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Which Hypercube Unfoldings Tile Space? (Patreon)

Published:
2021-05-11 21:35:31
Edited:
2021-05-14 20:14:13
Imported:
2026-06

Content

UPDATE: The video has now been released! https://www.youtube.com/watch?v=Yq3P-LhlcQo Thanks for your help everyone.

Ok, there is a lot going on here! First of all: this is the very-rough cut of my next video. It is still missing a bunch of edits/transitions and all of the graphics. Please do not share it.

If you want to follow along with the maths, I’m basing much of it on this 1984 paper. You can see the sorts of diagrams etc I’m referring to. https://unfolding.apperceptual.com/

Any comments you have about the video: please do leave them in the, ah, comments below.

Next up: we need to both test WHUTS.org and recruit moderators. Anyone can submit a tiling or request to be a ‘verifier’ but we need a team of ‘moderators’ behind the scenes. Basically: the sad truth is that any open submission project on the internet is going to be abused. But I refuse to let that stop me! So we need a team of moderators prepared to confirm which submissions are legitimate. To be on the safe side: I am only going to recruit Patreon supporters as moderators. And this does not involve checking the maths at all: merely confirming the upload is a legit submission. This job runs a very high risk of seeing some NSFW stuff, so isn’t for everyone. I’m going to be in there doing my best to weed out all malicious submissions. If you’d like to join me, here is the secret sign-up link: https://whuts.org/register?patreon=2d47f0bf3791

And of course everyone can sign up to be a verifier or submit a tiling. I think it is only fair that you all get the first chance to find some of the tilings because without your support the project wouldn’t even be happening! Head over to https://whuts.org

If you have any issues with the site, either leave a comment below or email matt+whuts@standupmaths.com The site has been built by your fellow Patreon supporter Oliver!

AOB:

- I don’t have any plans to animate the ‘Dali cross’ net unfolding into 2D and then tiling a plane. But if anyone does: pop it in the comments! We’ll keep one comment thread dedicated to that so people are not duplicating effort.

- The Maths Gear link to buy the models is not live yet, but when it’s ready I’ll post it here first. UPDATE: https://mathsgear.co.uk/products/cube-model

- Please don’t leak the unannounced findings in the video.

- Sorry this video also has a sponsor message. Trying to keep everything afloat during the pandemic.

- Hope you enjoy the video!

Files

Unfolded Hypercubes (preview rough cut)

This is a rough cut of the video with all of the graphics missing. You can follow along with the mathematics on this page: Peter Turney's 1984 paper Unfolding the Tesseract Journal of Recreational Mathematics, Vol. 17(1), 1984-85 https://unfolding.apperceptual.com/ And of course join in at WHUTS.org! Here is the stack exchange post: https://mathoverflow.net/questions/199097/which-unfoldings-of-the-hypercube-tile-3-space-how-to-check-for-isometric-space And the arXiv paper: https://arxiv.org/abs/1512.02086 Discuss this video on the Patreon post: https://www.patreon.com/posts/51144440

Comments

Adam Jensen

I personally really like the simplicity of 2D shapes. Any 2D shape can be "unfolded" into a single 1D net. There's just one possibility and it's the same for all shapes! :D

Joshua Randall

Thanks, Matt - great video! Looking forward to the final cut. Breaking math news! One thing to note is that the term "net" is new to me as used in the way it is in this video. After watching the video I now have an idea of what it might mean from context, but it continues to elude me exactly what its definition is. It seems to be some kind of way of "unfolding" a higher-dimensional shape into a lower dimensionality (is this always exactly one lower?). I wonder if it would be worth defining the term in the video since it seems pretty central and is used many times?

Matt Parker

Interesting point! I’ll see if ‘Future Matt’ is free to drop in and provide a definition. If anyone has a favourite definition let me know. Fun side fact: it has not been proven that all convex polyhedrons definitely have 2D nets! Maybe that should be a future video.

Adam Jensen

Signed up as a mod, though I suspect submissions won't show up until after the final video is published. Excited to see this project "unfold"!

Matt Parker

Thanks! I suspect we might get a few submissions from Patreons but we’ll see.

Daniel Powell

Now you have me imagining a dragon fractal of a single 1d line folded into 2d space and kinda..looks like it's trying to fill it, though of course, the overall 2d space itself is finite because it's a fractal, and it won't actually be filled until the line length is infinite. I have a feeling that's not in the scope of this video, tbh, and I don't really know the rules of dragon fractals anyway, except for the very basics a friend of mine tried explaining to me once, so maybe I'm completely out of my element..

