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* [28]1Escape time algorithm
(BUTTON) Toggle Escape time algorithm subsection
+ [29]1.1Unoptimized naïve escape time algorithm
+ [30]1.2Optimized escape time algorithms
+ [31]1.3Derivative Bailout or "derbail"
* [32]2Coloring algorithms
(BUTTON) Toggle Coloring algorithms subsection
+ [33]2.1Histogram coloring
+ [34]2.2Continuous (smooth) coloring
+ [35]2.3Exponentially mapped and cyclic iterations
+ [36]2.4Passing iterations into a color directly
o [37]2.4.1v refers to a normalized exponentially mapped
cyclic iter count
o [38]2.4.2f(v) refers to the sRGB transfer function
+ [39]2.5HSV coloring
+ [40]2.6LCH coloring
* [41]3Advanced plotting algorithms
(BUTTON) Toggle Advanced plotting algorithms subsection
+ [42]3.1Distance estimates
o [43]3.1.1Exterior distance estimation
o [44]3.1.2Interior distance estimation
+ [45]3.2Cardioid / bulb checking
+ [46]3.3Periodicity checking
+ [47]3.4Border tracing / edge checking
+ [48]3.5Rectangle checking
+ [49]3.6Symmetry utilization
+ [50]3.7Multithreading
+ [51]3.8Perturbation theory and series approximation
* [52]4References
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Plotting algorithms for the Mandelbrot set
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From Wikipedia, the free encyclopedia
Algorithms and methods of plotting the Mandelbrot set on a computing
device
This reads like a textbook's tone or style may not reflect the
[72]encyclopedic tone used on Wikipedia. See Wikipedia's [73]guide to
writing better articles for suggestions. (July 2021) ([74]Learn how and
when to remove this template message)
[75][220px-Fractal-zoom-1-03-Mandelbrot_Buzzsaw.png]
Still image of [76]a movie of increasing magnification on
0.001643721971153 − 0.822467633298876i
[77][220px-Mandelbrot_sequence_new_still.png]
Still image of [78]an animation of increasing magnification
There are many programs and [79]algorithms used to plot the
[80]Mandelbrot set and other [81]fractals, some of which are described
in [82]fractal-generating software. These programs use a variety of
algorithms to determine the color of individual [83]pixels efficiently.
Escape time algorithm[[84]edit]
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.
Unoptimized naïve escape time algorithm[[85]edit]
In both the unoptimized and optimized escape time algorithms, the x and
y locations of each point are used as starting values in a repeating,
or iterating calculation (described in detail below). The result of
each iteration is used as the starting values for the next. The values
are checked during each iteration to see whether they have reached a
critical "escape" condition, or "bailout". If that condition is
reached, the calculation is stopped, the pixel is drawn, and the next
x, y point is examined. For some starting values, escape occurs
quickly, after only a small number of iterations. For starting values
very close to but not in the set, it may take hundreds or thousands of
iterations to escape. For values within the Mandelbrot set, escape will
never occur. The programmer or user must choose how many iterations–or
how much "depth"–they wish to examine. The higher the maximal number of
iterations, the more detail and subtlety emerge in the final image, but
the longer time it will take to calculate the fractal image.
Escape conditions can be simple or complex. Because no complex number
with a real or imaginary part greater than 2 can be part of the set, a
common bailout is to escape when either coefficient exceeds 2. A more
computationally complex method that detects escapes sooner, is to
compute distance from the origin using the [86]Pythagorean theorem,
i.e., to determine the [87]absolute value, or modulus, of the complex
number. If this value exceeds 2, or equivalently, when the sum of the
squares of the real and imaginary parts exceed 4, the point has reached
escape. More computationally intensive rendering variations include the
[88]Buddhabrot method, which finds escaping points and plots their
iterated coordinates.
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.
To render such an image, the region of the complex plane we are
considering is subdivided into a certain number of [89]pixels. To color
any such pixel, let
[MATH: c
{\displaystyle c}
:MATH]
c be the midpoint of that pixel. We now iterate the critical point 0
under
[MATH: P c
{\displaystyle
P_{c}} :MATH]
P_{c} , checking at each step whether the orbit point has modulus
larger than 2. When this is the case, we know that
[MATH: c
{\displaystyle c}
:MATH]
c does not belong to the Mandelbrot set, and we color our pixel
according to the number of iterations used to find out. Otherwise, we
keep iterating up to a fixed number of steps, after which we decide
that our parameter is "probably" in the Mandelbrot set, or at least
very close to it, and color the pixel black.
In [90]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
[91]complex data type. The program may be simplified if the programming
language includes complex-data-type operations.
for each pixel (Px, Py) on the screen do
x0 := scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale
(-2.00, 0.47))
y0 := scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale
(-1.12, 1.12))
x := 0.0
y := 0.0
iteration := 0
max_iteration := 1000
while (x*x + y*y ≤ 2*2 AND iteration < max_iteration) do
xtemp := x*x - y*y + x0
y := 2*x*y + y0
x := xtemp
iteration := iteration + 1
color := palette[iteration]
plot(Px, Py, color)
Here, relating the pseudocode to
[MATH: c
{\displaystyle c}
:MATH]
c ,
[MATH: z
{\displaystyle z}
:MATH]
z and
[MATH: P c
{\displaystyle
P_{c}} :MATH]
P_{c} :
*
[MATH: z =
x + i y
{\displaystyle z=x+iy\ }
:MATH]
{\displaystyle z=x+iy\ }
*
[MATH: z 2 =
x 2
+ 2 i x y
{\displaystyle
z^{2}=x^{2}+2ixy} :MATH]
{\displaystyle z^{2}=x^{2}+2ixy} -
[MATH: y 2
{\displaystyle y^{2}\ }
:MATH]
{\displaystyle y^{2}\ }
*
[MATH: c =
x 0
+ i y 0
{\displaystyle c=x_{0}+iy_{0}\
} :MATH]
{\displaystyle c=x_{0}+iy_{0}\ }
and so, as can be seen in the pseudocode in the computation of x and y:
*
[MATH: x = R e
( z
2
+ c ) =
x 2
− y
2 + x 0
{\displaystyle
x=\mathop {\mathrm {Re} } (z^{2}+c)=x^{2}-y^{2}+x_{0}}
:MATH]
x={\mathop {\mathrm {Re} }}(z^{2}+c)=x^{2}-y^{2}+x_{0} and
[MATH: y = I m
( z
2
+ c ) =
2 x y + y 0 .
{\displaystyle y=\mathop {\mathrm {Im}
} (z^{2}+c)=2xy+y_{0}.\ } :MATH]
y={\mathop {\mathrm {Im} }}(z^{2}+c)=2xy+y_{0}.\
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 [92]palette 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.
Optimized escape time algorithms[[93]edit]
The code in the previous section uses an unoptimized inner while loop
for clarity. In the unoptimized version, one must perform five
multiplications per iteration. To reduce the number of multiplications
the following code for the inner while loop may be used instead:
x2:= 0
y2:= 0
w:= 0
while (x2 + y2 ≤ 4 and iteration < max_iteration) do
x:= x2 - y2 + x0
y:= w - x2 - y2 + y0
x2:= x * x
y2:= y * y
w:= (x + y) * (x + y)
iteration:= iteration + 1
The above code works via some algebraic simplification of the complex
multiplication:
[MATH:
( i y + x
)
2 = −
y 2
+ 2 i y x
+ x 2
=
x 2
− y
2 + 2 i y
x
{\displaystyle
{\begin{aligned}(iy+x)^{2}&=-y^{2}+2iyx+x^{2}\\&=x^{2}-y^{2}+2iyx\end{a
ligned}}} :MATH]
{\displaystyle
{\begin{aligned}(iy+x)^{2}&=-y^{2}+2iyx+x^{2}\\&=x^{2}-y^{2}+2iyx\end{a
ligned}}}
Using the above identity, the number of multiplications can be reduced
to three instead of five.
The above inner while loop can be further optimized by expanding w to
[MATH: w =
x 2
+ 2 x y +
y 2
{\displaystyle
w=x^{2}+2xy+y^{2}} :MATH]
{\displaystyle w=x^{2}+2xy+y^{2}}
Substituting w into
[MATH: y = w
− x 2
− y 2 +
y 0
{\displaystyle
y=w-x^{2}-y^{2}+y_{0}} :MATH]
{\displaystyle y=w-x^{2}-y^{2}+y_{0}} yields
[MATH: y = 2
x y + y 0
{\displaystyle
y=2xy+y_{0}} :MATH]
{\displaystyle y=2xy+y_{0}} and hence calculating w is no longer
needed.
The further optimized pseudocode for the above is:
x2:= 0
y2:= 0
while (x2 + y2 ≤ 4 and iteration < max_iteration) do
y:= 2 * x * y + y0
x:= x2 - y2 + x0
x2:= x * x
y2:= y * y
iteration:= iteration + 1
Note that in the above pseudocode,
[MATH: 2 x y
{\displaystyle 2xy}
:MATH]
2xy seems to increase the number of multiplications by 1, but since 2
is the multiplier the code can be optimized via
[MATH: (
x + x ) y
{\displaystyle (x+x)y}
:MATH]
{\displaystyle (x+x)y} .
Derivative Bailout or "derbail"[[94]edit]
[95][379px-Derbail_method_render.png]
An example of the fine detail possible with the usage of derbail,
rendered with 1024 samples
It is common to check the magnitude of z after every iteration, but
there is another method we can use that can converge faster and reveal
structure within [96]julia sets.
Instead of checking if the [97]magnitude of z after every iteration is
larger than a given value, we can instead check if the sum of each
[98]derivative of z up to the current iteration step is larger than a
given bailout value:
[MATH: z n ′ := ( 2 ∗ z
n − 1
′ ∗
z n −
1 ) +
1 {\displaystyle z_{n}^{\prime
}:=(2*z_{n-1}^{\prime }*z_{n-1})+1} :MATH]
{\displaystyle z_{n}^{\prime }:=(2*z_{n-1}^{\prime }*z_{n-1})+1}
The size of the dbail value can enhance the detail in the structures
revealed within the dbail method with very large values.
It is possible to find derivatives automatically by leveraging
[99]Automatic differentiation and computing the iterations using
[100]Dual numbers.
[101][253px-Example_of_derbail_precision_issues.png]
Hole caused by precision issues
Rendering fractals with the derbail technique can often require a large
number of samples per pixel, as there can be [102]precision issues
which lead to fine detail and can result in [103]noisy images even with
[104]samples in the hundreds or thousands.
Python code:
[105][255px-Derbail.png]
Derbail used on a julia set of the burning ship
def pixel(x: int, y: int, w: int, h: int) -> int:
def magn(a, b):
return a * a + b * b
dbail = 1e6
ratio = w / h
x0 = (((2 * x) / w) - 1) * ratio
y0 = ((2 * y) / h) - 1
dx_sum = 0
dy_sum = 0
iters = 0
limit = 1024
while magn(dx_sum, dy_sum) < dbail and iters < limit:
xtemp = x * x - y * y + x0
y = 2 * x * y + y0
x = xtemp
dx_sum += (dx * x - dy * y) * 2 + 1
dy_sum += (dy * x + dx * y) * 2
iters += 1
return iters
Coloring algorithms[[106]edit]
In addition to plotting the set, a variety of algorithms have been
developed to
* efficiently color the set in an aesthetically pleasing way
* show structures of the data (scientific visualisation)
Histogram coloring[[107]edit]
This section needs additional citations for [108]verification. Relevant
discussion may be found on the [109]talk page. Please help [110]improve
this article by [111]adding citations to reliable sources in this
section. Unsourced material may be challenged and removed. (June 2019)
([112]Learn how and when to remove this template message)
A more complex coloring method involves using a [113]histogram which
pairs each pixel with said pixel's maximum iteration count before
escape/bailout. This method will equally distribute colors to the same
overall area, and, importantly, is independent of the maximum number of
iterations chosen.^[114][1]
This algorithm has four passes. The first pass involves calculating the
iteration counts associated with each pixel (but without any pixels
being plotted). These are stored in an array: IterationCounts[x][y],
where x and y are the x and y coordinates of said pixel on the screen
respectively.
[115]Mandelbrot-no-histogram-coloring-10000-iterations.png
[116]Mandelbrot-no-histogram-coloring-1000-iterations.png
[117]Mandelbrot-no-histogram-coloring-100-iterations.png
[118]Mandelbrot-histogram-10000-iterations.png
[119]Mandelbrot-histogram-1000-iterations.png
[120]Mandelbrot-histogram-100-iterations.png
The top row is a series of plots using the escape time algorithm for
10000, 1000 and 100 maximum iterations per pixel respectively. The
bottom row uses the same maximum iteration values but utilizes the
histogram coloring method. Notice how little the coloring changes per
different maximum iteration counts for the histogram coloring method
plots.
