.e. in other words the Mandelbrot set is a set of kind of "well behaved" points that don't shoot away to infinity when we keep applying some operation to them over and over. Of course computers cannot evaluate infinitely many iterations of the series so they cannot compute the set 100% accurately, but we may very well [approximate](approximation.md) by performing many iterations (let's 100000) and seeing if the value we get is "very large" (let's say 1000000000) when we stop -- this will work correctly for most points and those few points near the set borders where we make a wrong guess won't really be noticed unless we zoom in very close -- in such cases we can simply perform more iterations to increase precision. To add **[colors](color.md)** to the visualization (so that we don't observe just the borders but also some kind of structure inside and outside of the set) we may simply assign different colors to the points depending e.g. on how big the absolute value is at the time we stop the evaluation, or how many iterations it took for the absolute value to exceed given limit (for points outside the set). Also note that for nice pictures we should apply [antialiasing](antialiasing.md). Additional fancy filters and [shaders](shader.md) such as some kind of postprocessing or fake 3D can also be applied to make the result even more impressive.
+I.e. taking any point *p* in the complex plane (whose real and imaginary parts we see as the *x* and *y* coordinates), plugging it into the above equation and iterating the series infinitely many times, if the absolute value of *z[n]* stays bounded under some finite value (even very large, just not infinitely large), the number belongs to the set, otherwise not (if the absolute value diverges towards infinity). I.e. in other words the Mandelbrot set is a set of kind of "well behaved" points that don't shoot away to infinity by repeated application of a particular equation. Of course computers cannot evaluate infinitely many iterations of the series so they cannot compute the set 100% accurately, but we may very well [approximate](approximation.md) by performing many iterations (let's 100000) and checking if the value we get is "very large" (let's say 1000000000) when we stop -- this will work correctly for most points and those few points near the set borders where we make a wrong guess won't really be noticed unless we zoom in very close -- in such cases we can simply perform more iterations to increase precision. To add **[colors](color.md)** to the visualization (so that we don't observe just the borders but also some kind of structure inside and outside of the set) we may simply assign different colors to the points depending e.g. on how big the absolute value is at the time we stop the evaluation, or how many iterations it took for the absolute value to exceed given limit (for points outside the set). Note also that high quality pictures should apply [antialiasing](antialiasing.md) such as [supersampling](supersampling.md). Additional fancy filters and [shaders](shader.md) such as some kind of postprocessing or fake 3D can also be applied to make the result even more impressive.
 
 There are further **[optimizations](optimization.md)** we may apply to calculate the set faster. The set itself also lies in the circle centered at [0,0] with radius 2, so points outside this area can be safely marked as lying outside the set. Furthermore it's proven that if absolute value of *z[n]* ever gets greater than 2, the point won't lie in the set (because getting absolute value greater than 2 basically means we start checking a point that inevitably lies outside the circle with radius 2 inside which the whole set lies, so we know the point won't lie in the set). A quick [bailout](bailout.md) check (not requiring square roots etc.) here can therefore be e.g. checking if either the real of imaginary component absolute value exceeds 2 (which implies the whole vector length must exceed 2 as well), or checking if the sum of squares of the components exceeds 4 (i.e. we squared both sides of the equation and got rid of the square root). [Symmetry](symmetry.md) of the set can also be used to skip computing some points. Further more complex optimizations exist, based e.g. on estimating distance of any given point to the set border etc.