The Mandelbrot set consists of the set of points $c$ in the complex plane for which the iteration

$$ z_{n+1} = z_{n}^2 + c{\mbox{,}} $$

where $z_{0}=0$ and $z~{\in}~{\mathbb{C}}$ remains bounded.

ckormanyos/mandelbrot implements a header-only C++ template library for extended-precision Mandelbrot fractal iterations. It is intended for deep dives in the Mandelbrot fractal world.

Fractal iterations are carried out with high-precision floating-point types such as those found in Boost.Multiprecision. Also fixed-point types having sufficient precision should work well (although these have not yet been extensively used or tested in this project).

This project uses Boost.Multiprecision to implement a high-precision Mandelbrot iteration and visualization. Graphic file creation uses Boost.Gil. Color-stretching in combination with the histogram method is used for creating vivid images. The default color scheme uses stretched, amplified and modulated black and white coloring. The Mandelbrot iteration is carried out with hardware concurrency with multiple threads. The multithreading dispatcher uses $(n-1)$ of the available CPU cores that can be found using hardware concurrency.

The following design goals have been incorporated.

• ckormanyos/mandelbrot is written in header-only C++14, and compatible through C++14, 17, 20, 23 and beyond.
• Support rectangular (including square) images having essentially any size and magnification, provided that the precision of the iteration type is set accordingly.
• The inner loop that performs the work of fractal iteration uses multiple, parallel CPU cores.
• C++ template design allows for flexible interchange of any appropriate kind of big-number type in the classes that implement fractal iteration.
• The iterative scheme is implemented as a flexible run-time polymorphic function. This allows for the implementation and interchange of multiple kinds of iterative methods. These can be specifically optimized for the particular iterative case at hand.
• Visualization of the result uses color-stretching techniques combined with the histogram method.
• Graphical representation uses Boost.Gil in combination with the JPEG-6b library (see below).
• Imaging also uses specialized versions of the libpng and the zlib libraries, in particular libpng-1.6.44 and zlib-1.3.1.1, with further notes on adaptions still pending.
• Color schemes can be easily adapted via straightforward creation (or modification) of a skinny derived class.

• Locate the solution file mandelbrot.sln in the root dierctory.
• Open the solution file in Visual Studio.
• Build the desired configuration and run in the usual way.
• The resulting JPEG image will be placed as a temporary file in the images\tmp folder.

The commands are as shown below, where you need to specialize --boost, --my_cc and --stdcc accordingly (if needed):

cd mandlebrot
./build_all.sh --boost=../boost-root --my_cc=g++ --stdcc=c++20
mkdir -p images/tmp
./test_mandelbrot.exe

In summary, the script build_all.sh is used to:

• Build the JPEG, ZLIB and PNG libraries with GNUmake.
• Compile and link test_mandelbrot.cpp to obtain the test program test_mandelbrot.exe.
• This build makes and uses its own specialized version of the JPEG-6b library (see below) and does not install or use a standard *nix version thereof.
• This build also makes and uses specialized versions of the libpng and the zlib libraries.

Simply go to the mandelbrot directory and run the build shell script build_all.sh. You may need to supply your system's (or your desired local) path to boost.

The build script does not execute the test program (i.e., via ./test_mandelbrot.exe). Test program execution must be performed manually. In order to do this, make the needed output image directory and run test_mandelbrot.exe as shown above.

To optionally use gmp_float, the GMP big-number backend, instead of cpp_dec_float, where both backends are from Boost.Multiprecision, simply use build_all_gmp.sh

cd mandlebrot
./build_all_gmp.sh --boost=../boost-root --my_cc=g++ --stdcc=c++20
mkdir -p images/tmp
./test_mandelbrot.exe

In other words, in *nix, it is also possible to optionally make straightforward use of GMP via Boost.Multiprecision's gmp_float backend (instead of the default cpp_dec_float backend). When doing so, define -DMANDELBROT_USE_GMP_FLOAT on the command line in the compile test_mandelbrot.cpp stage. In this case, you also need to link with -lgmp in the link stage.