Tahg

I've noticed there are several nets marked as pending. What does that mean in the context of the site, since there's nothing in either the moderation or verification queue? Also, a minor correction on the submission page: "Start by showing clearing..." should probably say "Start by showing clearly..."

Oliver Dunk

Hey! Pending means we have a tiling that passed moderation, but the maths still needs checking. If you're not seeing it, it's probably because you signed up with Matt's link and are a moderator. Maybe we'll change that up but that's how it works at the moment! Good spot on the typo. I'll fix it tomorrow morning!

Tahg

Not sure moderation is working. After I hit Accept it does something for a couple seconds, but nothing visibly changes. I expected some sort of confirmation. For example : for the page to update indicating my selection, or for it to no longer appear in the queue for me.

xje4

The WHUTS registration verification email got a somewhat high spam ranking from my provider. I assume it's because both subject and body are very simple short texts and because it's an HTML-only email. Maybe think about adjusting the email so it doesn't get sorted out as spam too often.

Martin Holeys

two things: 1) the sound is kind of quiet 2) im getting lost without the extra graphics you're pointing to :D

Bruce Garner

Why aren't the three you found Matt marked with your name on whuts.org? At least the one from the video should be ...

Tahg

The sound was fine for me. As for the graphics, that's what you get (or don't get) for seeing the extra early Patreon release of the video. TBH I'm not sure how much they are going to help though.

Martin Holeys

as for the sound, I could hear, but I had to put the volume up way up, in compare to other stuff I watch

Tahg

On 159, Image 6, you could simply rotate one of the sections 180 degrees to form a 4x4x2 block. That definitely tiles 3D space.

James Wootton

I have no useful constructive criticism. Only useless praise! This seems like an awesome basis for a citizen science project. I will be ready with my retweeting finger when this goes out. I'm also tickled by the surface similarities to a project of my own (flattening hypercubes to a square lattice) which I did a hacky generalization to the 3D structure of your Christmas tree.

Oliver Dunk

Thanks for the heads up, I'll investigate! I do plan on making the emails look more exciting, though it's unfortunate if they're getting designated like that. Do you mind me asking what provider you're using, just in case it helps to reproduce?

Oliver Dunk

This should be sorted now! Matt submitted them yesterday in the normal way. It looks like he got three approvals (turns out he knows what he's doing!) but I needed to give things a nudge for them to get published.

Oliver Dunk

You're right! If you could give it another try, that would be great. It looks like we were preventing you from moderating them because... they weren't moderated yet.

Neil McGovern

I was getting similar, the API call was giving 403 errors. However, this seems to be fixed now.

Oliver Dunk

Neil - nice! Just to confirm, I assume you wanted to approve those three submissions? The fix went out in two stages, and it looks like you got one of two - so your review was recorded ambiguously. I can set it correctly in the database based on your verdict.

Matt Parker

We haven't done any of the post production sound balancing etc yet. The final edit should have much better audio. I'll see if I can post it here first so you can check, just in case!

Matt Parker

And yes: this is what we call the "assembly edit" which is really just all the footage put in roughly the right order. I use it to make the graphics and plan any extra voice over I think might be required (I often don't write much of a script in advance, and so the editing process is still technically part of the script writing! It helps to see what we filmed all together before working out what needs to be added or removed.) And you get a sense of what it is like to be me filming with no diagrams next to me but imagining there are!

Matt Parker

Yes, I submitted them the same way anyone would so we could use them to test the moderation and verification process.

Matt Parker

Thanks James. That Christmas Tree is the project which keeps on giving. Hopefully we'll get the video done this week so you no longer have to hold your retweeting finger back.

xje4

Hi, I'm using mailbox.org And just to be clear, the email isn't actually categorized as spam, but the spam score is >5 which might be above the threshold of some other spam filters.

Martin Holeys

I'm happy that wait, I wasn't complaining, just giving feedback ;)

Tahg

fwiw, gmail which can sometimes be aggressive, likes it just fine.

Rick Rubenstein

Wonderful project! You say that two outcomes are possible: either we’ll show that they all tile space or learn which ones do and which don’t. But isn’t there a third possibility: one or more nets for which no one can either figure out a tiling or prove it impossible? I would imagine an impossibility proof might not be trivial.

