How many 3D nets does a 4D hypercube have?

Get to know Jane Street! www.janestreet.com/join-jane-...
Which hypercube unfoldings tile space? whuts.org/
Yes, you can buy one of the 261 models from this video and support the channel. All hand-numbered by me and with a signed certificate of authenticity. mathsgear.co.uk/products/cube...
Here is Giovanna Diaz and Joseph O’Rourke's paper: arxiv.org/abs/1512.02086
A091159 Number of distinct nets for the n-hypercube. oeis.org/A091159
The code Moritz used to find these values is here: github.com/google-research/go...
All of Moritz Firsching's 3D models: mo271.github.io/mo/198722/unf...
Their post on the number of unfoldings in higher dimensions. mathoverflow.net/questions/30...
This is the Math Overflow post which started it all: mathoverflow.net/questions/30...
Peter Turney's 1984 paper Unfolding the Tesseract unfolding.apperceptual.com/
The cubes I am using are called "mathlink" and I just bought a huge quantity from Amazon (because Learning Resources didn't answer my emails).
US: amzn.to/3bsWOZd
UK: amzn.to/3offAsr
The unfolding animation of the 'Dali cross' was kindly made by my Patreon supporter John Sawyer.
I actually put the rough-cut of this video out on Patreon earlier this week so they could provide feedback and help test the whuts.org site. Thanks so much for all of your help everyone! / 51144440
CORRECTIONS:
- I saw "288" at the end of the 8D number when it should be "228". The on-screen number is correct. I noticed too late to fix it!
- At 21:09 I say Diaz and O’Rourke found an unfolding of the Dali cross which tiles the plane. It’s actually a different 3D net they found which does this and the Dali is undetermined if it produces a tiling 2D net. (Thanks Dan L by email.)
- Let me know if you spot any more mistakes!
Filming and editing by Alex Genn-Bash
Maths graphics by Matt Parker
Music by Howard Carter
The song Hep Cats by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 licence. creativecommons.org/licenses/...
Source: incompetech.com/music/royalty-...
Artist: incompetech.com/
Yeah, I decided to replace the copyright-claimed Aerosmith.
Design by Simon Wright and Adam Robinson
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinrandomhouse.com/bo...
UK book: mathsgear.co.uk/collections/b...

[@jamesoh5497]

"You can be part of mathematical history!" *1 day later* "History is full. Go away"

[@hardwareful]

KZbinrs ruin everything, including surplus prices.

[@spaceshipable]

As a side-note, Minecraft turns out to be a really useful tool for playing with the tilings!

[@pedrobernardo5887]

I was here thinking I'd like to help but couldn't be bothered to learn modeling software or buy these blocks Matt is using. Great ide

[@jacobward7361]

All the submissions are just Minecraft screenshots

[@dirm12]

Presumably something on Geogebra too. But MC sounds great. Imma stack my inventory with coloured wools and get to work!

[@tamarpeer261]

That’s a really good idea

[@PromptCriticalJello]

next episode "The Mathematics Of Minecraft: Pi Equals 4"

[@MichaelWilliams-ow9ue]

The definition of a good fanbase: Literally the next day and every single tile has a submission on the website

[@benp2291]

More than that, I think they’ve now all been shown to tile 3D space! So we’ve now proved all nets of the 4D Hypercube tile 3D space!

[@RodelIturalde]

@@benp2291 the real question. Do all hypercube unfolding always fill the space in the dimension they are in? I sat, yes. And further, if you unfold again, they fill that space aswell. Actually, they will always fill the space thry unfold to (maybe since it is a cube and that fills the space). What if we have a shape that doesn't fill the space (like a dodecahedron(sp?), will that unfolding fill the 2d space. I think not.

[@CED99]

@@benp2291 time for Matt to publish with KZbin as a co-author

[@vigilantcosmicpenguin8721]

I think that just means they've all got time on their hands.

[@ComradeTiki]

@@RodelIturalde I've just confirmed that all 2D nets which tile the plane can be turned into 1D nets which tile the line

[@zinsch5588]

Matt Parker "It's not going to be quick." Narrator: It was going to be quick.

[@thewhitefalcon8539]

And quiet.

[@CameronWinters]

“Unfolding the Hypercube” sounds like part 1 of a prog-rock concept album.

[@chasm9557]

I think it sounds more like a mathcore album to me. That title makes me think of the album Calculating Infinity by the Dillinger Escape Plan.

[@FDog16]

Or sequel of film "Hypercube"

[@Arikayx13]

That’s got to be a psytrance or ambient track somewhere.

