Commons:Featured picture candidates/File:Mandelbrot Set Image 112.png

File:Mandelbrot Set Image 112.png

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Gallery: Commons:Featured pictures/Non-photographic_media/Computer-generated#Mathematics
Info Deep zoom near the extreme western tip of the Mandelbrot set at X ≈ -2.0 (specifically -1.999995...). View size: 1.15 × 10⁻¹¹⁹. The structure exhibits a striking symmetry resembling a dividing cell, or a stylized infinity symbol (∞). Rendered using perturbation theory with 8×8 SSAA (Super-Sampling Anti-Aliasing). C++ source code is included below. CPU AMD Phenom II X6 1055, 6 x Сore. Time spent: 21 hours. Created, uploaded and nominated by Aokoroko -- Aokoroko (talk) 10:32, 18 July 2026 (UTC)[reply]
Support -- Aokoroko (talk) 10:32, 18 July 2026 (UTC)[reply]
Support Wikipedian12512 (talk) 13:17, 18 July 2026 (UTC)[reply]

Comment Sorry, this vote has been struck because the voter does not yet meet the FPC voting requirements (minimum 100 Commons edits). Please feel free to participate again once the eligibility requirements are met. Thank you for your understanding. -- Radomianin (talk) 15:24, 18 July 2026 (UTC)[reply]
@Wikipedian12512: Please read Commons:Featured picture candidates#Voting -- George Chernilevsky ^talk 15:29, 18 July 2026 (UTC)[reply]

Comment For what it's worth, the top/bottom crop feels too tight. That is, I feel like there should be slightly more "blue" visible. Also, it's impressive that you can render detail at that level of 'zoom', but i'm curious if this particular structure at ('cell division') was already known or if you 'discovered' it while searching for an interesting structure. tyvm Pdanese (talk) 19:43, 18 July 2026 (UTC)[reply]
I'm doing more "blue" now! 21 hours!

Well, I was looking for interesting structure, yes. Aokoroko (talk) 20:22, 18 July 2026 (UTC)[reply]
Oppose This color combination is too vibrant and the image itself isn't that good either. Wolverine X-eye 09:00, 19 July 2026 (UTC)[reply]
Done Aokoroko (talk) 18:15, 19 July 2026 (UTC)[reply]
Support --The Cosmonaut (talk) 18:38, 19 July 2026 (UTC)[reply]
Support very nice, mathematically generated image -- George Chernilevsky ^talk 18:48, 19 July 2026 (UTC)[reply]
Support This is one of the images I like more. I wonder where within the Mandelbrot set we have a structure that connects both shapes. --PantheraLeo1359531 😺 (talk) 20:35, 19 July 2026 (UTC)[reply]
Addendum: I am glad I have an AMD TR 7970X, doing this with a Phenom must be a lot of pain :D --PantheraLeo1359531 😺 (talk) 20:36, 19 July 2026 (UTC)[reply]
Support It is very time-consuming to find such regions.--Majow (talk) 07:54, 20 July 2026 (UTC)[reply]
Support Юрий Д.К. 09:41, 20 July 2026 (UTC)[reply]
Comment @Aokoroko: It would be great to include in the image file a scheme showing the exact location, in Mandelbrot set, where the picture was calculated. I assume that no part of this image depicts parts of the Mandelbrot set itself, which is usuaaly painted black. Which would mean that this was taken from somme peripheric part and the differemt colours correspond to different numbers of iterations before the cycle diverged. Am I right? -- Alvesgaspar (talk) 12:47, 20 July 2026 (UTC)[reply]
Scheme showing the exact location, in Mandelbrot set - and how??? Is there one somewhere? At X ≈ -2.0 (specifically -1.999995...)!!! Aokoroko (talk) 13:05, 20 July 2026 (UTC)[reply]

Info See here: https://en.wikipedia.org/wiki/File:Mandelbrot-with-xy-grid.png -- Alvesgaspar (talk) 13:10, 20 July 2026 (UTC)[reply]
You are partly right! Yes, the vibrant colored areas represent the escape time (the number of iterations before the sequence diverges to infinity).

However, parts of the Mandelbrot set itself are actually present in this image. Inside those small 'infinity' structures, there are tiny, dark island copies of the main Mandelbrot set (micro-cardioids).