Karim Maassen

Visually pleasing and quite the interesting video this was. I'm excited to see what the collective mind will produce on WHUTS. There's one question that boggles me, which I'm not sure the video covered (but it could also be me): why would one want to find all these unfoldings? (Besides the obvious "why not?", which is always a good enough reason!)

Flik Frambozen

What are the blocks you used to make the models? Would like to get my hands on those so I can do some physical playtesting - doing it in my head is hurting my brain.

Matt Parker

Good point. I don't want to think about that third option. But you are right: proving it can't be done will not be trivial.

GwynBleidD

I was very, very disappointed that whichhypercubeunfoldingstilespace.org is not working... But that was an easy fix, so it works now! I'm willing to keep that domain up for the lifetime of whuts.org, but if for some reason you want to own that domain and do the redirection by yourself, just DM me, we can arrange something for sure.

Mik Scheper

GMail ALWAYS spam-trashes my emails, even though every person I send email to (hopefully) marks them as not-spam. I always suspected GMail users just have to manually greenlist everything they receive, that's not from Google. I feel targeted now. Are they really that pissed off that I didn't accept their job offer? 😏

Mik Scheper

I'm actually impressed by the way Matt can point to invisible things, floating in space, without looking even slightly like he's feeling awkward about it! 😉

xje4

Noticed a cosmetical mistake on WHUTS: The reset-password-email comes with the subject "Verify your account"

Adam Jensen

For moderation, I'm trying to just keep a tab open. I've noticed two things: 1) I have to refresh to see new tasks, and 2) I get signed out pretty often. I'm totally content with the current system. You don't have to change it. However, if it's easy to change, I'd love to see the task list refresh periodically on its own, and it would be nice if the token expiration were longer so I didn't get signed out so often. Thanks!

Gary Zubick (edited)

Comment edits

2026-09-08 08:00 Quick question: How many of the 10D's will unfold to the 2D plane?
2021-05-12 18:33

Gregor Shapiro

Perhaps the continuation will explain the Kalabi-Yau structure of string theory?

slugbiker

this is cool. makes me think of the Robert Heinlein short story: -And He Built a Crooked House.

Sullivan Figurskey

Ah, duh! Admittedly it was late by the time I found it, but that is certainly a more elegant solution

Matt Parker

Thanks! That is very kind of you! I have been trying to reduce my domain habit of just buying every variation for every silly project I think of. But that is a nice URL to have working. If you're ok to keep it redirecting to whuts.org while the project is ongoing that will be great. If it ends up being widely used, we can have a think about the best legacy option after.

Matt Parker

I think it is an open question if even one does that.

J.R. Hyde

I think I've solved #56. proof in progress.

J.R. Hyde

I think 8, 9, 56, 62, 67, 71, 83, 99, 117, 144, 177, 193, 220, and 225 comprise a family of unfoldings that can all be solved with a nearly identical arrangement. I've submitted #56 and #144, but now I'm writing up a more thorough explanation that applies to all of them.

Tahg

The following can all create 2 independent diagonal layers, 4 wide and infinitely long, and thus can easily be tiled. Listed here along with their tetromino name as viewed from their respective side. T shapes cannot be tiled in this way, and there's 1 or 2 in which the diagonals go in opposite directions which aren't in this list either. Of these, Matt has solved 73, and 120 was listed in the paper, but the other 6 work in exactly the same way. 31 (II) 72 (II) 73 (JL) 120 (II) 140 (SJ) 179 (JJ) 242 (LL) 243 (SL)

Matt Parker

That’s really nice! Could you make a version at 4K resolution (basically about 2160 pixels high) and on a black background?

Matt Parker

PS I can totally use what you’ve already done if you don’t have a chance to change it.

John Sawyer

I could do that but won't be able to get to it until Sunday. Would that work?

Tahg

I find it useful to define an "ideal" pattern, as one that can be repeated by shifting in a single axis at a time. This way you can treat it as a tiling of a cuboid in an axis aligned grid. It looks like this family of unfoldings can do that by stacking 1, 2, or 4 copies in the Z axis, shifting in the X axis appropriately.

Matt Parker

Maybe! The video will go out Friday or Monday. If we somehow turn it around for Friday I’ll use this version, no problem.

John Sawyer

Err alright let me see what I can do.

Matt Parker

Don’t go out of your way! Very happy to use what you’ve done already.