The first step of the second pass is to create an array of size n,
which is the maximum iteration count: NumIterationsPerPixel. Next, one
must iterate over the array of pixel-iteration count pairs,
IterationCounts[][], and retrieve each pixel's saved iteration count,
i, via e.g. i = IterationCounts[x][y]. After each pixel's iteration
count i is retrieved, it is necessary to index the
NumIterationsPerPixel by i and increment the indexed value (which is
initially zero) -- e.g. NumIterationsPerPixel[i] =
NumIterationsPerPixel[i] + 1 .
for (x = 0; x < width; x++) do
for (y = 0; y < height; y++) do
i:= IterationCounts[x][y]
NumIterationsPerPixel[i]++
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.
total: = 0
for (i = 0; i < max_iterations; i++) do
total += NumIterationsPerPixel[i]
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.
hue[][]:= 0.0
for (x = 0; x < width; x++) do
for (y = 0; y < height; y++) do
iteration:= IterationCounts[x][y]
for (i = 0; i <= iteration; i++) do
hue[x][y] += NumIterationsPerPixel[i] / total /* Must be floating-po
int division. */
...
color = palette[hue[m, n]]
...
Finally, the computed value is used, e.g. as an index to a color
palette.
This method may be combined with the smooth coloring method below for
more aesthetically pleasing images.
Continuous (smooth) coloring[[121]edit]
This image was rendered with the escape time algorithm. There are very
obvious "bands" of color
This image was rendered with the normalized iteration count algorithm.
The bands of color have been replaced by a smooth gradient. Also, the
colors take on the same pattern that would be observed if the escape
time algorithm were used.
The escape time algorithm is popular for its simplicity. However, it
creates bands of color, which, as a type of [122]aliasing, can detract
from an image's aesthetic value. This can be improved using an
algorithm known as "normalized iteration count",^[123][2]^[124][3]
which provides a smooth transition of colors between iterations. The
algorithm associates a real number
[MATH: ν
{\displaystyle \nu
} :MATH]
\nu with each value of z by using the connection of the iteration
number with the [125]potential function. This function is given by
[MATH: ϕ ( z )
= lim n
→ ∞
( log
| z n |
/
P n
) ,
{\displaystyle \phi
(z)=\lim _{n\to \infty }(\log |z_{n}|/P^{n}),}
:MATH]
{\displaystyle \phi (z)=\lim _{n\to \infty }(\log
|z_{n}|/P^{n}),}
where z[n] is the value after n iterations and P is the power for which
z is raised to in the Mandelbrot set equation (z[n+1] = z[n]^P + c, P
is generally 2).
If we choose a large bailout radius N (e.g., 10^100), we have that
[MATH: log
|
z
n |
/ P n =
log ( N
)
/ P ν (
z )
{\displaystyle
\log |z_{n}|/P^{n}=\log(N)/P^{\nu (z)}}
:MATH]
{\displaystyle \log |z_{n}|/P^{n}=\log(N)/P^{\nu (z)}}
for some real number
[MATH: ν ( z )
{\displaystyle \nu (z)}
:MATH]
\nu (z) , and this is
[MATH: ν ( z )
= n − log P
( log |
z n
|
/ log ( N )
) ,
{\displaystyle \nu
(z)=n-\log _{P}(\log |z_{n}|/\log(N)),}
:MATH]
{\displaystyle \nu (z)=n-\log _{P}(\log |z_{n}|/\log(N)),}
and as n is the first iteration number such that |z[n]| > N, the number
we subtract from n is in the interval [0, 1).
For the coloring we must have a cyclic scale of colors (constructed
mathematically, for instance) and containing H colors numbered from 0
to H − 1 (H = 500, for instance). We multiply the real number
[MATH: ν ( z )
{\displaystyle \nu (z)}
:MATH]
\nu (z) by a fixed real number determining the density of the colors in
the picture, take the integral part of this number modulo H, and use it
to look up the corresponding color in the color table.
For example, modifying the above pseudocode and also using the concept
of [126]linear interpolation would yield
for each pixel (Px, Py) on the screen do
x0:= scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (
-2.5, 1))
y0:= scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (
-1, 1))
x:= 0.0
y:= 0.0
iteration:= 0
max_iteration:= 1000
// Here N = 2^8 is chosen as a reasonable bailout radius.
while x*x + y*y ≤ (1 << 16) and iteration < max_iteration do
xtemp:= x*x - y*y + x0
y:= 2*x*y + y0
x:= xtemp
iteration:= iteration + 1
// Used to avoid floating point issues with points inside the set.
if iteration < max_iteration then
// sqrt of inner term removed using log simplification rules.
log_zn:= log(x*x + y*y) / 2
nu:= log(log_zn / log(2)) / log(2)
// Rearranging the potential function.
// Dividing log_zn by log(2) instead of log(N = 1<<8)
// because we want the entire palette to range from the
// center to radius 2, NOT our bailout radius.
iteration:= iteration + 1 - nu
color1:= palette[floor(iteration)]
color2:= palette[floor(iteration) + 1]
// iteration % 1 = fractional part of iteration.
color:= linear_interpolate(color1, color2, iteration % 1)
plot(Px, Py, color)
Exponentially mapped and cyclic iterations[[127]edit]
[128][220px-CyclicColoringLch.png]
Exponential Cyclic Coloring in LCH color space with shading
Typically when we render a fractal, the range of where colors from a
given palette appear along the fractal is static. If we desire to
offset the location from the border of the fractal, or adjust their
palette to cycle in a specific way, there are a few simple changes we
can make when taking the final iteration count before passing it along
to choose an item from our palette.
When we have obtained the iteration count, we can make the range of
colors non-linear. Raising a value normalized to the range [0,1] to a
power n, maps a linear range to an exponential range, which in our case
can nudge the appearance of colors along the outside of the fractal,
and allow us to bring out other colors, or push in the entire palette
closer to the border.
[MATH: v = ( ( i / m a
x i
) S N
) 1.5
mod N
{\displaystyle v=((\mathbf {i}
/max_{i})^{\mathbf {S} }\mathbf {N} )^{1.5}{\bmod {\mathbf {N}
}}} :MATH]
{\displaystyle v=((\mathbf {i} /max_{i})^{\mathbf {S} }\mathbf {N}
)^{1.5}{\bmod {\mathbf {N} }}}
where i is our iteration count after bailout, max_i is our iteration
limit, S is the exponent we are raising iters to, and N is the number
of items in our palette. This scales the iter count non-linearly and
scales the palette to cycle approximately proportionally to the zoom.
We can then plug v into whatever algorithm we desire for generating a
color.
Passing iterations into a color directly[[129]edit]
[130][320px-LCH_COLORING.png]
Example of exponentially mapped cyclic LCH coloring without shading
One thing we may want to consider is avoiding having to deal with a
palette or color blending at all. There are actually a handful of
methods we can leverage to generate smooth, consistent coloring by
constructing the color on the spot.
v refers to a normalized exponentially mapped cyclic iter count[[131]edit]
f(v) refers to the [132]sRGB transfer function[[133]edit]
A naive method for generating a color in this way is by directly
scaling v to 255 and passing it into RGB as such
rgb = [v * 255, v * 255, v * 255]
One flaw with this is that RGB is non-linear due to gamma; consider
linear sRGB instead. Going from RGB to sRGB uses an inverse companding
function on the channels. This makes the gamma linear, and allows us to
properly sum the colors for sampling.
srgb = [v * 255, v * 255, v * 255]
[134][320px-HSV_Gradient_Example.png]
HSV Gradient
HSV coloring[[135]edit]
HSV Coloring can be accomplished by mapping iter count from
[0,max_iter) to [0,360), taking it to the power of 1.5, and then modulo
360. [136]HSV Hue Calculation.png We can then simply take the
exponentially mapped iter count into the value and return
hsv = [powf((i / max) * 360, 1.5) % 360, 100, (i / max) * 100]
This method applies to HSL as well, except we pass a saturation of 50%
instead.
hsl = [powf((i / max) * 360, 1.5) % 360, 50, (i / max) * 100]
[137][320px-LCH_Gradient_Example.png]
LCH Gradient
LCH coloring[[138]edit]
One of the most perceptually uniform coloring methods involves passing
in the processed iter count into LCH. If we utilize the exponentially
mapped and cyclic method above, we can take the result of that into the
Luma and Chroma channels. We can also exponentially map the iter count
and scale it to 360, and pass this modulo 360 into the hue.
[MATH:
x ∈
Q +
s i
= ( i
/ m a x i ) x
v =
1.0 − c o
s 2
( π s i ) L
= 75 − ( 75 v ) C
= 28 + ( 75 − 75 v
) H
= ( 360
s i
) 1.5 mod
3 60
{\textstyle
{\begin{array}{lcl}x&\in &\mathbb {Q+} \\s_{i}&=&(i/max_{i})^{\mathbf
{x} }\\v&=&1.0-cos^{2}(\pi
s_{i})\\L&=&75-(75v)\\C&=&28+(75-75v)\\H&=&(360s_{i})^{1.5}{\bmod
{3}}60\end{array}}} :MATH]
{\textstyle {\begin{array}{lcl}x&\in &\mathbb {Q+}
\\s_{i}&=&(i/max_{i})^{\mathbf {x} }\\v&=&1.0-cos^{2}(\pi
s_{i})\\L&=&75-(75v)\\C&=&28+(75-75v)\\H&=&(360s_{i})^{1.5}{\bmod
{3}}60\end{array}}}
One issue we wish to avoid here is out-of-gamut colors. This can be
achieved with a little trick based on the change in in-gamut colors
relative to luma and chroma. As we increase luma, we need to decrease
chroma to stay within gamut.
s = iters/max_i;
v = 1.0 - powf(cos(pi * s), 2.0);
LCH = [75 - (75 * v), 28 + (75 - (75 * v)), powf(360 * s, 1.5) % 360];
Advanced plotting algorithms[[139]edit]
In addition to the simple and slow escape time algorithms already
discussed, there are many other more advanced algorithms that can be
used to speed up the plotting process.
Distance estimates[[140]edit]
One can compute the [141]distance from point c (in [142]exterior or
[143]interior) to nearest point on the [144]boundary of the Mandelbrot
set.^[145][4]^[146][5]
Exterior distance estimation[[147]edit]
The proof of the [148]connectedness of the Mandelbrot set in fact gives
a formula for the [149]uniformizing map of the [150]complement of
[MATH: M
{\displaystyle M}
:MATH]
M (and the [151]derivative of this map). By the [152]Koebe quarter
theorem, one can then estimate the distance between the midpoint of our
[153]pixel and the Mandelbrot set up to a factor of 4.
In other words, provided that the maximal number of iterations is
sufficiently high, one obtains a picture of the Mandelbrot set with the
following properties:
1. Every pixel that contains a point of the Mandelbrot set is colored
black.
2. Every pixel that is colored black is close to the Mandelbrot set.
[154][220px-Demm_2000_Mandelbrot_set.jpg]
Exterior distance estimate may be used to color whole complement of
Mandelbrot set
The upper bound b for the distance estimate of a pixel c (a complex
number) from the Mandelbrot set is given by^[155][6]^[156][7]^[157][8]
[MATH: b =
lim n →
∞ 2 ⋅
|
P
c n ( c )
|
⋅ ln |
P c n ( c )
| |
∂ ∂
c P
c n ( c )
|
,
{\displaystyle b=\lim
_{n\to \infty }{\frac {2\cdot |{P_{c}^{n}(c)|\cdot \ln
|{P_{c}^{n}(c)}}|}{|{\frac {\partial }{\partial
{c}}}P_{c}^{n}(c)|}},} :MATH]
{\displaystyle b=\lim _{n\to \infty }{\frac {2\cdot
|{P_{c}^{n}(c)|\cdot \ln |{P_{c}^{n}(c)}}|}{|{\frac {\partial
}{\partial {c}}}P_{c}^{n}(c)|}},}
where
*
[MATH: P c ( z )
{\displaystyle
P_{c}(z)\,} :MATH]
P_{c}(z)\, stands for [158]complex quadratic polynomial
*
[MATH: P c n ( c )
{\displaystyle
P_{c}^{n}(c)} :MATH]
P_{c}^{n}(c) stands for n iterations of
[MATH: P c ( z ) → z {\displaystyle P_{c}(z)\to
z} :MATH]
P_{c}(z)\to z or
[MATH: z 2 +
c → z
{\displaystyle z^{2}+c\to
z} :MATH]
z^{2}+c\to z , starting with
[MATH: z =
c {\displaystyle z=c}
:MATH]
z=c :
[MATH: P c 0 ( c )
= c {\displaystyle
P_{c}^{0}(c)=c} :MATH]
P_{c}^{0}(c)=c ,
[MATH: P c n + 1
( c ) = P c n ( c ) 2
+ c {\displaystyle
P_{c}^{n+1}(c)=P_{c}^{n}(c)^{2}+c} :MATH]
P_{c}^{n+1}(c)=P_{c}^{n}(c)^{2}+c ;
*
[MATH:
∂ ∂
c P
c n ( c )
{\displaystyle {\frac {\partial
}{\partial {c}}}P_{c}^{n}(c)} :MATH]
{\frac {\partial }{\partial {c}}}P_{c}^{n}(c) is the derivative of
[MATH: P c n ( c )
{\displaystyle
P_{c}^{n}(c)} :MATH]
P_{c}^{n}(c) with respect to c. This derivative can be found by
starting with
[MATH:
∂ ∂
c P
c 0 ( c )
= 1 {\displaystyle {\frac {\partial
}{\partial {c}}}P_{c}^{0}(c)=1} :MATH]
{\frac {\partial }{\partial {c}}}P_{c}^{0}(c)=1 and then
[MATH:
∂ ∂
c P
c n + 1
( c ) = 2 ⋅ P c n ( c )
⋅ ∂ ∂
c
P
c n
( c ) + 1
{\displaystyle {\frac
{\partial }{\partial {c}}}P_{c}^{n+1}(c)=2\cdot {}P_{c}^{n}(c)\cdot
{\frac {\partial }{\partial {c}}}P_{c}^{n}(c)+1}
:MATH]
{\frac {\partial }{\partial {c}}}P_{c}^{n+1}(c)=2\cdot
{}P_{c}^{n}(c)\cdot {\frac {\partial }{\partial {c}}}P_{c}^{n}(c)+1
. This can easily be verified by using the chain rule for the
derivative.