As mentioned in the description, this is a deep zoom at X ≈ -2.0 (specifically -1.999995...). This location is at the very tip of the main antenna (the extreme western needle) of the Mandelbrot set. Since the zoom is incredibly deep (1.15 × 10⁻¹¹⁹), making a visual map/scheme is practically impossible, as any standard resolution overview map would look like a single atom or completely disappear at this scale. The exact numerical coordinates provided are the only way to locate it. Aokoroko (talk) 13:15, 20 July 2026 (UTC)[reply]

Re = -1.99999543561201124623198345433951143502785679245726844745821388800402678

499411681518036306219179273434395557574279985918047221291197081186140687781560831995

and

Im = -0.00000000000000000000000026198152173811047783694060060607013913873144250

985383083459221663448338433592617272786772587281530484110756597337683912309313885172

Width = 0.76e-119 Aokoroko (talk) 13:31, 20 July 2026 (UTC)[reply]

And if you zoom in - not by 10e-119 but by 4.6e-162 (and 162 digits of Re and Im) - you can see a large Mandelbrot set! Aokoroko (talk) 15:37, 20 July 2026 (UTC)[reply]

Re = -1.999995435612011246231983454339511435027856792457268

4474582138880040267849941168151803630621917927343439555757

427998591804722129119708118614068778156083199446008941139

and

Im = -0.000000000000000000000000261981521738110477836940600

6060701391387314425098538308345922166344833843359261727278

677258728153048411075659733768391230931388517168205369880

4.6e-162 A large Mandelbrot set! https://commons.wikimedia.org/wiki/File:Mandelbrot_large.png Aokoroko (talk) 16:42, 20 July 2026 (UTC)[reply]
At this magnification, however, iteration counts higher than 50,000 are required. Majow (talk) 18:00, 20 July 2026 (UTC)[reply]
Yes! Aokoroko (talk) 18:27, 20 July 2026 (UTC)[reply]

I think it was already demonstrated that there are no disjointed parts of the Mandelbrot set. That is, all of them are interconnected. -- Alvesgaspar (talk) 18:39, 20 July 2026 (UTC)[reply]
Well, yes! Aokoroko (talk) 19:03, 20 July 2026 (UTC)[reply]

According to en:Mandelbrot set#Basic properties: Douady and Hubbard showed that the Mandelbrot set is connected. They constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk. Mandelbrot had originally conjectured that the Mandelbrot set is disconnected. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of $M$ . Upon further experiments, he revised his conjecture, deciding that $M$ should be connected. A topological proof of the connectedness was discovered in 2001 by Jeremy Kahn.^[1] Majow (talk) 23:10, 20 July 2026 (UTC)[reply]

Support --MZaplotnik^(talk) 16:45, 21 July 2026 (UTC)[reply]
Comment So where are they: Pdanese and Alvesgaspar? Aokoroko (talk) 19:11, 25 July 2026 (UTC)[reply]

Comment Indeed, I have a question. I assume that no point in this image belongs to the M-set, that is, iterations diverged to infinity in all of them. Which means calculations were made in the close periphery. Am I right? -- Alvesgaspar (talk) 21:20, 25 July 2026 (UTC)[reply]
No! The point in the middle (horizontal and vertical – both in the center) – in e-162 gives the Mandelbrot set!!! https://commons.wikimedia.org/wiki/File:Mandelbrot_large.png Aokoroko (talk) 23:07, 25 July 2026 (UTC)[reply]
If the point in the middle belongs to the Mandelbrot set, then there must be other M-set parts connected to it because the whole M set is connected. Were they caught by your algorithm? -- Alvesgaspar (talk) 23:21, 25 July 2026 (UTC)[reply]
I am familiar with the algorithm: the algorithm is based on a dot grid and smooths the calculated colors by averaging; consequently, it cannot capture these fine filaments. However, this is not a flaw, but an artistic decision. --Majow (talk) 06:37, 26 July 2026 (UTC)[reply]
You are absolutely right mathematically: the Mandelbrot set is connected, and these parts must be linked. However, there is a huge difference between mathematical truth and pixel resolution. At this extreme zoom (Width = 4.6 × 10⁻¹⁶²). When we look at the wider view (10⁻¹¹⁹), the grid of pixels is way too coarse. The connecting threads are billions of times smaller than a single pixel at that scale, so they completely "drop through the cracks" of the grid. They exist, but they are physically invisible to the rendering algorithm at that magnification level. It's like trying to see a 1-millimeter wire on a map of the entire solar system. Aokoroko (talk) 12:22, 26 July 2026 (UTC)[reply]

↑ Kahn, Jeremy (8 August 2001). The Mandelbrot Set is Connected: a Topological Proof.

[1] Kahn, Jeremy (8 August 2001). The Mandelbrot Set is Connected: a Topological Proof.

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