Mandelbrot & Co

explore the infinite beauty of fractals

FR

Mandelbrot & Co offers an immersive experience in the world of fractals, objects whose beauty fascinates mathematicians and artists.
Through a streamlined interface you explore in real time complex worlds for a unique contemplative experience. As with any trip, you can come back with photos and even a few short movies to share. Take the time and have fun!

Guided tour in Mandelbrot & Co

Some video examples from fractal journeys are in Gallery. We will be so delighted to publish the videos you made on Mandelbrot & Co. Please, make your own videos, publish on YouTube, send us the links !

Some aesthetic considerations

Each of these sets has a clean aesthetic. Very close to the sets of Julia, Mandelbrot or its variant in cos(z), The Burning Ship has a singular and particularly rich spectral beauty

Julia offers an infinite variation around the theme of autosimilarity, the journey may seem relatively monotonous.

Mandelbrot's cos(z) variant is characterized by “kaleidoscope” images and floral landscapes.

The Burning Ship features lace and distortions that allow the mind to recognize a variety of shapes : cats, crosses, Eiffel towers monsters, ethnic marks, ghosts and other nightmarish appearances. Probably the most amazing of these ensembles with areas reminiscent of the greatest artists from Van Gogh to Turner. A set not always easy to access but certainly the author's favorite;)

How does it work?

All calculations are done locally, there is no exchange with servers. Calculations are fast on newer computers: each image is divided into as many processors as your computer has (often 8 nowadays).
Each part of the image is therefore calculated in parallel. Calculation times are in the order of 50 to 400 milliseconds per frame with an Intel i7 process. Only the variant of the Mandelbrot set in cos (z) - that is, in complex cosine - of the point requires heavier calculations that are felt on the calculation time (from the order of the second to several seconds for deep zooms).

In practice we iterate (we repeat the calculation on the previous result) a function for each point and see if it diverges (it exceeds a certain value). At small scales, you may have to iterate more than 1000 times for each point and this on all the 640x640 = 409,600 points!

The program is written in Javascript. This language models floating numbers (decimal points) with a maximum of fifteen significant digits. This is why the image definition becomes bad beyond a magnification of the order of 1015. Which limits the magnification to just 1 million billion times...
When you get into the game of exploration it may be frustrating to reach the limit but going beyond that limit with Javascript would make the navigation much, much, much slower (several tens seconds for each image).

Browser Performance

The microprocessor and graphics card of the computer or tablet play a great deal on speed. But the browser used also. Chrome and Microsoft Edge give the best results (both based on the same Chromium platform). Firefox does not handle well beyond two processes (“worker”) in parallel, calculation times on the same machine are roughly 50% slower.

Where to start?

The easiest way is to let yourself be guided by your instinct. Some areas can be repetitive or even disappointing. To start your journey in the middle of fractals, you are offered for each set a few POI Points of Interest. You can access the POI list by clickingin the command bar.

Orienting

Points of Interests

For each set, you are offered a number of “places” for their interest and help you get started in your exploration. By clicking on the corresponding thumbnail, the area will be displayed directly and you will be able to continue exploring normally.

Center on a point

To find a view from the name of an image saved from our site - see the filename below - or because a specialized article mentioned a particularly interesting point, you can use the “center on a point” function.

Julia Set

For Julia set do not forget to indicate before starting the search the number “c”. Unlike other sets to explore, Julia's set, or more precisely “Julia's” sets depend on a parameter. Usage calls this parameter (a complex value) “c”.
You can enter the values of “c” in the fields below the image: rc indicates the actual part of “c” and ic its imaginary part.

How to zoom in Fractal Explorer?

Zoom with the mouse wheel

By pointing the mouse at a particular point, if you press the wheel you enlarge the scale, and vice versa. Each action on the wheel allows you to zoom in or out by a factor of 2.

Zoom finely

If you click on a point in the image a frame appears and follows your mouse. You can cancel by pressing Esc.
On the second click, you define the zoom frame.

Zoom on a touch screen

To zoom by a factor of 2, just press twice quickly. The image will be centered on the affected point. To zoom out, tap the “Show previous view” icon

Logarithmic scale

A logarithmic scale allows you to know the zoom level. The scale turns red around 10-13, area from which accuracy may begin to become insufficient. This results in a coarser pixelation of the image.

Filters

Mandelbrot & Co offers some filters to apply on the main view. Filters apply on videos and image backups except on Safari. For videos, this significantly lengthens the calculation time.
You can also play on hue, saturation and brightness. In some cases the filters may slightly tarnish the image, this effect can be compensated for by increasing the brightness.
The filters used are convolution filters (see links to wikipedia below). For example the Contour Lines filter is the combination of the Sharpen filter and the Laplace filter, used for the Edge Detection filter.
Applying the Contour Lines, Edge Detection or 3D shading gives effects very interesting as you can see below.

No Filter
Edge Detection
Contour Lines
3D - Phong Shading

Favorites

By clicking on theYou add a thumbnail to your favorites in one of the 12 slots on the left side of the screen.
By clicking on the trash canin the top left, you can select the favorites you want to delete. The removal is effective by recliking on the garbage can.

Favorites are very useful for memorizing a “place” during your explorations. Your favorites are stored locally in your browser. You find them when you return to the site.

Save Images

You have the option to save the image in png format on your hard drive. The image will be saved in your Downloads folder of the browser you are using.

The file name allows you to find the image by its coordinates using the centering system.

The first letter indicates the set (M: Mandelbrot, B: Burning Ship, J: Julia and C: CoSZ). The following numbers are used to locate the image according to the nomenclature: (rz) +i (iz) ± (Δ) with the real part (rz), the imaginary part (iz) and the Δ

Example: M-0.632173+I0.451271±5.11E-4.png indicates that this is an image of the Mandelbrot (M) set, centered on rz=-0.632173 and iz=0.451271 and for a Δ=5.11E-4

For the whole of Julia, we also need to know the reference point “c”. The file name is completed according to the nomenclature: (rz) +i (iz) ± (Δ) rc (rc) ic (ic) .png with rc real part and ic imaginary part.

To infinity and beyond.

With the diving optionyou can create a small movie that will zoom in on the currently displayed area. For Chrome and Micosoft Edge browsers a free WebM movie will be available for download. If you use Firefox or Safari browsers the encoding process is not supported. The site offers for these two browsers only a view of a succession of images (technically it is simply a slide show) with an equally striking effect but unfortunately not downloadable.

In any case the process of creating images and, if necessary, encoding in WebM format, may be quite long depending on the desired area and zoom level. For the moment, films of reduced size (480 pixels x 270 pixels) are proposed to ensure an acceptable calculation time, which can nevertheless be in the order of ten minutes. As an option you can opt for a full-HD movie (1920 x 1080) but you will have to be patient!

On small screens, such as 9" or 10" tablets, it is not possible to make movies.

Useful Links

Wikipedia's must-see

To go further

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Contact

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