The idea behind this formula is simple: When the [159]equipotential
lines for the potential function
[MATH: ϕ ( z )
{\displaystyle \phi (z)}
:MATH]
\phi (z) lie close, the number
[MATH: | ϕ ′
( z )
|
{\displaystyle |\phi '(z)|}
:MATH]
|\phi '(z)| is large, and conversely, therefore the equipotential lines
for the function
[MATH: ϕ ( z ) / |
ϕ ′ ( z
) | {\displaystyle \phi (z)/|\phi
'(z)|} :MATH]
\phi (z)/|\phi '(z)| should lie approximately regularly.
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 [160]Julia set. In this approach,
pixels that are sufficiently close to M are drawn using a different
color. This creates drawings where the thin "filaments" of the
Mandelbrot set can be easily seen. This technique is used to good
effect in the B&W images of Mandelbrot sets in the books "The Beauty of
Fractals^[161][9]" and "The Science of Fractal Images".^[162][10]
Here is a sample B&W image rendered using Distance Estimates:
[163][220px-Mandel_zoom_06_double_hook_B%26W_DE.jpg]
This is a B&W image of a portion of the Mandelbrot set rendered using
Distance Estimates (DE)
Distance Estimation can also be used to render [164]3D images of
Mandelbrot and Julia sets
Interior distance estimation[[165]edit]
[166][220px-Mandelbrot_Interior_600.png]
Pixels colored according to the estimated interior distance
It is also possible to estimate the distance of a limitly periodic
(i.e., [167]hyperbolic) point to the boundary of the Mandelbrot set.
The upper bound b for the distance estimate is given by^[168][4]
[MATH: b = 1 −
| ∂
∂ z
P
c p
( z
0 ) | 2
| ∂
∂ c
∂ ∂
z P
c p ( z 0 ) + ∂
∂ z
∂ ∂
z P
c p ( z 0 )
∂ ∂
c P
c p ( z 0 ) 1 −
∂ ∂
z P
c p ( z 0 )
| ,
{\displaystyle b={\frac
{1-\left|{{\frac {\partial }{\partial
{z}}}P_{c}^{p}(z_{0})}\right|^{2}}{\left|{{\frac {\partial
}{\partial {c}}}{\frac {\partial }{\partial
{z}}}P_{c}^{p}(z_{0})+{\frac {\partial }{\partial {z}}}{\frac
{\partial }{\partial {z}}}P_{c}^{p}(z_{0}){\frac {{\frac
{\partial }{\partial {c}}}P_{c}^{p}(z_{0})}{1-{\frac {\partial
}{\partial {z}}}P_{c}^{p}(z_{0})}}}\right|}},}
:MATH]
{\displaystyle b={\frac {1-\left|{{\frac {\partial }{\partial
{z}}}P_{c}^{p}(z_{0})}\right|^{2}}{\left|{{\frac {\partial
}{\partial {c}}}{\frac {\partial }{\partial
{z}}}P_{c}^{p}(z_{0})+{\frac {\partial }{\partial {z}}}{\frac
{\partial }{\partial {z}}}P_{c}^{p}(z_{0}){\frac {{\frac
{\partial }{\partial {c}}}P_{c}^{p}(z_{0})}{1-{\frac {\partial
}{\partial {z}}}P_{c}^{p}(z_{0})}}}\right|}},}
where
*
[MATH: p
{\displaystyle
p} :MATH]
p is the period,
*
[MATH: c
{\displaystyle
c} :MATH]
c is the point to be estimated,
*
[MATH: P c ( z )
{\displaystyle P_{c}(z)}
:MATH]
P_{c}(z) is the [169]complex quadratic polynomial
[MATH: P c ( z )
= z
2 + c
{\displaystyle
P_{c}(z)=z^{2}+c} :MATH]
P_{c}(z)=z^{2}+c
*
[MATH: P c p ( z 0 ) {\displaystyle
P_{c}^{p}(z_{0})} :MATH]
P_{c}^{p}(z_{0}) is the
[MATH: p
{\displaystyle
p} :MATH]
p -fold iteration of
[MATH: P c ( z ) → z {\displaystyle P_{c}(z)\to
z} :MATH]
P_{c}(z)\to z , starting with
[MATH: P c 0 ( z )
= z
0 {\displaystyle
P_{c}^{0}(z)=z_{0}} :MATH]
P_{c}^{0}(z)=z_{0}
*
[MATH: z 0
{\displaystyle
z_{0}} :MATH]
z_{0} is any of the
[MATH: p
{\displaystyle
p} :MATH]
p points that make the [170]attractor of the iterations of
[MATH: P c ( z ) → z {\displaystyle P_{c}(z)\to
z} :MATH]
P_{c}(z)\to z starting with
[MATH: P c 0 ( z )
= c {\displaystyle
P_{c}^{0}(z)=c} :MATH]
P_{c}^{0}(z)=c ;
[MATH: z 0
{\displaystyle
z_{0}} :MATH]
z_{0} satisfies
[MATH: z 0 =
P c
p
( z 0 ) {\displaystyle
z_{0}=P_{c}^{p}(z_{0})} :MATH]
z_{0}=P_{c}^{p}(z_{0}) ,
*
[MATH:
∂ ∂
c ∂
∂ z
P
c p
( z
0 ) {\displaystyle {\frac {\partial
}{\partial {c}}}{\frac {\partial }{\partial
{z}}}P_{c}^{p}(z_{0})} :MATH]
{\frac {\partial }{\partial {c}}}{\frac {\partial }{\partial
{z}}}P_{c}^{p}(z_{0}) ,
[MATH:
∂ ∂
z ∂
∂ z
P
c p
( z
0 ) {\displaystyle {\frac {\partial
}{\partial {z}}}{\frac {\partial }{\partial
{z}}}P_{c}^{p}(z_{0})} :MATH]
{\frac {\partial }{\partial {z}}}{\frac {\partial }{\partial
{z}}}P_{c}^{p}(z_{0}) ,
[MATH:
∂ ∂
c P
c p ( z 0 ) {\displaystyle {\frac {\partial
}{\partial {c}}}P_{c}^{p}(z_{0})} :MATH]
{\frac {\partial }{\partial {c}}}P_{c}^{p}(z_{0}) and
[MATH:
∂ ∂
z P
c p ( z 0 ) {\displaystyle {\frac {\partial
}{\partial {z}}}P_{c}^{p}(z_{0})} :MATH]
{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0}) are various
derivatives of
[MATH: P c p ( z )
{\displaystyle
P_{c}^{p}(z)} :MATH]
P_{c}^{p}(z) , evaluated at
[MATH: z 0
{\displaystyle
z_{0}} :MATH]
z_{0} .
Analogous to the exterior case, once b is found, we know that all
points within the distance of b/4 from c are inside the Mandelbrot set.
There are two practical problems with the interior distance estimate:
first, we need to find
[MATH: z 0
{\displaystyle
z_{0}} :MATH]
z_{0} precisely, and second, we need to find
[MATH: p
{\displaystyle p}
:MATH]
p precisely. The problem with
[MATH: z 0
{\displaystyle
z_{0}} :MATH]
z_{0} is that the convergence to
[MATH: z 0
{\displaystyle
z_{0}} :MATH]
z_{0} by iterating
[MATH: P c ( z )
{\displaystyle P_{c}(z)}
:MATH]
P_{c}(z) requires, theoretically, an infinite number of operations. The
problem with any given
[MATH: p
{\displaystyle p}
:MATH]
p is that, sometimes, due to rounding errors, a period is falsely
identified to be an integer multiple of the real period (e.g., a period
of 86 is detected, while the real period is only 43=86/2). In such
case, the distance is overestimated, i.e., the reported radius could
contain points outside the Mandelbrot set.
[171][220px-MandelbrotOrbitInfimum.png]
3D view: smallest absolute value of the orbit of the interior points of
the Mandelbrot set
Cardioid / bulb checking[[172]edit]
One way to improve calculations is to find out beforehand whether the
given point lies within the cardioid or in the period-2 bulb. Before
passing the complex value through the escape time algorithm, first
check that:
[MATH: p = (
x −
1 4 )
2
+ y 2
{\displaystyle p={\sqrt
{\left(x-{\frac {1}{4}}\right)^{2}+y^{2}}}}
:MATH]
p={\sqrt {\left(x-{\frac {1}{4}}\right)^{2}+y^{2}}} ,
[MATH: x ≤
p − 2 p 2 +
1 4
{\displaystyle x\leq
p-2p^{2}+{\frac {1}{4}}} :MATH]
{\displaystyle x\leq p-2p^{2}+{\frac {1}{4}}} ,
[MATH: (
x + 1 )
2 + y 2 ≤
1 16
{\displaystyle (x+1)^{2}+y^{2}\leq
{\frac {1}{16}}} :MATH]
{\displaystyle (x+1)^{2}+y^{2}\leq {\frac {1}{16}}} ,
where x represents the real value of the point and y the imaginary
value. The first two equations determine that the point is within the
cardioid, the last the period-2 bulb.
The cardioid test can equivalently be performed without the square
root:
[MATH: q =
( x − 1 4
) 2 +
y 2
, {\displaystyle q=\left(x-{\frac
{1}{4}}\right)^{2}+y^{2},} :MATH]
{\displaystyle q=\left(x-{\frac {1}{4}}\right)^{2}+y^{2},}
[MATH: q
( q + (
x −
1 4 )
) ≤ 1 4
y
2 .
{\displaystyle
q\left(q+\left(x-{\frac {1}{4}}\right)\right)\leq {\frac
{1}{4}}y^{2}.} :MATH]
{\displaystyle q\left(q+\left(x-{\frac {1}{4}}\right)\right)\leq
{\frac {1}{4}}y^{2}.}
3rd- and higher-order buds do not have equivalent tests, because they
are not perfectly circular.^[173][11] However, it is possible to find
whether the points are within circles inscribed within these
higher-order bulbs, preventing many, though not all, of the points in
the bulb from being iterated.
Periodicity checking[[174]edit]
To prevent having to do huge numbers of iterations for points inside
the set, one can perform periodicity checking, which checks whether a
point reached in iterating a pixel has been reached before. If so, the
pixel cannot diverge and must be in the set. Periodicity checking is a
trade-off, as the need to remember points costs data management
instructions and memory, but saves computational instructions. However,
checking against only one previous iteration can detect many periods
with little performance overhead. For example, within the while loop of
the pseudocode above, make the following modifications:
xold:= 0
yold:= 0
period:= 0
while (x*x + y*y ≤ 2*2 and iteration < max_iteration) do
xtemp:= x*x - y*y + x0
y:= 2*x*y + y0
x:= xtemp
iteration:= iteration + 1
if x ≈ xold and y ≈ yold then
iteration:= max_iteration /* Set to max for the color plotting */
break /* We are inside the Mandelbrot set, leave the while loop *
/
period:= period + 1
if period > 20 then
period:= 0
xold:= x
yold:= y
The above code stores away a new x and y value on every 20^th
iteration, thus it can detect periods that are up to 20 points long.
Border tracing / edge checking[[175]edit]
[176][220px-Mandelbrot_DEM_Sobel.png]
Edge detection using Sobel filter of hyperbolic components of
Mandelbrot set
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Because the Mandelbrot set is a [184]simply connected set,^[185][12]
any point enclosed by a closed shape whose borders lie entirely within
the Mandelbrot set must itself be in the Mandelbrot set. Border tracing
works by following the [186]lemniscates of the various iteration levels
(colored bands) all around the set, and then filling the entire band at
once. This can be a good speed increase, because it means that large
numbers of points can be skipped.^[187][13] Note that border tracing
can't be used to identify bands of pixels outside the set if the plot
computes DE (Distance Estimate) or potential (fractional iteration)
values.
Border tracing is especially beneficial for skipping large areas of a
plot that are parts of the Mandelbrot set (in M), since determining
that a pixel is in M requires computing the maximum number of
iterations.
Below is an example of a Mandelbrot set rendered using border tracing:
This is a 400x400 pixel plot using simple escape-time rendering with a
maximum iteration count of 1000 iterations. It only had to compute
6.84% of the total iteration count that would have been required
without border tracing. It was rendered using a slowed-down rendering
engine to make the rendering process slow enough to watch, and took
6.05 seconds to render. The same plot took 117.0 seconds to render with
border tracing turned off with the same slowed-down rendering engine.
