Wikipedia:Reference desk/Archives/Mathematics/2020 May 10

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May 10

Mandelbrot set

In Mandelbrot set#Image_gallery_of_a_zoom_sequence there are 17 images, 0 thru 16. I can locate image "N+1" in image "N" in most of the images - though there are a couple that have multiple possible locations. But I cannot locate image 16 within image 15 anywhere. Image 16 has a lot of dark blue to the W, N & E and traces of dark red in the NE & NW corners. I have not managed anything using the interactive viewer. Can anyone locate 16 in 15? Thanks. -- SGBailey (talk) 18:09, 10 May 2020 (UTC)

The links to the Deviant Art source images suggest that 15 & 16 have been reversed.

• Our File:Mandel zoom 15 one island.jpg

Source: Mandelbrot-set-Step-16-100663910

Coordinates of the center: Re(c) = 0.7436438870357766659094695 Im(c) = 0.1318259042126211322095025

Magnification relative to the initial image: 2.4942758E14

• Our File:Mandel zoom 16 spiral island.jpg

Source: Mandelbrot-set-Step-15-100663743

Coordinates of the center: Re(c) = 0.743643887035762993093485 Im(c) = 0.13182590421259917627678

Magnification relative to the initial image: 3.4426202E13

-- ToE 19:03, 10 May 2020 (UTC)

Hmm... They look to me like they're in the right order. Image 16 has link to open it in an interactive viewer, so probably the easiest thing is to open it there and zoom out a few times. From what I could see, 16 is a very small rectangle near the top, just to the right of center, of 15. It can be hard to tell for certain though; two different locations on the set can appear virtually identical. --RDBury (talk) 19:07, 10 May 2020 (UTC)

Hmm^2. I don't know what you call the shape in WP:16, but it is a light blue trilobite on a dark blue background. If WP:16 should actually be found in WP:14, I can see none of those either. (I think I can see WP:15 in WP:14 though) And then I cannot see WP:15 in WP:16 - the colours don't match. -- SGBailey (talk) 19:46, 10 May 2020 (UTC)

What I want to be able to do is take a copy of 14, crop a rectangle from it and enlarge it a bit and have it look the same (bar resolution) as 15. And then do the same from 15 to 16. -- SGBailey (talk) 19:48, 10 May 2020 (UTC)

I agree with RDB now, though the zoom appears to be about three times more than the 7X suggested by the DA magnification numbers. Looks like e-mc3 at DA just swapped the two images in their upload, and our User:Lanthanum-138 (AKA User:Double sharp?) corrected their order here. SGB, look again at the projection at the top, above and just barely to the left of 15's center spiral, but look for a stronger zoom than you might have at first, so a mid-sized (or small but not tiny) spiral is just barely cropped out of the bottom of 16. (The right side of 16's crop has about the same x-(Re-)coordinate as the center of 15's center spiral.) See it? -- ToE 19:56, 10 May 2020 (UTC)

I tried doing what SGB was thinking about (since I just happen to have GIMP installed). Again, it's really hard to be sure because of the quasi-self-similarity going on, but my best guest is starting with [1], image 16 is in the 16x12 pixel box with upper left corner at (581, 50) (counting left to right and top down as they do in most graphics editing software). At that level of magnification you just get a pixelated mess so I'm not that sure how useful that is. A better approach might be to check the parameters given in the source files, the algebra would be straightforward but the extra precision needed would be cumbersome. --RDBury (talk) 22:38, 10 May 2020 (UTC)

Hmm indeed. I was looking at something a bit larger, roughly the 56x42 box with a (561, 50) upper left corner, but looking closer I see that no, it isn't a perfect match. (The relative height of the bottom two significant spirals in my crop differ from those in 16.) Given the self-similarity, yours seems possible, but like you said, it is a pixelated mess. This suggests that the 7X zoom from the DA parameters are farther off than I earlier suggested. -- ToE

Yeah that larger rectangle seem pretty close too; maybe if you rotated it a bit it would work. Now I'm starting to doubt whether the link for the interactive viewer is giving the right position; the coordinates don't seem to match those given by e-mc3 anyway. This thread is starting to look like the Mandelbrot set, only instead of getting more detail as you zoom in you get more confusion. --RDBury (talk) 01:28, 11 May 2020 (UTC)

PS. I think [2] is a better match for the location of 16 in the interactive viewer. If anyone agrees and no one disagrees I'll go ahead and put that in the article. I got this with a bit of URL hacking using the e-mc3 coordinates from the original web page. --RDBury (talk) 01:55, 11 May 2020 (UTC)

Double-good job! Not only did you nail that interactive viewer URL, zooming out shows you were right about the ~ 16x12 box. You rock! (Looks to be ~ 64X zoom between 15 & 16.) -- ToE 02:53, 11 May 2020 (UTC)

Wow, thanks. It's nice to know that all those hours I spent as a youth playing with Fractint weren't a waste of time :) --RDBury (talk) 20:45, 11 May 2020 (UTC)

I'm not 100% convinced by the 16*12 box, but at least it looks plausible, so I'll take that as a "resolved". Thank you both very much. Is one of you going to update the main article in some fashion - I am not confident enough to do so. -- SGBailey (talk) 06:30, 11 May 2020 (UTC)