Several images are highlighted in the paragraphs below. Please see the gallery of ckormanyos/mandelbrot for a more comprehensive collection of images.

In these particular images presented from the gallery of ckormanyos/mandelbrot, we have concentrated on various beautiful iteration results represented in pure black-and-white/gray-scale tones.

A black-and-white rendition of the well-known, classic full Mandelbrot iteration is shown below.

This image has the following features and parameters.

• The square image has $2048 {\times} 2048$ pixels.
• Centered at the point $(-0.75+i~0.0)$.
• Uses a half-width of $1.35$.
• Computed with $2,000$ iteratons using the cpp_dec_float type from Boost.Multiprecision instantiated with $37$ decimal digits of precision.

A beautiful iteration result showing swirly seahorses and a mini Mandelbrot image in black-and-white color representation is shown below.

This image has the following features and parameters.

• The square image has $2048 {\times} 2048$ pixels.
• Centered at the point $(-0.7453983606667815+i~0.1125046349959942)$.
• Uses a half-width of $1.76{\times}10^{-12}$.
• Computed with $2,000$ iteratons using the cpp_dec_float type from Boost.Multiprecision instantiated with $37$ decimal digits of precision.

The following deep dive has a scale of approximately $2.55{\times}10^{-55}$. The result of this deep dive iteration is represented by the image below.

This image has the the following features and parameters.

• The square image has $2048 {\times} 2048$ pixels.
• Centered at the point $(-1.2951890821477774570170641771856819267{\ldots}+i~0.44093698267832013888090367835626261211{\ldots})$.
• See the configuration code mandelbrot_cfg_MANDELBROT_09_DEEP_DIVE_02.h for the full precision of the center point.
• Uses a half-width of $2.55{\times}10^{-55}$.
• Computed with $15,000$ iteratons using the cpp_dec_float type from Boost.Multiprecision instantiated with $95$ decimal digits of precision.

The following deep dive has a scale of approximately $8.1{\times}10^{-95}$. The result of this deep dive iteration is represented by the image below.

This image has the the following features and parameters.

• The square image has $2048 {\times} 2048$ pixels.
• Centered at the point $(-1.7683830049648722692890015019043698317{\ldots}-i~0.0035564756051580419085635598914228270{\ldots})$.
• See the configuration code mandelbrot_cfg_MANDELBROT_60_SATELITE_REGION_01.h for the full precision of the center point.
• Uses a half-width of $8.1{\times}10^{-95}$.
• Computed with $60,000$ iteratons using the cpp_dec_float type from Boost.Multiprecision instantiated with $95$ decimal digits of precision.

One of my personal best deep dives has a scale of approximately $10^{-311}$. This is a huge magnification factor having a scale of approximately $1$ followed by $311$ zeros. The result of this very deep dive iteration is represented by the image below.

This image has the the following features and parameters.

• The square image has $2048 {\times} 2048$ pixels.
• Centered at the point $(0.3602404434376143632361252444495453084{\ldots}-i~0.6413130610648031748603750151793020665{\ldots})$.
• See the configuration code mandelbrot_cfg_MANDELBROT_20_ZOOM_VERY_DEEP_00.h for the full precision of the center point provided with slightly more than $1,000$ decimal places.
• Uses a half-width of $4.4 {\times} 10^{-311}$.
• Computed with $60,000$ iteratons using the cpp_dec_float type from Boost.Multiprecision in combination with low-order perturbative iteration having $24$ decimal digits of precision. The central point of the calculation is instantiated with $365$ decimal digits of precision.

ckormanyos/mandelbrot uses configuration files to specify the point, size and depth of the iterative fractal calculation. The configuration files are stored in the mandelbrot/cfg directory.

The code responsible for controlling the iterative calculation, performing color-stretching and writing the output file is written in a generic and configurable way.

To characterize a new Mandelbrot fractal iteration, simply write a header-configuration-file with content similar to that shown below. Include the configuration file in test_mandelbrot.cpp and run the calculation.