Note that even when the settings are changed to calculate fractional
iteration values (which prevents border tracing from tracing
non-Mandelbrot points) the border tracing algorithm still renders this
plot in 7.10 seconds because identifying Mandelbrot points always
requires the maximum number of iterations. The higher the maximum
iteration count, the more costly it is to identify Mandelbrot points,
and thus the more benefit border tracing provides.
That is, even if the outer area uses smooth/continuous coloring then
border tracing will still speed up the costly inner area of the
Mandelbrot set. Unless the inner area also uses some smooth coloring
method, for instance [188]interior distance estimation.
Rectangle checking[[189]edit]
An older and simpler to implement method than border tracing is to use
rectangles. There are several variations of the rectangle method. All
of them are slower than border tracing because they end up calculating
more pixels.
The basic method is to calculate the border pixels of a box of say 8x8
pixels. If the entire box border has the same color, then just fill in
the 36 pixels (6x6) inside the box with the same color, instead of
calculating them. (Mariani's algorithm.)^[190][14]
A faster and slightly more advanced variant is to first calculate a
bigger box, say 25x25 pixels. If the entire box border has the same
color, then just fill the box with the same color. If not, then split
the box into four boxes of 13x13 pixels, reusing the already calculated
pixels as outer border, and sharing the inner "cross" pixels between
the inner boxes. Again, fill in those boxes that has only one border
color. And split those boxes that don't, now into four 7x7 pixel boxes.
And then those that "fail" into 4x4 boxes. (Mariani-Silver algorithm.)
Even faster is to split the boxes in half instead of into four boxes.
Then it might be optimal to use boxes with a 1.4:1 [191]aspect ratio,
so they can be split like [192]how A3 papers are folded into A4 and A5
papers. (The DIN approach.)
One variant just calculates the corner pixels of each box. However this
causes damaged pictures more often than calculating all box border
pixels. Thus it only works reasonably well if only small boxes of say
6x6 pixels are used, and no recursing in from bigger boxes.
([193]Fractint method.)
As with border tracing, rectangle checking only works on areas with one
discrete color. But even if the outer area uses smooth/continuous
coloring then rectangle checking will still speed up the costly inner
area of the Mandelbrot set. Unless the inner area also uses some smooth
coloring method, for instance [194]interior distance estimation.
Symmetry utilization[[195]edit]
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.
Julia sets have symmetry around the origin. This means that quadrant 1
and quadrant 3 are symmetric, and quadrants 2 and quadrant 4 are
symmetric. Supporting symmetry for both Mandelbrot and Julia sets
requires handling symmetry differently for the two different types of
graphs.
Multithreading[[196]edit]
Escape-time rendering of Mandelbrot and Julia sets lends itself
extremely well to parallel processing. On multi-core machines the area
to be plotted can be divided into a series of rectangular areas which
can then be provided as a set of tasks to be rendered by a pool of
rendering threads. This is an [197]embarrassingly parallel^[198][15]
computing problem. (Note that one gets the best speed-up by first
excluding symmetric areas of the plot, and then dividing the remaining
unique regions into rectangular areas.)^[199][16]
Here is a short video showing the Mandelbrot set being rendered using
multithreading and symmetry, but without boundary following:
This is a short video showing rendering of a Mandelbrot set using
multi-threading and symmetry, but with boundary following turned off.
Finally, here is a video showing the same Mandelbrot set image being
rendered using multithreading, symmetry, and boundary following:
This is a short video showing rendering of a Mandelbrot set using
boundary following, multi-threading, and symmetry
Perturbation theory and series approximation[[200]edit]
Very highly magnified images require more than the standard 64–128 or
so bits of precision that most hardware [201]floating-point units
provide, requiring renderers to use slow "BigNum" or
"[202]arbitrary-precision" math libraries to calculate. However, this
can be sped up by the exploitation of [203]perturbation theory. Given
[MATH: z n + 1
= z n 2
+ c {\displaystyle
z_{n+1}=z_{n}^{2}+c} :MATH]
z_{n+1}=z_{n}^{2}+c
as the iteration, and a small epsilon and delta, it is the case that
[MATH: (
z n
+ ϵ )
2 + ( c + δ ) = z n 2
+ 2 z n ϵ
+ ϵ
2 + c +
δ , {\displaystyle (z_{n}+\epsilon
)^{2}+(c+\delta )=z_{n}^{2}+2z_{n}\epsilon +\epsilon
^{2}+c+\delta ,} :MATH]
{\displaystyle (z_{n}+\epsilon )^{2}+(c+\delta
)=z_{n}^{2}+2z_{n}\epsilon +\epsilon ^{2}+c+\delta ,}
or
[MATH: =
z n +
1 + 2
z n
ϵ + ϵ 2 +
δ , {\displaystyle
=z_{n+1}+2z_{n}\epsilon +\epsilon ^{2}+\delta ,}
:MATH]
{\displaystyle =z_{n+1}+2z_{n}\epsilon +\epsilon ^{2}+\delta ,}
so if one defines
[MATH: ϵ n + 1
= 2 z n
ϵ n
+ ϵ n 2
+ δ , {\displaystyle \epsilon
_{n+1}=2z_{n}\epsilon _{n}+\epsilon _{n}^{2}+\delta
,} :MATH]
{\displaystyle \epsilon _{n+1}=2z_{n}\epsilon _{n}+\epsilon
_{n}^{2}+\delta ,}
one can calculate a single point (e.g. the center of an image) using
high-precision arithmetic (z), giving a reference orbit, and then
compute many points around it in terms of various initial offsets delta
plus the above iteration for epsilon, where epsilon-zero is set to 0.
For most iterations, epsilon does not need more than 16 significant
figures, and consequently hardware floating-point may be used to get a
mostly accurate image.^[204][17] There will often be some areas where
the orbits of points diverge enough from the reference orbit that extra
precision is needed on those points, or else additional local
high-precision-calculated reference orbits are needed. By measuring the
orbit distance between the reference point and the point calculated
with low precision, it can be detected that it is not possible to
calculate the point correctly, and the calculation can be stopped.
These incorrect points can later be re-calculated e.g. from another
closer reference point.
Further, it is possible to approximate the starting values for the
low-precision points with a truncated [205]Taylor series, which often
enables a significant amount of iterations to be skipped.^[206][18]
Renderers implementing these techniques are [207]publicly available and
offer speedups for highly magnified images by around two orders of
magnitude.^[208][19]
An alternate explanation of the above:
For the central point in the disc
[MATH: c
{\displaystyle c}
:MATH]
c and its iterations
[MATH: z n
{\displaystyle
z_{n}} :MATH]
{\displaystyle z_{n}} , and an arbitrary point in the disc
[MATH: c + δ
{\displaystyle c+\delta }
:MATH]
{\displaystyle c+\delta } and its iterations
[MATH: z n ′
{\displaystyle z'_{n}}
:MATH]
{\displaystyle z'_{n}} , it is possible to define the following
iterative relationship:
[MATH: z n ′
= z n +
ϵ n
{\displaystyle
z'_{n}=z_{n}+\epsilon _{n}} :MATH]
{\displaystyle z'_{n}=z_{n}+\epsilon _{n}}
With
[MATH: ϵ 1 =
δ {\displaystyle \epsilon _{1}=\delta
} :MATH]
{\displaystyle \epsilon _{1}=\delta } . Successive iterations of
[MATH: ϵ n
{\displaystyle \epsilon
_{n}} :MATH]
\epsilon _{n} can be found using the following:
[MATH: z n + 1
′ = z n ′
2
+ (
c + δ )
{\displaystyle
z'_{n+1}={z'_{n}}^{2}+(c+\delta )}
:MATH]
{\displaystyle z'_{n+1}={z'_{n}}^{2}+(c+\delta )}
[MATH: z n + 1
′ = ( z n +
ϵ n
) 2 +
c + δ {\displaystyle
z'_{n+1}=(z_{n}+\epsilon _{n})^{2}+c+\delta }
:MATH]
{\displaystyle z'_{n+1}=(z_{n}+\epsilon _{n})^{2}+c+\delta }
[MATH: z n + 1
′ = z n
2
+ c + 2 z
n
ϵ n
+
ϵ n
2
+ δ
{\displaystyle
z'_{n+1}={z_{n}}^{2}+c+2z_{n}\epsilon _{n}+{\epsilon
_{n}}^{2}+\delta } :MATH]
{\displaystyle z'_{n+1}={z_{n}}^{2}+c+2z_{n}\epsilon
_{n}+{\epsilon _{n}}^{2}+\delta }
[MATH: z n + 1
′ = z n + 1
+ 2 z n
ϵ n
+
ϵ n
2
+ δ
{\displaystyle
z'_{n+1}=z_{n+1}+2z_{n}\epsilon _{n}+{\epsilon _{n}}^{2}+\delta
} :MATH]
{\displaystyle z'_{n+1}=z_{n+1}+2z_{n}\epsilon _{n}+{\epsilon
_{n}}^{2}+\delta }
Now from the original definition:
[MATH: z n + 1
′ = z n + 1
+ ϵ n + 1
{\displaystyle
z'_{n+1}=z_{n+1}+\epsilon _{n+1}}
:MATH]
{\displaystyle z'_{n+1}=z_{n+1}+\epsilon _{n+1}} ,
It follows that:
[MATH: ϵ n + 1
= 2 z n
ϵ n
+
ϵ n
2
+ δ
{\displaystyle \epsilon
_{n+1}=2z_{n}\epsilon _{n}+{\epsilon _{n}}^{2}+\delta
} :MATH]
{\displaystyle \epsilon _{n+1}=2z_{n}\epsilon _{n}+{\epsilon
_{n}}^{2}+\delta }
As the iterative relationship relates an arbitrary point to the central
point by a very small change
[MATH: δ
{\displaystyle \delta
} :MATH]
\delta , then most of the iterations of
[MATH: ϵ n
{\displaystyle \epsilon
_{n}} :MATH]
\epsilon _{n} are also small and can be calculated using floating point
hardware.
However, for every arbitrary point in the disc it is possible to
calculate a value for a given
[MATH: ϵ n
{\displaystyle \epsilon
_{n}} :MATH]
{\displaystyle \epsilon _{n}} without having to iterate through the
sequence from
[MATH: ϵ 0
{\displaystyle \epsilon
_{0}} :MATH]
\epsilon_0 , by expressing
[MATH: ϵ n
{\displaystyle \epsilon
_{n}} :MATH]
\epsilon _{n} as a power series of
[MATH: δ
{\displaystyle \delta
} :MATH]
\delta .
[MATH: ϵ n =
A n
δ + B n
δ 2
+ C n
δ 3
+ … {\displaystyle \epsilon
_{n}=A_{n}\delta +B_{n}\delta ^{2}+C_{n}\delta ^{3}+\dotsc
} :MATH]
{\displaystyle \epsilon _{n}=A_{n}\delta +B_{n}\delta
^{2}+C_{n}\delta ^{3}+\dotsc }
With
[MATH: A 1 =
1 , B
1 = 0 ,
C 1
= 0 , …
{\displaystyle
A_{1}=1,B_{1}=0,C_{1}=0,\dotsc } :MATH]
{\displaystyle A_{1}=1,B_{1}=0,C_{1}=0,\dotsc } .
Now given the iteration equation of
[MATH: ϵ
{\displaystyle \epsilon
} :MATH]
\epsilon , it is possible to calculate the coefficients of the power
series for each
[MATH: ϵ n
{\displaystyle \epsilon
_{n}} :MATH]
\epsilon _{n} :
[MATH: ϵ n + 1
= 2 z n
ϵ n
+
ϵ n
2
+ δ
{\displaystyle \epsilon
_{n+1}=2z_{n}\epsilon _{n}+{\epsilon _{n}}^{2}+\delta
} :MATH]
{\displaystyle \epsilon _{n+1}=2z_{n}\epsilon _{n}+{\epsilon
_{n}}^{2}+\delta }
[MATH: ϵ n + 1
= 2 z n ( A n δ
+ B
n δ 2 +
C n
δ
3 + … ) + (
A n
δ + B n
δ 2
+ C n
δ 3
+ … )
2
+ δ {\displaystyle \epsilon
_{n+1}=2z_{n}(A_{n}\delta +B_{n}\delta ^{2}+C_{n}\delta
^{3}+\dotsc )+(A_{n}\delta +B_{n}\delta ^{2}+C_{n}\delta
^{3}+\dotsc )^{2}+\delta } :MATH]
{\displaystyle \epsilon _{n+1}=2z_{n}(A_{n}\delta +B_{n}\delta
^{2}+C_{n}\delta ^{3}+\dotsc )+(A_{n}\delta +B_{n}\delta
^{2}+C_{n}\delta ^{3}+\dotsc )^{2}+\delta }
[MATH: ϵ n + 1
= (
2 z
n A n +
1 ) δ + ( 2 z n
B n
+
A n
2
) δ
2
+ ( 2
z n
C
n + 2
A n
B
n )
δ 3
+ … {\displaystyle \epsilon
_{n+1}=(2z_{n}A_{n}+1)\delta +(2z_{n}B_{n}+{A_{n}}^{2})\delta
^{2}+(2z_{n}C_{n}+2A_{n}B_{n})\delta ^{3}+\dotsc }
:MATH]
{\displaystyle \epsilon _{n+1}=(2z_{n}A_{n}+1)\delta
+(2z_{n}B_{n}+{A_{n}}^{2})\delta
^{2}+(2z_{n}C_{n}+2A_{n}B_{n})\delta ^{3}+\dotsc }
Therefore, it follows that:
[MATH: A n + 1
= 2 z n
A n
+ 1 {\displaystyle
A_{n+1}=2z_{n}A_{n}+1} :MATH]
{\displaystyle A_{n+1}=2z_{n}A_{n}+1}
[MATH: B n + 1
= 2 z n
B n
+
A n
2
{\displaystyle
B_{n+1}=2z_{n}B_{n}+{A_{n}}^{2}}
:MATH]
{\displaystyle B_{n+1}=2z_{n}B_{n}+{A_{n}}^{2}}
[MATH: C n + 1
= 2 z n
C n
+ 2 A n
B n
{\displaystyle
C_{n+1}=2z_{n}C_{n}+2A_{n}B_{n}}
:MATH]
{\displaystyle C_{n+1}=2z_{n}C_{n}+2A_{n}B_{n}}
[MATH: ⋮
{\displaystyle
\vdots } :MATH]
\vdots
The coefficients in the power series can be calculated as iterative
series using only values from the central point's iterations
[MATH: z
{\displaystyle z}
:MATH]
z , and do not change for any arbitrary point in the disc. If
[MATH: δ
{\displaystyle \delta
} :MATH]
\delta is very small,
[MATH: ϵ n
{\displaystyle \epsilon
_{n}} :MATH]
\epsilon _{n} should be calculable to sufficient accuracy using only a
few terms of the power series. As the Mandelbrot Escape Contours are
'continuous' over the complex plane, if a points escape time has been
calculated, then the escape time of that points neighbours should be
similar. Interpolation of the neighbouring points should provide a good
estimation of where to start in the
[MATH: ϵ n
{\displaystyle \epsilon
_{n}} :MATH]
\epsilon _{n} series.