SGB, the onscreen documentation for that interactive browser is "Web Mandelbrot - click any point to zoom in, click near sides to zoom out.", and each zoom step is 2X. When I took RDB's starting link (which sure looks like 16) and then zoomed out 6 times (64X), the result sure looks like a panned 15. (I don't see how to pan other than zooming out one too far, then zooming in at your desired center.) The different pallet doesn't help comparison, and with the time to render, it is easy to get lost in all the spirals. So I did it again, but by copying each step into a new browser tab before zooming out until I had 7 tabs ranging from 1x to 16X. Stepping through tabs then made following the path of the shrinking box a bit easier. Plus, it was only after I first found what appeared to be 15 did I compare zooms and find the 64X, which agrees with RDB's box size: 64X 16*12 = 1024*768, matching the file size of 15 RDB was using. Please give it a try and see if your eyes agree with mine. -- ToE 11:04, 11 May 2020 (UTC)

The interactive viewer in the link sides heavily with ease of use in the trade-off with functionality, so zoom in or zoom out are about the only things you can do. But for an article that's meant to accessible to the general public it's probably best to have the simplest controls possible. There are many Mandelbrot/fractal viewers out there though, and I certainly haven't tested all of them. What I meant by URL hacking above was that I took the URL from the link, pasted it into a text editor, then copied the coordinates from the e-mc3 page (allowing for a missing minus sign) and pasted them into the URL to replace the ones from the original, and finally pasted the updated URL into my browser. So the center at least of the new link should match the center of 16, but the magnification factor is a different matter. On the e-mc3 page it gives the magnification level relative to the original window, and not knowing the size of the original window I took a guess which would be correct up to an order of magnitude. Taking the reciprocal, since the interactive viewer uses the size of the window instead of the magnification level, I pasted this value into the URL in a process similar to that above, that gave me correct coordinates up to level of magnification. Finally, since the viewer only allows zooms by a factor of 2, I manually adjusted the magnification level in the URL to get the best fit with 16. The upshot is that the center, at least, is accurate, and the level of magnification is accurate to about 1 and a half significant digits.

The coordinates in the original link to the viewer are off by about a millionth, but given that the level of magnification is around 10¹³, it's really not even close. My theory as to what happened was the anonymous editor who added it used the viewer to follow the steps given in the gallery without using the actual coordinates. This process worked surprisingly well, but somewhere around image 10 there are two nearly identical "islands" and he/she picked the wrong one. It's an understandable mistake since the two islands appear to be, feature for feature, the same; there is only a slight difference in size and in escape times (which would affect color), and those are variables you'd be ignoring to allow for differences in viewers. It turns out that the similarities in the islands carry over when you zoom in, so the end result was two nearly identical images from different locations. Anyway. I'm going to go ahead and update the article with the new coordinates for the link. --RDBury (talk) 20:45, 11 May 2020 (UTC)

15 & 16 have a different source than 0 - 14. 0 - 14 were "own works" of User:Wolfgangbeyer, uploaded in September 2005. Wolfgangbeyer added the sequence to our article in December 2006. 15 & 16 were uploaded by User:Lanthanum-138 in February 2011, sourced from e-mc3's Deviant Art page, but weren't added to Mandelbrot set#Image gallery of a zoom sequence until July 2016 when 49.144.196.203 unsuccessfully attempted to add 15 on 29 July and User:Eleuther properly added them both the following day, remarking "added last 2 images of sequence ... not sure why they were left out", presumably after noting the previous day's change. The Commons user page commons:User:Wolfgangbeyer includes those first 15 images and also 14 "zoom to" images with a box in one image showing the borders of the subsequent image. -- ToE 00:49, 12 May 2020 (UTC)

It's a bit confusing (not surprising) as to when the link was added because there was a similar link added earlier for image 6. So I thought it was an anonymous user, but apparently it was actually this revision by User:Fluoranium hexafluorostibanuide. It looks like many of his/her other additions to the article were removed shortly after. If what you're worried about is inconsistency between the last two images and the rest they look ok to me; I've been staring at these things so long that I think I can tell which of the islands in question an image is taken from (usually). One thing I noticed though after putting the corrected coordinates into the article is that it's at the maximum zoom level for the that particular viewer. So you can zoom out from there but not in, and maybe it might be better to use one of the previous zooms instead. Btw, the zoom sequence is very similar to p81-86 of The Beauty of Fractals; it's been nagging at me that I'd seen it before. --RDBury (talk) 09:04, 12 May 2020 (UTC)

Former Lanthanum-138 here (now Double sharp): indeed, I got steps 15 and 16 from e-mc3's DeviantArt gallery. As you can see, his steps 0 through 14 match the WP articles' ones; it being over nine years ago, I cannot remember if I thought e-mc3 was the same user who created our step 0 through 14 images then. You can see on that gallery a 2012 comment explaining that they are two different people, linking to commons:User_talk:Wolfgangbeyer (the uploader of 0 through 14), but I wouldn't have seen that when I uploaded these in 2011...anyway, yes, I reversed 15 and 16 compared to e-mc3's numbering, because it was obviously wrong just from the colour schemes (and also, look at the magnifications listed on e-mc3's DeviantArt pages). Double sharp (talk) 14:26, 13 May 2020 (UTC)