[@badradish2116]

their follow up "orbiting the glome" wasn't as good.

[@doctorscoot]

Collapsing into the Singularity

[@bloodgain]

"Classic mathematician" An engineer, a physicist and a mathematician spend the night in the same hotel. At midnight, the engineer is awakened by the smell of smoke. He takes a step down the hall and sees a small fire. Thinking fast, he dumps his wastebasket, fills it with water, and puts out the flames. Satisfied, he goes back to bed. Later on, the physicist is also awakened by the smell of smoke. When he investigates, he finds a second fire in the hall. He runs to the end of the hall and picks up a fire extinguisher. In the time it takes to do this, the fire grows in size, but he is able to do some quick mental calculations and find the ideal place to aim the fire extinguisher nozzle. He manages to put out the fire and returns the extinguisher to its support. Satisfied, he goes back to bed. Finally, near dawn, the mathematician wakes up to the smell of smoke. He walks down the hall and - you guessed it - finds a fire. Looking up and down the hall, he finds the fire extinguisher in its holder. "Aha!" , he says triumphantly, "there is a solution!" Satisfied, he goes back to bed.

[@segmentsAndCurves]

Somebody have just said my name

[@schizoframia4874]

Are there no smoke alarms

[@mayfield3314]

Nonsense, any physicist will know never to put a used fire extinguisher back on the wall!

[@AexisRai]

the chemist arson who keeps setting the fires in the hotel finally won

[@karlcole5617]

that joke makes me happy and angry

[@s_gaming71]

holy crap i dont know if its just me but that music that suddenly starts at 10:07 was sooooo loud it almost blew my entire house into orbit !!!

[@alexandermirdzveli3200]

And folded my eardrums into the 4th dimension

[@setharnold9764]

Thanks for the warning

[@dsample]

It's not that bit which is too loud, it's the rest of the video which is too quiet.

[@MrDannyDetail]

@@dsample I was going to say the same thing. I had to put both the youtube voulme and my laptop's master volume up to 100 to hear Matt (the master volume rarely goes above around 50, and even at that level facebook videos are already way too loud for my liking), then nearly fell out of my chair when that music came on! I think something unexpected happened to the sound balance in rendering or during uploading and Matt may not have realised. Alternatively it may that for some reason (perhaps mic failure) the video is using audio from the cameras microphone instead of the clip on mic we can see him wearing, in which case he is obviously further away from the camera one, and hence quieter, though if this was the explanation then Matt would normally mention this in the description (it has actually happened to him at least once before (off the top of my head I think it was a video about a GPS device?).

[@Simply_CH23]

my heart leaped into the 4th dimension when that came up.

[@joesankar]

And less than 24 hours later, there are multiple submissions for every 4D net... I'm genuinely impressed, well done internet

[@reeper147]

I remember way back in like 2007-2009, there were millions of pictures of galaxies, and they had a hard task with classifying them as elliptical, spiral, etc. AI wasn't at the point where it could handle such a task, and there simply weren't enough scientists for it. So they turned to the internet, and had volunteers just be shown hundreds of pictures, classifying what type of galaxy they were, and then hundreds of people would each analyze the same picture to get good results. I spent most of my lunches in highschool doing that in the library.

[@jotha885]

18:07 There was actually a different third emotion in my mind: "Wait, the Parker tilling is actually a real tilling??!?" ;)

[@tim40gabby25]

'Parker tiling' is less incorrect

[@ELYESSS]

you can't name it that if it actually works

[@hamiltonianpathondodecahed5236]

Maybe it should be called "Matt Tiling"

[@rohitraghunathan]

Yes. Kept waiting for the Parker moment that never came

[@HerodotusVon]

Hamiltonian Path on dodecahedron nah, it should be called a Parker Tiling. That way, Parker can also become a Parker adjective

[@MariaVlasiou]

You could have a complete taskmaster series with standup mathematicians

[@rewrose2838]

Matt Parker, James Grime, Ben Sparks.. who else?

[@MariaVlasiou]

@@rewrose2838 one might get inspiration from the 24 hours maths magic show with hannah fry (and tons of others)

[@pabloagsutinnavavieyra2308]

I would pay to see that

[@leonardocgomide9488]

@@rewrose2838 hannah fry and tom crawford ?

[@durvsh]

@@rewrose2838 Steve Mould too

[@StormeeSkyes]

As a secondary school teacher I can only dream of owning that many multilink cubes!