Further, separate interpolation of both real axis points and imaginary
axis points should provide both an upper and lower bound for the point
being calculated. If both results are the same (i.e. both escape or do
not escape) then the difference
[MATH: Δ
n {\displaystyle \Delta n}
:MATH]
{\displaystyle \Delta n} can be used to recuse until both an upper and
lower bound can be established. If floating point hardware can be used
to iterate the
[MATH: ϵ
{\displaystyle \epsilon
} :MATH]
\epsilon series, then there exists a relation between how many
iterations can be achieved in the time it takes to use BigNum software
to compute a given
[MATH: ϵ n
{\displaystyle \epsilon
_{n}} :MATH]
\epsilon _{n} . If the difference between the bounds is greater than
the number of iterations, it is possible to perform binary search using
BigNum software, successively halving the gap until it becomes more
time efficient to find the escape value using floating point hardware.
References[[209]edit]
1. [210]^ [211]"Newbie: How to map colors in the Mandelbrot set?".
www.fractalforums.com. May 2007. [212]Archived from the original on
9 September 2022. Retrieved 11 February 2020.
2. [213]^ García, Francisco; Ángel Fernández; Javier Barrallo; Luis
Martín. [214]"Coloring Dynamical Systems in the Complex Plane"
(PDF). [215]Archived (PDF) from the original on 30 November 2019.
Retrieved 21 January 2008. {{[216]cite journal}}: Cite journal
requires |journal= ([217]help)
3. [218]^ Linas Vepstas. [219]"Renormalizing the Mandelbrot Escape".
[220]Archived from the original on 14 February 2020. Retrieved 11
February 2020.
4. ^ [221]^a [222]^b Albert Lobo. [223]"Interior and exterior distance
bounds for the Mandelbrot set". [224]Archived from the original on
9 September 2022. Retrieved 29 April 2021.
5. [225]^ Wilson, Dr. Lindsay Robert (2012). [226]"Distance estimation
method for drawing Mandelbrot and Julia sets" (PDF). [227]Archived
(PDF) from the original on 3 May 2021. Retrieved 3 May 2021.
6. [228]^ [229]Chéritat, Arnaud (2016). [230]"Boundary detection
methods via distance estimators". [231]Archived from the original
on 18 December 2022. Retrieved 2 January 2023.
7. [232]^ Christensen, Mikael Hvidtfeldt (2011). [233]"Distance
Estimated 3D Fractals (V): The Mandelbulb & Different DE
Approximations". [234]Archived from the original on 13 May 2021.
Retrieved 10 May 2021.
8. [235]^ Dang, Yumei; [236]Louis Kauffman; [237]Daniel Sandin (2002).
"Chapter 3.3: The Distance Estimation Formula". [238]Hypercomplex
Iterations: Distance Estimation and Higher Dimensional Fractals
(PDF). World Scientific. pp. 17–18. [239]Archived (PDF) from the
original on 23 March 2021. Retrieved 29 April 2021.
9. [240]^ Peitgen, Heinz-Otto; Richter Peter (1986). [241]The Beauty
of Fractals. Heidelberg: Springer-Verlag.
[242]ISBN [243]0-387-15851-0.
10. [244]^ Peitgen, Heinz-Otto; Saupe Dietmar (1988). [245]The Science
of Fractal Images. New York: Springer-Verlag. p. 202.
[246]ISBN [247]0-387-96608-0.
11. [248]^ [249]"Mandelbrot Bud Maths". [250]Archived from the original
on 14 February 2020. Retrieved 11 February 2020.
12. [251]^ Douady, Adrien; Hubbard, John (2009). [252]"Exploring the
Mandelbrot set. Exploring the Mandelbrot set. The Orsay Notes".
Retrieved 9 April 2023. {{[253]cite journal}}: Cite journal
requires |journal= ([254]help)
13. [255]^ [256]"Boundary Tracing Method". Archived from [257]the
original on 20 February 2015.
14. [258]^ Dewdney, A. K. (1989). "Computer Recreations, February 1989;
A tour of the Mandelbrot set aboard the Mandelbus". Scientific
American. p. 111. [259]JSTOR [260]24987149. (subscription required)
15. [261]^
[262]http://courses.cecs.anu.edu.au/courses/COMP4300/lectures/embPa
rallel.4u.pdf [263]Archived 27 January 2020 at the [264]Wayback
Machine^[[265]bare URL PDF]
16. [266]^
[267]http://cseweb.ucsd.edu/groups/csag/html/teaching/cse160s05/lec
tures/Lecture14.pdf [268]Archived 26 January 2020 at the
[269]Wayback Machine^[[270]bare URL PDF]
17. [271]^ [272]"Superfractalthing - Arbitrary Precision Mandelbrot Set
Rendering in Java". [273]Archived from the original on 30 June
2020. Retrieved 11 February 2020.
18. [274]^ K. I. Martin. [275]"Superfractalthing Maths" (PDF). Archived
from [276]the original (PDF) on 28 June 2014. Retrieved 11 February
2020. {{[277]cite journal}}: Cite journal requires |journal=
([278]help)
19. [279]^ [280]"Kalles Fraktaler 2". [281]Archived from the original
on 24 February 2020. Retrieved 11 February 2020.
* [282]v
* [283]t
* [284]e
[285]Fractals
Characteristics
* [286]Fractal dimensions
+ [287]Assouad
+ [288]Box-counting
o [289]Higuchi
+ [290]Correlation
+ [291]Hausdorff
+ [292]Packing
+ [293]Topological
* [294]Recursion
* [295]Self-similarity
[296]Iterated function
system
* [297]Barnsley fern
* [298]Cantor set
* [299]Koch snowflake
* [300]Menger sponge
* [301]Sierpinski carpet
* [302]Sierpinski triangle
* [303]Apollonian gasket
* [304]Fibonacci word
* [305]Space-filling curve
+ [306]Blancmange curve
+ [307]De Rham curve
o [308]Minkowski
+ [309]Dragon curve
+ [310]Hilbert curve
+ [311]Koch curve
+ [312]Lévy C curve
+ [313]Moore curve
+ [314]Peano curve
+ [315]Sierpiński curve
+ [316]Z-order curve
* [317]String
* [318]T-square
* [319]n-flake
* [320]Vicsek fractal
* [321]Hexaflake
* [322]Gosper curve
* [323]Pythagoras tree
* [324]Weierstrass function
[325]Strange attractor
* [326]Multifractal system
[327]L-system
* [328]Fractal canopy
* [329]Space-filling curve
+ [330]H tree
[331]Escape-time
fractals
* [332]Burning Ship fractal
* [333]Julia set
+ [334]Filled
+ [335]Newton fractal
+ [336]Douady rabbit
* [337]Lyapunov fractal
* [338]Mandelbrot set
+ [339]Misiurewicz point
* [340]Multibrot set
* [341]Newton fractal
* [342]Tricorn
* [343]Mandelbox
* [344]Mandelbulb
[345]Rendering techniques
* [346]Buddhabrot
* [347]Orbit trap
* [348]Pickover stalk
[349]Random fractals
* [350]Brownian motion
+ [351]Brownian tree
+ [352]Brownian motor
* [353]Fractal landscape
* [354]Lévy flight
* [355]Percolation theory
* [356]Self-avoiding walk
People
* [357]Michael Barnsley
* [358]Georg Cantor
* [359]Bill Gosper
* [360]Felix Hausdorff
* [361]Desmond Paul Henry
* [362]Gaston Julia
* [363]Helge von Koch
* [364]Paul Lévy
* [365]Aleksandr Lyapunov
* [366]Benoit Mandelbrot
* [367]Hamid Naderi Yeganeh
* [368]Lewis Fry Richardson
* [369]Wacław Sierpiński
Other
* "[370]How Long Is the Coast of Britain?"