Oleh Shvets

Where can I find a *.stl model for a cube, you’ve used in a video?

John Sawyer

Oh and it says it is still processing the higher res on a 4K version will be available eventually. You Tube eh? Anyway happy to help a fellow Aussie. :D

Matt Parker

Cheers mate! I’ll credit you as “John Sawyer” unless you prefer otherwise.

Oliver Dunk

This is fixed now - thanks again.

Georgios Papoutsis

Have a (computer generated) solution for most of the first 100 so far and counting... (did not find one only for 76). Do you have a preferred method for sending in a batch of solutions instead of me spamming the whuts.org site (I manually double checked and submitted the one for number 2)?

Georgios Papoutsis

I let the simulations finish, and also run 76 again at a higher depth, and I can announce: I have a tiling for every single unfolding! Pretty exciting! I hope I did not mess up anything :) Will try to verify and double check, and find some way to share the results!

Tahg

Wow, is this going to be over pretty much before it's begun? I'm not really surprised to be honest.

Matt Parker

This is great! Don’t worry: I’m not going to abruptly cancel the project. Send me the details over to matt+whuts@standupmaths.com

Georgios Papoutsis

Great! I just sent out an email. I also wrote a verifier in the meantime and run the solutions through it, they look good from my side. I hope somebody else can independently verify these!

Matt Parker

I'm inclined to say that some 3D images would be useful. What do other verifiers think?

Karim Maassen

The just released public youtube video still has quite low sound levels. Had to crank up my laptop in order to hear it properly.

Karim Maassen

I'm afraid the sound it still quiet in the final cut.

Matt Parker

Yes, I saw that in a bunch of comments on the video as well. I asked and people seemed to say it was annoying but still fine to watch the video once the volume was up. I'm so sorry; I personally get irrationally angry when I need to ride the volume up and down watching youtube video. My video editor is doing 100 push-ups as penance and promises to make the next one 'marching band loud'.

Karim Maassen

Looking forward to having my eardrums getting destroyed next time! But yeah, it's still fine to watch for sure

John Sawyer

Thanks good to know it wasn't me it really was annoying but my amp coped okay. :D

Simon Pain

Great video & astounding effort hanging up all those models. I spotted another correction to list: for the 6D number you said 502,111 where the on-screen number is 502,110.

John Sawyer

Matt two questions the paper by Diaz and O'Rourke only says that can almost title the plane not totally title it. Did I miss something else that says it can title the plane. The other question I have is can all the Octocubes actually be unfolded into non-intersecting nets or is that a stupid questions. Sorry Engineer here not mathematician.

xje4

With many submissions coming in now, it looks like a new bug appeared. In some submissions (for example submission ID 78, 86 and 88) some pictures are missing

Kieran

I've only noticed it once, but I found refreshing the page worked.

Kieran

Quick suggestion for the WHUTS website - it might be useful to link to Firsching's models on the home page (perhaps also the submissions page). There was a submission for #7 I had to reject because the submitter had misinterpreted the shape of the net, likely due to three of the cubes being hidden in the WHUTS image. If people were directed to Firsching's rotating models, the shape of the net would likely be clearer.

xje4

Had it many times now and each time even after a reload, the same pictures are missing

Matt Parker

All the pictures on the WHUTS site are exactly Moritz’s rotating models. Are people not noticing that they can be moved around?

John L

Question, is it okay to reject someone’s submission made out of plastic bricks because they have called the legos? Americans need to take that extra s from the end of Lego and put it on maths where it belongs

Tahg

Probably not! I only managed to figure it out by dragging one by accident.

Tahg

Another Patron has come up with solutions for all of them too. I'm not sure about all solutions for all of them though. I saw the github data but I couldn't make heads or tails of how that translated into tilings.

Tahg

No, but the proper term in America is still lego, no matter how many of them you have. How you shorten a word by taking out the middle though I will never understand. Maybe everyone should just use the full term of mathematics.

sim1

Fantastic to see graph theory being used like this. Even better that it is free trees which are my favourite bit in graph theory.

slugbiker

just watched the new version with all the graphics. one suggestion: the relative sizes of numbers of unfoldings would be easier to see if they were right justified. e.g. (may not line up in this comment, but you get the idea). #unfoldings : Dimensions 1 : 2D 11 : 3D 261 : 4D 9,694 : 5D 502,110 : 6D

GwynBleidD

I suggest changing a mouse pointer to "grab" or "move" when hovering over the model.