[@standupmaths]

They were not cheap! I’m selling them to recoup my costs.

[@londonalicante]

First site I saw was £59.99+tax for 500. 261x 8 =2088. So thats five packs including 412 spares, about £300+ 20%tax = £360. Matt needs to sell 8 models at £45 to recoup costs. If he sells all 261 at £45 he makes.. £11,745!

[@markusklemm4516]

@@londonalicante But what about the duplicates he made for demonstrating 3D tiling?

[@__________________________hi52]

550 dollars

[@Solitaan]

"It's not going to be quick"... yeah, later the same day and there's a submission for each one! Good job yall!

[@alexanderf8451]

And also a computer proof that all of them have at least one tiling.

[@deamon6681]

I'm confused, there are still multiple pending.

[@error.418]

@@deamon6681 Pending, but still submissions for all of them.

[@KevanV2]

I like how some of these are even modeled using minecraft.

[@bwayagnes2452]

WOAH

[@brandonmack111]

I love that, at the mathsgear link, it says that the nets can't be folded back into a 4D hypercube without highly specialized equipment.

[@faielgila7375]

What, you don't have a hypernet folding plate?

[@Lukionest]

@@faielgila7375 I found a hypernet folding plate on Amazon, but it was ridiculously expensive, so I think I'll wait to see if they go on sale during Black Friday.

[@michaelharrison1093]

I find that a tee shirt folder does the job if you are willing to practice a bit and get your technique honed

[@brandonmack111]

@@Lukionest yeah, probably a good idea. Though you might be able to use it to create artificial wormholes, and become filthy rich.. just a thought.

[@TomE1248]

This video was quieter than the adverts by quite a margin

[@DavidLindes]

I’ve got adverts disabled, but indeed, much quieter than what I was just watching, and a little quiet for my tastes... I tried raising the volume, but even at full blast, I strained a bit to hear. I managed, just not easily. Since I did manage, though... cool stuff, Matt! Thanks! (And thanks in advance for incorporating this feedback into your future process for future videos to hopefully have them be a bit louder.)

[@Tinil0]

Yeah, no ads here but 10:09 hits and blasted out my eardrums at the volume I was trying to actually hear at

[@BRP42]

Yeah the music blasted. Unfortunate.

[@vincentplr]

The higher the dimensions, the lower the volume ?

[@bluerizlagirl]

That must be because the intensity of sound travelling through four dimensional space would diminish according to an inverse cube law, since each joule of energy now has to be distributed throughout the volume of a sphere as opposed to over its surface area in three dimensional space.

[@RedHair651]

The fact that you can unfold a 4D cube into a 3D shape that perfectly tiles a 3D space, that can be unfolded into a 2D shape that perfectly tiles a 2D plane. This blew my mind. I wonder if it works with dimensions 5-10

[@atimholt]

The 120-cell is my favorite shape, so you got me curious about its nets. I googled “120-cell number of nets” and found a paper from 1998 called “The number of nets of the regular convex polytopes in dimension ≤ 4”. Notably, the authors are named F. Buekenhout and *M. Parker*. Thatʼs nothing to do with you, right? EDIT: Hereʼs the abstract: Classifying the nets (also called unfoldings or developments or patterns) of the regular convex polytopes under the isometry group of the polytope is equivalent to classifying the spanning trees of the facet-adjacency graph under its automorphism group. This is done for all such polytopes of dimension at most 4. © 1998 Elsevier Science B.V. All rights reserved

[@andrewprahst2529]

He be wildin

[@colin1727]

Interestingly one the website I viewed the paper on m. Parker has nothing else on his profile

[@bow-tiedengineer4453]

come on Matt, you need to confirm or deny this!

[@alphazero924]

My previous post got removed. Likely due to personal information reasons. Long story short, it's not Matt.

[@stevegoodson9022]

This reminds me of an incident in my early education, when I was about 8 or so. Our teacher demonstrateed sticking together squares of paper and folding them into a cube, then just before a break asked us to think about how many different ways you could stick the squares together and still get a cube. I spent the break furiously scribbling away on graph paper, and after the break confidently strode back into the classroom and showed my answer - 20. The teacher took sadistic delight in pointing out to me that I'd come up with 9 pairs of reflected shapes, so 9 of my solutions didn't count. Spent the rest of the lesson passionately arguing that while rotations obviously weren't different shapes, reflections were. I think this was the point where my school realised I was going to be a problematic student to teach. Sadly I lost my intuition for maths over the years, would probably take me hours to come up with the wrong answer now.