+ [371]Coastline paradox
* [372]Fractal art
* [373]List of fractals by Hausdorff dimension
* [374]The Fractal Geometry of Nature (1982 book)
* [375]The Beauty of Fractals (1986 book)
* [376]Chaos: Making a New Science (1987 book)
* [377]Kaleidoscope
* [378]Chaos theory
* [379]v
* [380]t
* [381]e
[382]Mathematics and art
Concepts
* [383]Algorithm
* [384]Catenary
* [385]Fractal
* [386]Golden ratio
* [387]Hyperboloid structure
* [388]Minimal surface
* [389]Paraboloid
* [390]Perspective
+ [391]Camera lucida
+ [392]Camera obscura
* [393]Plastic number
* [394]Projective geometry
* Proportion
+ [395]Architecture
+ [396]Human
* [397]Symmetry
* [398]Tessellation
* [399]Wallpaper group
[400]Fibonacci word: detail of artwork by Samuel Monnier, 2009
Forms
* [401]Algorithmic art
* [402]Anamorphic art
* [403]Architecture
+ [404]Geodesic dome
+ [405]Islamic
+ [406]Mughal
+ [407]Pyramid
+ [408]Vastu shastra
* [409]Computer art
* [410]Fiber arts
* [411]4D art
* [412]Fractal art
* [413]Islamic geometric patterns
+ [414]Girih
+ [415]Jali
+ [416]Muqarnas
+ [417]Zellij
* [418]Knotting
+ [419]Celtic knot
+ [420]Croatian interlace
+ [421]Interlace
* [422]Music
* [423]Origami
* [424]Sculpture
* [425]String art
* [426]Tiling
Artworks
* [427]List of works designed with the golden ratio
* [428]Continuum
* [429]Mathemalchemy
* [430]Mathematica: A World of Numbers... and Beyond
* [431]Octacube
* [432]Pi
* [433]Pi in the Sky
[434]Buildings
* [435]Cathedral of Saint Mary of the Assumption
* [436]Hagia Sophia
* [437]Pantheon
* [438]Parthenon
* [439]Pyramid of Khufu
* [440]Sagrada Família
* [441]Sydney Opera House
* [442]Taj Mahal
[443]Artists
Renaissance
* [444]Paolo Uccello
* [445]Piero della Francesca
* [446]Leonardo da Vinci
+ [447]Vitruvian Man
* [448]Albrecht Dürer
* [449]Parmigianino
+ [450]Self-portrait in a Convex Mirror
19th–20th
Century
* [451]William Blake
+ [452]The Ancient of Days
+ [453]Newton
* [454]Jean Metzinger
+ [455]Danseuse au café
+ [456]L'Oiseau bleu
* [457]Giorgio de Chirico
* [458]Man Ray
* [459]M. C. Escher
+ [460]Circle Limit III
+ [461]Print Gallery
+ [462]Relativity
+ [463]Reptiles
+ [464]Waterfall
* [465]René Magritte
+ [466]La condition humaine
* [467]Salvador Dalí
+ [468]Crucifixion
+ [469]The Swallow's Tail
* [470]Crockett Johnson
Contemporary
* [471]Max Bill
* [472]Martin and [473]Erik Demaine
* [474]Scott Draves
* [475]Jan Dibbets
* [476]John Ernest
* [477]Helaman Ferguson
* [478]Peter Forakis
* [479]Susan Goldstine
* [480]Bathsheba Grossman
* [481]George W. Hart
* [482]Desmond Paul Henry
* [483]Anthony Hill
* [484]Charles Jencks
+ [485]Garden of Cosmic Speculation
* [486]Andy Lomas
* [487]Robert Longhurst
* [488]Jeanette McLeod
* [489]Hamid Naderi Yeganeh
* [490]István Orosz
* [491]Hinke Osinga
* [492]Antoine Pevsner
* [493]Tony Robbin
* [494]Alba Rojo Cama
* [495]Reza Sarhangi
* [496]Oliver Sin
* [497]Hiroshi Sugimoto
* [498]Daina Taimiņa
* [499]Roman Verostko
* [500]Margaret Wertheim
Theorists
Ancient
* [501]Polykleitos
+ Canon
* [502]Vitruvius
+ [503]De architectura
Renaissance
* [504]Filippo Brunelleschi
* [505]Leon Battista Alberti
+ [506]De pictura
+ [507]De re aedificatoria
* [508]Piero della Francesca
+ [509]De prospectiva pingendi
* [510]Luca Pacioli
+ [511]De divina proportione
* [512]Leonardo da Vinci
+ [513]A Treatise on Painting
* [514]Albrecht Dürer
+ Vier Bücher von Menschlicher Proportion
* [515]Sebastiano Serlio
+ Regole generali d'architettura
* [516]Andrea Palladio
+ [517]I quattro libri dell'architettura
Romantic
* [518]Samuel Colman
+ Nature's Harmonic Unity
* [519]Frederik Macody Lund
+ Ad Quadratum
* [520]Jay Hambidge
+ The Greek Vase
Modern
* [521]Owen Jones
+ [522]The Grammar of Ornament
* [523]Ernest Hanbury Hankin
+ The Drawing of Geometric Patterns in Saracenic Art
* [524]G. H. Hardy
+ [525]A Mathematician's Apology
* [526]George David Birkhoff
+ Aesthetic Measure
* [527]Douglas Hofstadter
+ [528]Gödel, Escher, Bach
* [529]Nikos Salingaros
+ The 'Life' of a Carpet
Publications
* [530]Journal of Mathematics and the Arts
* [531]Lumen Naturae
* [532]Making Mathematics with Needlework
* [533]Rhythm of Structure
* [534]Viewpoints: Mathematical Perspective and Fractal Geometry in
Art
Organizations
* [535]Ars Mathematica
* [536]The Bridges Organization
* [537]European Society for Mathematics and the Arts
* [538]Goudreau Museum of Mathematics in Art and Science
* [539]Institute For Figuring
* [540]Mathemalchemy
* [541]National Museum of Mathematics
Related
* [542]Droste effect
* [543]Mathematical beauty
* [544]Patterns in nature
* [545]Sacred geometry
* [546]Category
* [547]v
* [548]t
* [549]e
[550]Visualization of technical information
Fields
* [551]Biological data visualization
* [552]Chemical imaging
* [553]Crime mapping
* [554]Data visualization
* [555]Educational visualization
* [556]Flow visualization
* [557]Geovisualization
* [558]Information visualization
* [559]Mathematical visualization
* [560]Medical imaging
* [561]Molecular graphics
* [562]Product visualization
* [563]Scientific visualization
* [564]Software visualization
* [565]Technical drawing
* [566]User interface design
* [567]Visual culture
* [568]Volume visualization
Image types
* [569]Chart
* [570]Diagram
* [571]Engineering drawing
* [572]Graph of a function
* [573]Ideogram
* [574]Map
* [575]Photograph
* [576]Pictogram
* [577]Plot
* [578]Sankey diagram
* [579]Schematic
* [580]Skeletal formula
* [581]Statistical graphics
* [582]Table
* [583]Technical drawings
* [584]Technical illustration
People
Pre-19th century
* [585]Edmond Halley
* [586]Charles-René de Fourcroy
* [587]Joseph Priestley
* [588]Gaspard Monge
19th century
* [589]Charles Dupin
* [590]Adolphe Quetelet
* [591]André-Michel Guerry
* [592]William Playfair
* [593]August Kekulé
* [594]Charles Joseph Minard
* [595]Luigi Perozzo
* [596]Francis Amasa Walker
* [597]John Venn
* [598]Oliver Byrne
* [599]Matthew Sankey
* [600]Charles Booth
* [601]Georg von Mayr
* [602]John Snow
* [603]Florence Nightingale
* [604]Karl Wilhelm Pohlke
* [605]Toussaint Loua
* [606]Francis Galton
Early 20th century
* [607]Edward Walter Maunder
* [608]Otto Neurath
* [609]W. E. B. Du Bois
* [610]Henry Gantt
* [611]Arthur Lyon Bowley
* [612]Howard G. Funkhouser
* [613]John B. Peddle
* [614]Ejnar Hertzsprung
* [615]Henry Norris Russell
* [616]Max O. Lorenz
* [617]Fritz Kahn
* [618]Harry Beck
* [619]Erwin Raisz
Mid 20th century
* [620]Jacques Bertin
* [621]Rudolf Modley
* [622]Arthur H. Robinson
* [623]John Tukey
* [624]Mary Eleanor Spear
* [625]Edgar Anderson
* [626]Howard T. Fisher
Late 20th century
* [627]Borden Dent
* [628]Nigel Holmes
* [629]William S. Cleveland
* [630]George G. Robertson
* [631]Bruce H. McCormick
* [632]Catherine Plaisant
* [633]Stuart Card
* [634]Pat Hanrahan
* [635]Edward Tufte
* [636]Ben Shneiderman
* [637]Michael Friendly
* [638]Howard Wainer
* [639]Clifford A. Pickover
* [640]Lawrence J. Rosenblum
* [641]Thomas A. DeFanti
* [642]George Furnas
* [643]Sheelagh Carpendale
* [644]Cynthia Brewer
* [645]Miriah Meyer
* [646]Jock D. Mackinlay
* [647]Alan MacEachren
* [648]David Goodsell
* [649]Michael Maltz
* [650]Leland Wilkinson
* [651]Alfred Inselberg
Early 21st century
* [652]Ben Fry
* [653]Hans Rosling
* [654]Christopher R. Johnson
* [655]David McCandless
* [656]Mauro Martino
* [657]John Maeda
* [658]Tamara Munzner
* [659]Jeffrey Heer
* [660]Gordon Kindlmann
* [661]Hanspeter Pfister
* [662]Manuel Lima
* [663]Aaron Koblin
* [664]Martin Krzywinski
* [665]Bang Wong
* [666]Jessica Hullman
* [667]Hadley Wickham
* [668]Polo Chau
* [669]Fernanda Viégas
* [670]Martin Wattenberg
* [671]Claudio Silva
* [672]Ade Olufeko
* [673]Moritz Stefaner
Related topics
* [674]Cartography
* [675]Chartjunk
* [676]Computer graphics
+ [677]in computer science
* [678]CPK coloring
* [679]Graph drawing
* [680]Graphic design
* [681]Graphic organizer
* [682]Imaging science
* [683]Information graphics
* [684]Information science
* [685]Misleading graph
* [686]Neuroimaging
* [687]Patent drawing
* [688]Scientific modelling
* [689]Spatial analysis
* [690]Visual analytics
* [691]Visual perception
* [692]Volume cartography
* [693]Volume rendering
* [694]Information art
* [695]v
* [696]t
* [697]e
[698]Computer science
Note: This template roughly follows the 2012 [699]ACM Computing
Classification System.
[700]Hardware
* [701]Printed circuit board
* [702]Peripheral
* [703]Integrated circuit
* [704]Very Large Scale Integration
* [705]Systems on Chip (SoCs)
* [706]Energy consumption (Green computing)
* [707]Electronic design automation
* [708]Hardware acceleration
[709]Computer Retro.svg
Computer systems organization
* [710]Computer architecture
* [711]Embedded system
* [712]Real-time computing
* [713]Dependability
[714]Networks
* [715]Network architecture
* [716]Network protocol
* [717]Network components
* [718]Network scheduler
* [719]Network performance evaluation
* [720]Network service
Software organization
* [721]Interpreter
* [722]Middleware
* [723]Virtual machine
* [724]Operating system
* [725]Software quality
[726]Software notations and [727]tools
* [728]Programming paradigm
* [729]Programming language
* [730]Compiler
* [731]Domain-specific language
* [732]Modeling language
* [733]Software framework
* [734]Integrated development environment
* [735]Software configuration management
* [736]Software library
* [737]Software repository
[738]Software development
* [739]Control variable
* [740]Software development process
* [741]Requirements analysis
* [742]Software design
* [743]Software construction
* [744]Software deployment
* [745]Software engineering
* [746]Software maintenance
* [747]Programming team
* [748]Open-source model
[749]Theory of computation
* [750]Model of computation
* [751]Formal language
* [752]Automata theory
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30. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Optimized_escape_time_algorithms
31. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Derivative_Bailout_or_"derbail"
32. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Coloring_algorithms
33. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Histogram_coloring
34. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Continuous_(smooth)_coloring
35. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Exponentially_mapped_and_cyclic_iterations
36. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Passing_iterations_into_a_color_directly
37. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#v_refers_to_a_normalized_exponentially_mapped_cyclic_iter_count
38. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#f(v)_refers_to_the_sRGB_transfer_function
39. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#HSV_coloring
40. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#LCH_coloring
41. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Advanced_plotting_algorithms
42. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Distance_estimates
43. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Exterior_distance_estimation
44. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Interior_distance_estimation
45. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Cardioid_/_bulb_checking
46. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Periodicity_checking
47. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Border_tracing_/_edge_checking
48. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Rectangle_checking
49. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Symmetry_utilization
50. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Multithreading
51. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#Perturbation_theory_and_series_approximation
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69. https://www.wikidata.org/wiki/Special:EntityPage/Q85793700
70. https://en.wikipedia.org/w/index.php?title=Special:DownloadAsPdf&page=Plotting_algorithms_for_the_Mandelbrot_set&action=show-download-screen
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86. https://en.wikipedia.org/wiki/Pythagorean_theorem
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88. https://en.wikipedia.org/wiki/Buddhabrot
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92. https://en.wikipedia.org/wiki/Palette_(computing)
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95. https://en.wikipedia.org/wiki/File:Derbail_method_render.png
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98. https://en.wikipedia.org/wiki/Derivative
