# Restoring Bill Thurston's figures in an answer

The question Floating polyhedra with fair equilibria was answered by Bill Thurston, but his (typically amazing) answer contains links to 7 figures, all now broken. I don't know how to retrieve those images from the past, but if someone does know how and is willing to spend the time & effort, it would be a nice contribution to the MathOverflow community.

Here are the other MO posts by Bill Thurston with missing pictures:

Explicit metrics

Perron number distribution

https://mathoverflow.net/a/38025

https://mathoverflow.net/a/39304

https://mathoverflow.net/a/39551

https://mathoverflow.net/a/39571

https://mathoverflow.net/a/39607

https://mathoverflow.net/a/50607

https://mathoverflow.net/a/51042

https://mathoverflow.net/a/52875

https://mathoverflow.net/a/52973

https://mathoverflow.net/a/53183

https://mathoverflow.net/a/53848

• I added a list of all the Thurston posts with missing pictures that I could find: feel free to edit if I missed any or if any get fixed!
– j.c.
Jun 24, 2019 at 23:40
• perhaps someone close to Bill Thurston can restore these images from his Dropbox folder? Jun 25, 2019 at 11:15
• It seems that many of those links were pointing to DropBox. Why not contact DropBox directly, explain who Bill Thurston was and aske them if those files can be restored and offered to the MO community? Jun 25, 2019 at 13:02
• In other news, we really need to auto-convert image links to stack.imgur. Jun 26, 2019 at 11:02
• @NoahSnyder: He may have access to Bill's computer, though; the account isn't necessarily the easiest place to look. Jun 29, 2019 at 9:42
• It might be useful to chase for links to figures, maybe starting from most upvoted answers, beyond the case of Thurston, so as to anticipate on possible missing figures in the future.
– YCor
Jul 1, 2019 at 19:03
• @YCor Doing something like that might be a reasonable topic for a separate discussion. I have made at least a small experiment and left a few comments on archiving images, dead links. Jul 14, 2019 at 12:34

I couldn't find the images, but the Mathematica notebook he posted still seems to be accessible via the Wayback Machine!

I've put a copy temporarily at https://tqft.net/Buoyancy.nb. If someone wants to do the remaining steps, and then let me know that I can take the file down, that would be great.

• I've just posted a comment on the answer with a link to the Wayback Machine copy of the Mathematica notebook.
– j.c.
Jun 24, 2019 at 23:36
• Thanks, @‍ScottMorrison and @j.c.! I have added links to as many of the figures as I could figure (ha!) out. I'm not sure what RightingTorque is. There is a picture of an icosahedron, but none of a dodecahedron; and two figures labelled DodecaFloat, so I wasn't sure which belongs. Jun 25, 2019 at 1:14

I have reproduced two more figures for which the Mathematica code was given [(1) and (2)], and I have written to Dropbox Support, to inquire if they can restore the broken links.

[Most likely the files are still there, but Dropbox removed support for public links in 2017. See this message.]

UPDATE: Dropbox customer support answered: "As I do appreciate your situation and concern, I apologize that we will not be able to assist with this issue."

For the record, here is the collection of broken links:

http://dl.dropbox.com/u/5390048/NonconvexNonsingular.jpg
http://dl.dropbox.com/u/5390048/TilingSlabs.jpg
http://dl.dropbox.com/u/5390048/CurveEmbellishment.jpg
http://dl.dropbox.com/u/5390048/TetrahedronPoints.jpg
http://dl.dropbox.com/u/5390048/chippedsquare.jpg
http://dl.dropbox.com/u/5390048/CornerCloseup.jpg
http://dl.dropbox.com/u/5390048/OnionPeels.png
http://dl.dropbox.com/u/5390048/Chebyshev10.jpg
http://dl.dropbox.com/u/5390048/Chebyshev6.jpg
http://dl.dropbox.com/u/5390048/MonobumpFlower.jpg
http://dl.dropbox.com/u/5390048/Chebyshev5.jpg
http://dl.dropbox.com/u/5390048/InterpolatingFunctions.jpg
http://dl.dropbox.com/u/5390048/CutLocus.jpg
http://dl.dropbox.com/u/5390048/ConjugateLocus.jpg
http://dl.dropbox.com/u/5390048/Tree%20pictures.jpg
http://dl.dropbox.com/u/5390048/splines.jpg
http://dl.dropbox.com/u/5390048/Tree%20pictures.jpg
http://dl.dropbox.com/u/5390048/BernsteinImages.jpg
http://dl.dropbox.com/u/5390048/PositivePolynomials.jpg
http://dl.dropbox.com/u/5390048/genus_2.gif
http://dl.dropbox.com/u/5390048/orange_torus.gif
http://dl.dropbox.com/u/5390048/PerronPoints5%2C3.jpg
http://dl.dropbox.com/u/5390048/PerronPoints21.jpg
http://dl.dropbox.com/u/5390048/CDF21.jpg
http://dl.dropbox.com/u/5390048/PerronPoints21%281.5%2C2%29.jpg
http://dl.dropbox.com/u/5390048/PerronPoints21%283%2C4%29.jpg
http://dl.dropbox.com/u/5390048/NegativeMetrics.jpg
http://dl.dropbox.com/u/5390048/4point.jpg
http://dl.dropbox.com/u/5390048/6point.jpg
http://dl.dropbox.com/u/5390048/FloatingSquare.jpg
http://dl.dropbox.com/u/5390048/FloatingDiamond.jpg
http://dl.dropbox.com/u/5390048/FloatingSquareEnergy.jpg
http://dl.dropbox.com/u/5390048/FloatingCubeEnergy.jpg
http://dl.dropbox.com/u/5390048/RightingTorque.jpg
http://dl.dropbox.com/u/5390048/Dodecahedron.jpg
http://dl.dropbox.com/u/5390048/DodecaFloatEnergy.jpg

• I've added spaces to force linebreaks and make the list easier to track.
– Asaf Karagila Mod
Jun 28, 2019 at 8:08
• Thanks. It's quite unfortunate that the links are broken, I had hoped that Dropbox would have been willing to restore the "public link" functionality for a brief period of time and only for this particular account, but it seems that's not going to happen. Jun 28, 2019 at 8:13
• @CarloBeenakker Dropbox, inc. being a company with a market valuation well in the billion-dollar-range, I'd have been surprised if they would have made such exception -- companies of that size tend to act somewhat similar to authorities. At least you would need to convince them that making the suggested exception would yield sufficient positive- or not making it would yield sufficient negative publicity for them.
– Stefan Kohl Mod
Jun 28, 2019 at 16:37
• I talked to a friend at DropBox and unfortunately data is likely gone now due to automatic deletion of free accounts when an account is inactive for a certain amount of time. They're going to look to see if there's a copy anywhere, but it's unlikely. Jun 28, 2019 at 19:04
• At least we can all be grateful that Dropbox considered the request sufficiently carefully to write a nonsensical reply. Jun 28, 2019 at 21:46
• I wish posts/answers like this would be archived by archive.org (with deep links). Unfortunately it's not the case, it seems. Jun 29, 2019 at 8:18
• Is it possible that Thurston's computer/computer account with a copy of his Dropbox is still available? Jun 29, 2019 at 8:20
• @DimaPasechnik: Sometimes archive.org does provide the Dropbox images: see for example Thurston's Is there an underlying explanation for the magical powers of the Schwarzian derivative?, Can we add two matrices by performing an operation on their eigenvalues & eigenvectors?, A random walk on random lines, and Volumes of n-balls: what is so special about n=5?. Unfortunately this did not happen for the OP's list of 16 posts.
– jeq
Jun 29, 2019 at 14:23
• whether or not archive.org has crawled an MO page appears to be a random event; I have over 300 images in MO answers and about 50 of those have been copied to archive.org; perhaps there is a way to instruct archive.org to systematically copy the whole MO site, but probably this has to be arranged at some higher level; I am now instructing archive.org "by hand" to copy all of these 300 images, one-by-one: I have not found out how to do this in batch. Jun 29, 2019 at 15:21
• IMHO it looks like a good idea if Mathoverflow (and perhaps other stackexchange sites) join archive-it.org (putting on Wayback machine sufficiently highly scored questions and answer looks like a good idea) Jun 29, 2019 at 22:36