Consider as a sample, for instance, the relevant content of the configuration file mandelbrot_cfg_MANDELBROT_01_FULL.h.

  // This is the classic full immage.

  constexpr char MANDELBROT_FILENAME_STRING[]       = "MANDELBROT_01_FULL";

  constexpr int  MANDELBROT_COORD_PNT_DIGITS10      =      37;
  constexpr int  MANDELBROT_ITERATION_DIGITS10      =      37;
  constexpr int  MANDELBROT_CALCULATION_PIXELS_X    =    2048;
  constexpr int  MANDELBROT_CALCULATION_PIXELS_Y    =    2048;
  constexpr int  MANDELBROT_CALCULATION_ITERATIONS  =    2000;

  constexpr char MANDELBROT_POINT_DX_HALF[]         = "1.35";
  constexpr char MANDELBROT_POINT_DY_HALF[]         = "1.35";
  constexpr char MANDELBROT_POINT_CENTER_X[]        = "-0.75";
  constexpr char MANDELBROT_POINT_CENTER_Y[]        = "+0.00";

  #define MANDELBROT_GENERATOR_TYPE mandelbrot_generator_trivial // NOLINT(cppcoreguidelines-macro-usage)

  #include <mandelbrot/cfg/mandelbrot_cfg.h>

Clearly named definitions are used to specify:

• the output file name,
• the number of decimal digits of precision (in the big-number type used in the iteration),
• the numbers of pixels and iterations,
• and the coordinate center and half-width of the magnification region,
• the iterative method (such as mandelbrot_generator_trivial or mandelbrot_generator_perturbative),
• where the dimensions MANDELBROT_POINT_DX_HALF, MANDELBROT_POINT_DY_HALF, MANDELBROT_POINT_CENTER_X, and MANDELBROT_POINT_CENTER_Y are provided as character strings, which remain exact even if the magnification scale requires multiple precision,
• and MANDELBROT_CALCULATION_ITERATIONS sets the default maximum number of iterations which can be modified at run time (i.e., for the Mandelbrot Discovery program).

Let's find some proven deep dives!

Discovering the Mandelbrot set leads to fascinating research that is mathematically complicated. Investigations in this domain offer enriching findings. Yet it can sometimes be challenging obtain results. In particular, it can be difficult to find deep dive points within the infinitely-complex (and complex-valued) Mandelbrot area. Miniscule coordinates that offer interesting deep dives are not intuitively easy to guess or find (even with dedicated searches).

In this section, we specifically list several points having proven deep dives to $10^{51}$ magnification. Once a point is found, a picture has been created with full $51$-decimal digit precision. Images are presented in the corresponding links for the points.

Details:

• The listed points have approximately $54$ decimal digits of precision.
• Calculations use a half-width of $1.4{\times}10^{-51}$.
• Iterations are carried out with a big-float representation having $76$ decimal digits of precision.
• See the table below for these points and links to the corresponding images.

Deep dives are mathematically challenging and computationally difficult. Even when an interesting point can be found, iterative calculations tend to be hampered by the very large floating-point types needed to represent the point in the complex plane.

A magnification of $10^{100}$, in classical iteration for instance, requires a floating-point type having more than $100$ decimal digits of precision, such as $124$ digits. This is the only way to trivially resolve the minute differences in the iterations. Huge number calculations are, however, time-consuming and weigh heavily on CPU resources.

Computation-time of classical iteration (with full precision) scales with the computational complexity of multiplication of the underlying big-numbers. This is non-linear. Typical multiplication complexities are quadratic Order $N^{2}$ for school-multiplciation or around Order $N^{1.6}$ for higher order multiplication schemes like Karatsuba. This non-linear increase in complexity is also compounded by the fact that deeper dives require higher iteration counts.

There are, nonetheless, perturbative schemes that can reduce the width of the floating-point types used in iterative caluclations. These can significantly accelerate high-precision deep dives. One such scheme was contributed by S-Streulicht in the Speed Gain of PR-100 effort.