99. https://en.wikipedia.org/wiki/Automatic_differentiation
100. https://en.wikipedia.org/wiki/Dual_number
101. https://en.wikipedia.org/wiki/File:Example_of_derbail_precision_issues.png
102. https://en.wikipedia.org/wiki/Floating-point_arithmetic
103. https://en.wikipedia.org/wiki/Image_noise
104. https://en.wikipedia.org/wiki/Multisample_anti-aliasing
105. https://en.wikipedia.org/wiki/File:Derbail.png
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134. https://en.wikipedia.org/wiki/File:HSV_Gradient_Example.png
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137. https://en.wikipedia.org/wiki/File:LCH_Gradient_Example.png
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143. https://en.wikipedia.org/wiki/Interior_(topology)
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155. https://en.wikipedia.org/wiki/Plotting_algorithms_for_the_Mandelbrot_set#cite_note-Cheritat-6
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538. https://en.wikipedia.org/wiki/Goudreau_Museum_of_Mathematics_in_Art_and_Science
539. https://en.wikipedia.org/wiki/Institute_For_Figuring
540. https://en.wikipedia.org/wiki/Mathemalchemy
541. https://en.wikipedia.org/wiki/National_Museum_of_Mathematics
542. https://en.wikipedia.org/wiki/Droste_effect
543. https://en.wikipedia.org/wiki/Mathematical_beauty
544. https://en.wikipedia.org/wiki/Patterns_in_nature
545. https://en.wikipedia.org/wiki/Sacred_geometry
546. https://en.wikipedia.org/wiki/Category:Mathematics_and_art
547. https://en.wikipedia.org/wiki/Template:Visualization
548. https://en.wikipedia.org/wiki/Template_talk:Visualization
549. https://en.wikipedia.org/w/index.php?title=Template:Visualization&action=edit
550. https://en.wikipedia.org/wiki/Visualization_(graphics)
551. https://en.wikipedia.org/wiki/Biological_data_visualization
552. https://en.wikipedia.org/wiki/Chemical_imaging
553. https://en.wikipedia.org/wiki/Crime_mapping
554. https://en.wikipedia.org/wiki/Data_visualization
555. https://en.wikipedia.org/wiki/Visualization_(graphics)
556. https://en.wikipedia.org/wiki/Flow_visualization
557. https://en.wikipedia.org/wiki/Geovisualization
558. https://en.wikipedia.org/wiki/Information_visualization
559. https://en.wikipedia.org/wiki/Mathematical_diagram
560. https://en.wikipedia.org/wiki/Medical_imaging
561. https://en.wikipedia.org/wiki/Molecular_graphics
562. https://en.wikipedia.org/wiki/Visualization_(graphics)
563. https://en.wikipedia.org/wiki/Scientific_visualization
564. https://en.wikipedia.org/wiki/Software_visualization
565. https://en.wikipedia.org/wiki/Technical_drawing
566. https://en.wikipedia.org/wiki/User_interface_design
567. https://en.wikipedia.org/wiki/Visual_culture
568. https://en.wikipedia.org/wiki/Volume_rendering
569. https://en.wikipedia.org/wiki/Chart
570. https://en.wikipedia.org/wiki/Diagram
571. https://en.wikipedia.org/wiki/Engineering_drawing
572. https://en.wikipedia.org/wiki/Graph_of_a_function
573. https://en.wikipedia.org/wiki/Ideogram
574. https://en.wikipedia.org/wiki/Map
575. https://en.wikipedia.org/wiki/Photograph
576. https://en.wikipedia.org/wiki/Pictogram
577. https://en.wikipedia.org/wiki/Plot_(graphics)
578. https://en.wikipedia.org/wiki/Sankey_diagram
579. https://en.wikipedia.org/wiki/Schematic
580. https://en.wikipedia.org/wiki/Skeletal_formula
581. https://en.wikipedia.org/wiki/Statistical_graphics
582. https://en.wikipedia.org/wiki/Table_(information)
583. https://en.wikipedia.org/wiki/Technical_drawing
584. https://en.wikipedia.org/wiki/Technical_illustration
585. https://en.wikipedia.org/wiki/Edmond_Halley
586. https://en.wikipedia.org/wiki/Charles-René_de_Fourcroy
587. https://en.wikipedia.org/wiki/Joseph_Priestley
588. https://en.wikipedia.org/wiki/Gaspard_Monge
589. https://en.wikipedia.org/wiki/Charles_Dupin
590. https://en.wikipedia.org/wiki/Adolphe_Quetelet
591. https://en.wikipedia.org/wiki/André-Michel_Guerry
592. https://en.wikipedia.org/wiki/William_Playfair
593. https://en.wikipedia.org/wiki/August_Kekulé
594. https://en.wikipedia.org/wiki/Charles_Joseph_Minard
595. https://en.wikipedia.org/w/index.php?title=Luigi_Perozzo&action=edit&redlink=1
596. https://en.wikipedia.org/wiki/Francis_Amasa_Walker
597. https://en.wikipedia.org/wiki/John_Venn
598. https://en.wikipedia.org/wiki/Oliver_Byrne_(mathematician)
599. https://en.wikipedia.org/wiki/Matthew_Henry_Phineas_Riall_Sankey
600. https://en.wikipedia.org/wiki/Charles_Booth_(social_reformer)
601. https://en.wikipedia.org/w/index.php?title=Georg_von_Mayr&action=edit&redlink=1
602. https://en.wikipedia.org/wiki/John_Snow
603. https://en.wikipedia.org/wiki/Florence_Nightingale
604. https://en.wikipedia.org/wiki/Karl_Wilhelm_Pohlke
605. https://en.wikipedia.org/wiki/Toussaint_Loua
606. https://en.wikipedia.org/wiki/Francis_Galton
607. https://en.wikipedia.org/wiki/Edward_Walter_Maunder
608. https://en.wikipedia.org/wiki/Otto_Neurath
609. https://en.wikipedia.org/wiki/W._E._B._Du_Bois
610. https://en.wikipedia.org/wiki/Henry_Gantt
611. https://en.wikipedia.org/wiki/Arthur_Lyon_Bowley
612. https://en.wikipedia.org/wiki/Howard_G._Funkhouser
613. https://en.wikipedia.org/wiki/John_B._Peddle
614. https://en.wikipedia.org/wiki/Ejnar_Hertzsprung
615. https://en.wikipedia.org/wiki/Henry_Norris_Russell
616. https://en.wikipedia.org/wiki/Max_O._Lorenz
617. https://en.wikipedia.org/wiki/Fritz_Kahn
618. https://en.wikipedia.org/wiki/Harry_Beck
619. https://en.wikipedia.org/wiki/Erwin_Raisz
620. https://en.wikipedia.org/wiki/Jacques_Bertin
621. https://en.wikipedia.org/wiki/Rudolf_Modley
622. https://en.wikipedia.org/wiki/Arthur_H._Robinson
623. https://en.wikipedia.org/wiki/John_Tukey
624. https://en.wikipedia.org/wiki/Mary_Eleanor_Spear
625. https://en.wikipedia.org/wiki/Edgar_Anderson
626. https://en.wikipedia.org/wiki/Howard_T._Fisher
627. https://en.wikipedia.org/wiki/Borden_Dent
628. https://en.wikipedia.org/wiki/Nigel_Holmes
629. https://en.wikipedia.org/wiki/William_S._Cleveland
630. https://en.wikipedia.org/wiki/George_G._Robertson
631. https://en.wikipedia.org/wiki/Bruce_H._McCormick
632. https://en.wikipedia.org/wiki/Catherine_Plaisant
633. https://en.wikipedia.org/wiki/Stuart_Card
634. https://en.wikipedia.org/wiki/Pat_Hanrahan
635. https://en.wikipedia.org/wiki/Edward_Tufte
636. https://en.wikipedia.org/wiki/Ben_Shneiderman
637. https://en.wikipedia.org/wiki/Michael_Friendly
638. https://en.wikipedia.org/wiki/Howard_Wainer
639. https://en.wikipedia.org/wiki/Clifford_A._Pickover
640. https://en.wikipedia.org/wiki/Lawrence_J._Rosenblum
641. https://en.wikipedia.org/wiki/Thomas_A._DeFanti
642. https://en.wikipedia.org/wiki/George_Furnas
643. https://en.wikipedia.org/wiki/Sheelagh_Carpendale
644. https://en.wikipedia.org/wiki/Cynthia_Brewer
645. https://en.wikipedia.org/wiki/Miriah_Meyer
646. https://en.wikipedia.org/wiki/Jock_D._Mackinlay
647. https://en.wikipedia.org/wiki/Alan_MacEachren
648. https://en.wikipedia.org/wiki/David_Goodsell
649. https://en.wikipedia.org/wiki/Michael_Maltz
650. https://en.wikipedia.org/wiki/Leland_Wilkinson
651. https://en.wikipedia.org/wiki/Alfred_Inselberg
652. https://en.wikipedia.org/wiki/Ben_Fry
653. https://en.wikipedia.org/wiki/Hans_Rosling
654. https://en.wikipedia.org/wiki/Christopher_R._Johnson
655. https://en.wikipedia.org/wiki/David_McCandless
656. https://en.wikipedia.org/wiki/Mauro_Martino
657. https://en.wikipedia.org/wiki/John_Maeda
658. https://en.wikipedia.org/wiki/Tamara_Munzner
659. https://en.wikipedia.org/wiki/Jeffrey_Heer
660. https://en.wikipedia.org/wiki/Gordon_Kindlmann
661. https://en.wikipedia.org/wiki/Hanspeter_Pfister
662. https://en.wikipedia.org/wiki/Manuel_Lima
663. https://en.wikipedia.org/wiki/Aaron_Koblin
664. https://en.wikipedia.org/w/index.php?title=Martin_Krzywinski&action=edit&redlink=1
665. https://en.wikipedia.org/wiki/Bang_Wong
666. https://en.wikipedia.org/wiki/Jessica_Hullman
667. https://en.wikipedia.org/wiki/Hadley_Wickham
668. https://en.wikipedia.org/w/index.php?title=Polo_Chau&action=edit&redlink=1
669. https://en.wikipedia.org/wiki/Fernanda_Viégas
670. https://en.wikipedia.org/wiki/Martin_M._Wattenberg
671. https://en.wikipedia.org/wiki/Claudio_Silva_(computer_scientist)
672. https://en.wikipedia.org/wiki/Ade_Olufeko
673. https://en.wikipedia.org/wiki/Moritz_Stefaner
674. https://en.wikipedia.org/wiki/Cartography
675. https://en.wikipedia.org/wiki/Chartjunk
676. https://en.wikipedia.org/wiki/Computer_graphics
677. https://en.wikipedia.org/wiki/Computer_graphics_(computer_science)
678. https://en.wikipedia.org/wiki/CPK_coloring
679. https://en.wikipedia.org/wiki/Graph_drawing
680. https://en.wikipedia.org/wiki/Graphic_design
681. https://en.wikipedia.org/wiki/Graphic_organizer
682. https://en.wikipedia.org/wiki/Imaging_science
683. https://en.wikipedia.org/wiki/Infographic
684. https://en.wikipedia.org/wiki/Information_science
685. https://en.wikipedia.org/wiki/Misleading_graph
686. https://en.wikipedia.org/wiki/Neuroimaging
687. https://en.wikipedia.org/wiki/Patent_drawing
688. https://en.wikipedia.org/wiki/Scientific_modelling
689. https://en.wikipedia.org/wiki/Spatial_analysis
690. https://en.wikipedia.org/wiki/Visual_analytics
691. https://en.wikipedia.org/wiki/Visual_perception
692. https://en.wikipedia.org/wiki/Volume_cartography
693. https://en.wikipedia.org/wiki/Volume_rendering
694. https://en.wikipedia.org/wiki/Information_art
695. https://en.wikipedia.org/wiki/Template:Computer_science
696. https://en.wikipedia.org/wiki/Template_talk:Computer_science
697. https://en.wikipedia.org/w/index.php?title=Template:Computer_science&action=edit
698. https://en.wikipedia.org/wiki/Computer_science
699. https://en.wikipedia.org/wiki/ACM_Computing_Classification_System
700. https://en.wikipedia.org/wiki/Computer_hardware
701. https://en.wikipedia.org/wiki/Printed_circuit_board
702. https://en.wikipedia.org/wiki/Peripheral
703. https://en.wikipedia.org/wiki/Integrated_circuit
704. https://en.wikipedia.org/wiki/Very_Large_Scale_Integration
705. https://en.wikipedia.org/wiki/System_on_a_chip
706. https://en.wikipedia.org/wiki/Green_computing
707. https://en.wikipedia.org/wiki/Electronic_design_automation
708. https://en.wikipedia.org/wiki/Hardware_acceleration
709. https://en.wikipedia.org/wiki/File:Computer_Retro.svg
710. https://en.wikipedia.org/wiki/Computer_architecture
711. https://en.wikipedia.org/wiki/Embedded_system
712. https://en.wikipedia.org/wiki/Real-time_computing
713. https://en.wikipedia.org/wiki/Dependability
714. https://en.wikipedia.org/wiki/Computer_network
715. https://en.wikipedia.org/wiki/Network_architecture
716. https://en.wikipedia.org/wiki/Network_protocol
717. https://en.wikipedia.org/wiki/Networking_hardware
718. https://en.wikipedia.org/wiki/Network_scheduler
719. https://en.wikipedia.org/wiki/Network_performance
720. https://en.wikipedia.org/wiki/Network_service
721. https://en.wikipedia.org/wiki/Interpreter_(computing)
722. https://en.wikipedia.org/wiki/Middleware
723. https://en.wikipedia.org/wiki/Virtual_machine
724. https://en.wikipedia.org/wiki/Operating_system
725. https://en.wikipedia.org/wiki/Software_quality
726. https://en.wikipedia.org/wiki/Programming_language_theory
727. https://en.wikipedia.org/wiki/Programming_tool
728. https://en.wikipedia.org/wiki/Programming_paradigm
729. https://en.wikipedia.org/wiki/Programming_language
730. https://en.wikipedia.org/wiki/Compiler_construction
731. https://en.wikipedia.org/wiki/Domain-specific_language
732. https://en.wikipedia.org/wiki/Modeling_language
733. https://en.wikipedia.org/wiki/Software_framework
734. https://en.wikipedia.org/wiki/Integrated_development_environment
735. https://en.wikipedia.org/wiki/Software_configuration_management
736. https://en.wikipedia.org/wiki/Library_(computing)
737. https://en.wikipedia.org/wiki/Software_repository
738. https://en.wikipedia.org/wiki/Software_development
739. https://en.wikipedia.org/wiki/Control_variable_(programming)
740. https://en.wikipedia.org/wiki/Software_development_process
741. https://en.wikipedia.org/wiki/Requirements_analysis
742. https://en.wikipedia.org/wiki/Software_design
743. https://en.wikipedia.org/wiki/Software_construction
744. https://en.wikipedia.org/wiki/Software_deployment
745. https://en.wikipedia.org/wiki/Software_engineering
746. https://en.wikipedia.org/wiki/Software_maintenance
747. https://en.wikipedia.org/wiki/Programming_team
748. https://en.wikipedia.org/wiki/Open-source_software