Perturbative schemes are still computationally intensive. But the strong dependence on the width of the underlying central point is removed. This is traded for a slightly more complicated iteration scheme and a bit more pre-iterative preparational effort. The advantages for deep dives can, however, be stark. The work for deeper and deeper dives scales more or less linearly instead of quadratically and increases only with iteration count. This is a very equitable trade.

For this lower-order perturbation expansion, begin with the core functionality of the original formula

$$ z_{k+1} = z_{k}^2 + c $$

and perform a perturbative delta transformation on the coordinates. See also Deep Zoom Theory: Perturbation.

In other words

$$ z_{k+1}~{\rightarrow}~z_{k+1} + e_{k+1} $$

$$ z_{k}~{\rightarrow}~z_{k} + e_{k} $$

$$ c~{\rightarrow}~c + d{\mbox{.}} $$

Plugging this into the original formula results in

$$ z_{k+1} + e_{k+1} = z_{k}^{2} + c + e_{k}^2 + 2 z_{k} e_{k} + d{\mbox{.}} $$

Thereby, we replace the original formula with

$$ e_{k+1} = e_{k}^2 + 2 z_{k} e_{k} + d{\mbox{,}} $$

where the $z_{k}$ are the pre-calculated values.

The transformed coordinates can be iterated with drastically decreased precision such as $24$ decimal digits. This can result in saved calculation time.

Further details of the perturbative calculation are briefly described in the following.

• We also evaluate $z_{k+1}+e_{k+1}$ for the escape check. Alternatively it could be sufficient to check only $e_{k}$. If a given $e_{k}$ escapes at some iteration, then the $z_{k}$ are also expected to escape, albeit at a different iteration step.
• For clarity, we emphasize that the $z_{k}$ are initially calculated with full precision. But they are iterated with low precision such as $24$ decimal digits.
• $z_{k}$ is calculated for one reference point (i.e., the center of the picture) with high precision and used for the transformation.
• Constrains on that central point: The number of iterations prior to its escape needs to be higher than those for the orbital points. Interestingly enough, however, the point does necessarily need to be bound.

A small test program exercises a variety fractal iterations that can be selected individually at compile-time. The default configuration uses the relatively quick-to-calculate seahorse image configured in mandelbrot_cfg_MANDELBROT_05_SEAHORSES.h.

Continuous integration runs on Ubuntu and MacOS with both GCC/clang and also runs on Windows with MSVC. GCC's run-time sanitizers are also used in CI in order to help assure dynamic quality. CI uses the develop branch of modular-boost for multiprecision types.

Improvements on the TODO list include (among others) the following.

• Experiment with (and implement) massive inner loop parallelization via GPU programming.
• Find/use an acceleration technique (intended to reduce required precision of deep dives) such as orbit perturbation, series expansion, etc.
• Discover more interesting coordinates for iteration (help wanted).

We use a modified version of jpeg-6b. It is located here. According to license agreement, this work is referred to as the Independent JPEG Group's software. Please see their original README in its entirety for more information about and licensing terms of the Independent JPEG Group's software.

The following adaptions have been performed.

• Test/compile only the subset of functions empirically found to be needed for use with Boost.Gil.
• Adapt to VS compiler and VS solution workspace (when building with MSVC).
• Use GNUmake to build the JPEG library (when building on *nix).
• Run the *.c/*.h files through the Artistic Style automatic code formatter, using a version of AStyle from somewhere around 2015.
• Remove several unused preprocessor options such as short names for linkers.
• Eliminate all uses of NEED_FAR_POINTERS and NEED_SHORT_EXTERNAL_NAMES.
• Handle Level-3 warnings found in MSVC.
• Handle GCC warnings from -Wall, -Wextra, -Wpedantic, -Wconversion and -Wsign-conversion.

The Mandelbrot Discovery program aims to provide semi-automatic assistance to find interesting points for fractal deep dives.

This project is work in progress which is being undertaken in the MandelbrotDiscovery directory.