749. https://en.wikipedia.org/wiki/Theory_of_computation
750. https://en.wikipedia.org/wiki/Model_of_computation
751. https://en.wikipedia.org/wiki/Formal_language
752. https://en.wikipedia.org/wiki/Automata_theory
753. https://en.wikipedia.org/wiki/Computability_theory
754. https://en.wikipedia.org/wiki/Computational_complexity_theory
755. https://en.wikipedia.org/wiki/Logic_in_computer_science
756. https://en.wikipedia.org/wiki/Semantics_(computer_science)
757. https://en.wikipedia.org/wiki/Algorithm
758. https://en.wikipedia.org/wiki/Algorithm_design
759. https://en.wikipedia.org/wiki/Analysis_of_algorithms
760. https://en.wikipedia.org/wiki/Algorithmic_efficiency
761. https://en.wikipedia.org/wiki/Randomized_algorithm
762. https://en.wikipedia.org/wiki/Computational_geometry
763. https://en.wikipedia.org/wiki/Discrete_mathematics
764. https://en.wikipedia.org/wiki/Probability
765. https://en.wikipedia.org/wiki/Statistics
766. https://en.wikipedia.org/wiki/Mathematical_software
767. https://en.wikipedia.org/wiki/Information_theory
768. https://en.wikipedia.org/wiki/Mathematical_analysis
769. https://en.wikipedia.org/wiki/Numerical_analysis
770. https://en.wikipedia.org/wiki/Theoretical_computer_science
771. https://en.wikipedia.org/wiki/Information_system
772. https://en.wikipedia.org/wiki/Database
773. https://en.wikipedia.org/wiki/Computer_data_storage
774. https://en.wikipedia.org/wiki/Enterprise_information_system
775. https://en.wikipedia.org/wiki/Social_software
776. https://en.wikipedia.org/wiki/Geographic_information_system
777. https://en.wikipedia.org/wiki/Decision_support_system
778. https://en.wikipedia.org/wiki/Process_control
779. https://en.wikipedia.org/wiki/Multimedia_database
780. https://en.wikipedia.org/wiki/Data_mining
781. https://en.wikipedia.org/wiki/Digital_library
782. https://en.wikipedia.org/wiki/Computing_platform
783. https://en.wikipedia.org/wiki/Digital_marketing
784. https://en.wikipedia.org/wiki/World_Wide_Web
785. https://en.wikipedia.org/wiki/Information_retrieval
786. https://en.wikipedia.org/wiki/Computer_security
787. https://en.wikipedia.org/wiki/Cryptography
788. https://en.wikipedia.org/wiki/Formal_methods
789. https://en.wikipedia.org/wiki/Security_service_(telecommunication)
790. https://en.wikipedia.org/wiki/Intrusion_detection_system
791. https://en.wikipedia.org/wiki/Computer_security_compromised_by_hardware_failure
792. https://en.wikipedia.org/wiki/Network_security
793. https://en.wikipedia.org/wiki/Information_security
794. https://en.wikipedia.org/wiki/Application_security
795. https://en.wikipedia.org/wiki/Human–computer_interaction
796. https://en.wikipedia.org/wiki/Interaction_design
797. https://en.wikipedia.org/wiki/Social_computing
798. https://en.wikipedia.org/wiki/Ubiquitous_computing
799. https://en.wikipedia.org/wiki/Visualization_(graphics)
800. https://en.wikipedia.org/wiki/Computer_accessibility
801. https://en.wikipedia.org/wiki/Synthography
802. https://en.wikipedia.org/wiki/Concurrency_(computer_science)
803. https://en.wikipedia.org/wiki/Concurrent_computing
804. https://en.wikipedia.org/wiki/Parallel_computing
805. https://en.wikipedia.org/wiki/Distributed_computing
806. https://en.wikipedia.org/wiki/Multithreading_(computer_architecture)
807. https://en.wikipedia.org/wiki/Multiprocessing
808. https://en.wikipedia.org/wiki/Artificial_intelligence
809. https://en.wikipedia.org/wiki/Natural_language_processing
810. https://en.wikipedia.org/wiki/Knowledge_representation_and_reasoning
811. https://en.wikipedia.org/wiki/Computer_vision
812. https://en.wikipedia.org/wiki/Automated_planning_and_scheduling
813. https://en.wikipedia.org/wiki/Mathematical_optimization
814. https://en.wikipedia.org/wiki/Control_theory
815. https://en.wikipedia.org/wiki/Philosophy_of_artificial_intelligence
816. https://en.wikipedia.org/wiki/Distributed_artificial_intelligence
817. https://en.wikipedia.org/wiki/Machine_learning
818. https://en.wikipedia.org/wiki/Supervised_learning
819. https://en.wikipedia.org/wiki/Unsupervised_learning
820. https://en.wikipedia.org/wiki/Reinforcement_learning
821. https://en.wikipedia.org/wiki/Multi-task_learning
822. https://en.wikipedia.org/wiki/Cross-validation_(statistics)
823. https://en.wikipedia.org/wiki/Computer_graphics
824. https://en.wikipedia.org/wiki/Computer_animation
825. https://en.wikipedia.org/wiki/Rendering_(computer_graphics)
826. https://en.wikipedia.org/wiki/Photograph_manipulation
827. https://en.wikipedia.org/wiki/Graphics_processing_unit
828. https://en.wikipedia.org/wiki/Mixed_reality
829. https://en.wikipedia.org/wiki/Virtual_reality
830. https://en.wikipedia.org/wiki/Image_compression
831. https://en.wikipedia.org/wiki/Solid_modeling
832. https://en.wikipedia.org/wiki/E-commerce
833. https://en.wikipedia.org/wiki/Enterprise_software
834. https://en.wikipedia.org/wiki/Computational_mathematics
835. https://en.wikipedia.org/wiki/Computational_physics
836. https://en.wikipedia.org/wiki/Computational_chemistry
837. https://en.wikipedia.org/wiki/Computational_biology
838. https://en.wikipedia.org/wiki/Computational_social_science
839. https://en.wikipedia.org/wiki/Computational_engineering
840. https://en.wikipedia.org/wiki/Health_informatics
841. https://en.wikipedia.org/wiki/Digital_art
842. https://en.wikipedia.org/wiki/Electronic_publishing
843. https://en.wikipedia.org/wiki/Cyberwarfare
844. https://en.wikipedia.org/wiki/Electronic_voting
845. https://en.wikipedia.org/wiki/Video_game
846. https://en.wikipedia.org/wiki/Word_processor
847. https://en.wikipedia.org/wiki/Operations_research
848. https://en.wikipedia.org/wiki/Educational_technology
849. https://en.wikipedia.org/wiki/Document_management_system
850. https://en.wikipedia.org/wiki/Category:Computer_science
851. https://en.wikipedia.org/wiki/Outline_of_computer_science
852. https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Computer_science
853. https://commons.wikimedia.org/wiki/Category:Computer_science
854. https://en.wikipedia.org/wiki/Template:Computer_graphics
855. https://en.wikipedia.org/wiki/Template_talk:Computer_graphics
856. https://en.wikipedia.org/w/index.php?title=Template:Computer_graphics&action=edit
857. https://en.wikipedia.org/wiki/Computer_graphics_(computer_science)
858. https://en.wikipedia.org/wiki/Vector_graphics
859. https://en.wikipedia.org/wiki/Diffusion_curve
860. https://en.wikipedia.org/wiki/Pixel
861. https://en.wikipedia.org/wiki/2D_computer_graphics
862. https://en.wikipedia.org/wiki/2.5D
863. https://en.wikipedia.org/wiki/Isometric_video_game_graphics
864. https://en.wikipedia.org/wiki/Mode_7
865. https://en.wikipedia.org/wiki/Parallax_scrolling
866. https://en.wikipedia.org/wiki/Ray_casting
867. https://en.wikipedia.org/wiki/Skybox_(video_games)
868. https://en.wikipedia.org/wiki/Alpha_compositing
869. https://en.wikipedia.org/wiki/Layers_(digital_image_editing)
870. https://en.wikipedia.org/wiki/Text-to-image
871. https://en.wikipedia.org/wiki/3D_computer_graphics
872. https://en.wikipedia.org/wiki/3D_projection
873. https://en.wikipedia.org/wiki/3D_rendering
874. https://en.wikipedia.org/wiki/Image-based_modeling_and_rendering
875. https://en.wikipedia.org/wiki/Spectral_rendering
876. https://en.wikipedia.org/wiki/Unbiased_rendering
877. https://en.wikipedia.org/wiki/Aliasing
878. https://en.wikipedia.org/wiki/Anisotropic_filtering
879. https://en.wikipedia.org/wiki/Cel_shading
880. https://en.wikipedia.org/wiki/Computer_graphics_lighting
881. https://en.wikipedia.org/wiki/Global_illumination
882. https://en.wikipedia.org/wiki/Hidden-surface_determination
883. https://en.wikipedia.org/wiki/Polygon_mesh
884. https://en.wikipedia.org/wiki/Triangle_mesh
885. https://en.wikipedia.org/wiki/Shading
886. https://en.wikipedia.org/wiki/Deferred_shading
887. https://en.wikipedia.org/wiki/Surface_triangulation
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889. https://en.wikipedia.org/wiki/Affine_transformation
890. https://en.wikipedia.org/wiki/Back-face_culling
891. https://en.wikipedia.org/wiki/Clipping_(computer_graphics)
892. https://en.wikipedia.org/wiki/Collision_detection
893. https://en.wikipedia.org/wiki/Planar_projection
894. https://en.wikipedia.org/wiki/Rendering_(computer_graphics)
895. https://en.wikipedia.org/wiki/Rotation_(mathematics)
896. https://en.wikipedia.org/wiki/Scaling_(geometry)
897. https://en.wikipedia.org/wiki/Shadow_mapping
898. https://en.wikipedia.org/wiki/Shadow_volume
899. https://en.wikipedia.org/wiki/Shear_matrix
900. https://en.wikipedia.org/wiki/Translation_(geometry)
901. https://en.wikipedia.org/wiki/List_of_computer_graphics_algorithms
902. https://en.wikipedia.org/w/index.php?title=Plotting_algorithms_for_the_Mandelbrot_set&oldid=1148917111
903. https://en.wikipedia.org/wiki/Help:Category
904. https://en.wikipedia.org/wiki/Category:Fractals
905. https://en.wikipedia.org/wiki/Category:Complex_dynamics
906. https://en.wikipedia.org/wiki/Category:Graphics_software
907. https://en.wikipedia.org/wiki/Category:Computer_graphics
908. https://en.wikipedia.org/wiki/Category:Algorithms
909. https://en.wikipedia.org/wiki/Category:CS1_errors:_missing_periodical
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918. https://en.wikipedia.org/wiki/Category:All_articles_with_style_issues
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920. https://en.wikipedia.org/wiki/Category:Articles_needing_additional_references_from_June_2019
921. https://en.wikipedia.org/wiki/Category:All_articles_needing_additional_references
922. https://en.wikipedia.org/wiki/Category:Copied_and_pasted_articles_and_sections_with_url_provided_from_July_2022
923. https://en.wikipedia.org/wiki/Category:All_copied_and_pasted_articles_and_sections
924. https://en.wikipedia.org/wiki/Category:Articles_with_example_pseudocode
925. https://en.wikipedia.org/wiki/Category:Articles_containing_video_clips
926. https://en.wikipedia.org/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_3.0_Unported_License
927. https://foundation.wikimedia.org/wiki/Terms_of_Use
928. https://foundation.wikimedia.org/wiki/Privacy_policy
929. https://www.wikimediafoundation.org/
930. https://foundation.wikimedia.org/wiki/Privacy_policy
931. https://en.wikipedia.org/wiki/Wikipedia:About
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937. https://foundation.wikimedia.org/wiki/Cookie_statement
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Hidden links:
941. https://en.wikipedia.org/wiki/File:Fractal-zoom-1-03-Mandelbrot_Buzzsaw.png
942. https://en.wikipedia.org/wiki/File:Mandelbrot_sequence_new_still.png
943. https://en.wikipedia.org/wiki/File:Derbail_method_render.png
944. https://en.wikipedia.org/wiki/File:Example_of_derbail_precision_issues.png
945. https://en.wikipedia.org/wiki/File:Derbail.png
946. https://en.wikipedia.org/wiki/File:Question_book-new.svg
947. https://en.wikipedia.org/wiki/File:Mandelbrot-no-histogram-coloring-10000-iterations.png
948. https://en.wikipedia.org/wiki/File:Mandelbrot-no-histogram-coloring-1000-iterations.png
949. https://en.wikipedia.org/wiki/File:Mandelbrot-no-histogram-coloring-100-iterations.png
950. https://en.wikipedia.org/wiki/File:Mandelbrot-histogram-10000-iterations.png
951. https://en.wikipedia.org/wiki/File:Mandelbrot-histogram-1000-iterations.png
952. https://en.wikipedia.org/wiki/File:Mandelbrot-histogram-100-iterations.png
953. https://en.wikipedia.org/wiki/File:Escape_Time_Algorithm_bands.png
954. https://en.wikipedia.org/wiki/File:Normalized_Iteration_Count_Algorithm_1.png
955. https://en.wikipedia.org/wiki/File:CyclicColoringLch.png
956. https://en.wikipedia.org/wiki/File:LCH_COLORING.png
957. https://en.wikipedia.org/wiki/File:HSV_Gradient_Example.png
958. https://en.wikipedia.org/wiki/File:LCH_Gradient_Example.png
959. https://en.wikipedia.org/wiki/File:Demm_2000_Mandelbrot_set.jpg
960. https://en.wikipedia.org/wiki/File:Mandel_zoom_06_double_hook_B%26W_DE.jpg
961. https://en.wikipedia.org/wiki/File:Mandelbrot_Interior_600.png
962. https://en.wikipedia.org/wiki/File:MandelbrotOrbitInfimum.png
963. https://en.wikipedia.org/wiki/File:Mandelbrot_DEM_Sobel.png
964. https://en.wikipedia.org/wiki/File:Boundary_following_movie.webm
965. https://en.wikipedia.org/wiki/File:Multi-threaded_mandelbrot_render_w_symmetry.webm
966. https://en.wikipedia.org/wiki/File:Multi-threaded_mandelbrot_render_w_bounary_following_%26_symmetry.webm
967. https://creativecommons.org/licenses/by-sa/3.0/