[@NoobixCube]

Of course reflections should count. Is a left hand just a right hand flipped over? No, it’s its chiral partner. Most of the biochemistry going on in our bodies wouldn’t function if we ate foods with reflected geometry in their molecular bonds, because the enzymes that dissolve them just physically wouldn’t fit.

[@_okedata]

​@@NoobixCube yeah but this isn't biochemistry, its nets of a cube. and in this case the foldings are equivalent. a left hand might not be a right hand flipped over buts that's because they are 3d objects and your flipping them in 3d. 2d shapes have no 3d component and flipping them in the 3rd dimension creates another 2d shape which is indistinguishable from another net

[@skyjoe55]

"Would probably take me hours to come up with the wrong answer now" is such a great way of describing math

[@temmiemew]

well hey! you still got em all!

[@flexico64]

"Teacher took sadistic pleasure in telling a student they were wrong" I struggle to express my feelings towards this without learning several new dimensions of profanity

[@k0pstl939]

"Tetris induced fever dream" Matt Parker 2021

[@k0pstl939]

"""Tetris induced fever dream" Matt Parker 2021" K0PSTL 2021" lyle 2021"

[@heitzd1]

r/brandnewsentance

[@daphenomenalz4100]

Noivern 👋

[@k0pstl939]

@@daphenomenalz4100 my old name was shinynoivern99

[@orsonzedd]

@@heitzd1 no I've had these before this isn't a novel idea

[@KiloOscarZulu]

If 8D shapes cost 70 cents to make, Jeff Bezos has enough money to make every one of the 9D unfoldings! And have some money left over.

[@ronansuperfrog8425]

That's the strangest defining of wealth I've ever seen

[@dropit7694]

@@ronansuperfrog8425 having that much wealth is strange. We all know it’s real, but we can’t comprehend having anything close to it

[@z-beeblebrox]

@@ronansuperfrog8425 Stranger still since if Kilo had chosen 74 cents, they wouldn't have had to say "and have some money left over"

[@sleepylilturtle760]

youtube's AI definitely thinks this is, at the very least, a tetris adjacent video

[@the1exnay]

Tetris uses tetrominoes, and the cube section of this video was talking about a subset of hexominoes. So i see the relation

[@Volcanic_Rave]

The video also just straight up used Korobeiniki (the song most commonly known as the Tetris Type A theme) during the animation at 7:23. Probably just detected that.

[@aaron670]

I really hope someone is able to find an unfolding that almost tiles 3d space. Naming of course would need to be something like the Parker tiling.

[@KingofJ95]

Something like a huge arrangement of the nets in wildly different orientations to create a perfectly tiling pattern, but there's one singular gap in there.

[@sebastianjost]

@@KingofJ95 exactly what I was thinking 😅👍 That would actually be pretty difficult to find I believe. A single gap is difficult to achieve with 8-block nets

[@TheDGomezzi]

Wow, aside from the 7 that were found prior to the video, 19 more have already been found and verified! A bunch more are pending too. So cool!

[@kindoflame]

Its at 138 now. :)

[@kostyr13]

@@kindoflame Decades of progess in just a few hours :)

[@mrakoslav7057]

18:08 I went "I could find one", and I did! Thanks for the opportunity!!

[@squibble311]

WHUTS gang

[@blueqmusics]

8:29 Props to Matt and the team for buying themselves over a thousand toy cubes for assembling hypercube nets.

[@wojciechszmyt3360]

9:37 this reminded me of a fever dream I had once as a kid: it was about circular Lego blocks. The pins on them couldn't fit the geometry and it was blowing my already burning mind 😅

[@niklasjockel1515]

Experiencing non-euclidian-geometry-induced lovecraftian fear of the not-understandable as a kid. Reminds me of the good old days

[@revenevan11]

Lol, that's a straight up trippy fever dream, nice! (Sorry for how uncomfortable you must have been at the time though, and thanks for commenting about it.)

[@definesigint2823]

Oh, I remember fever dreams; for a long time I feared that state / it felt like torturous insanity. I'm glad you made it out the other side.

[@metamorphiczeolite]

Matt Parker, you make the world a better place. Thanks!

[@thepip3599]

I remember trying to figure out how many possible nets of a cube there are while I was doodling in math class in high school! I love it when it turns out that my random ideas are actually serious mathematical stuff.

[@WarcowUshi]

So the question that I have for this is actually, how many 2D nets exist of the 3D nets for the 4D hypercube. And how many of those tile the plane? And if all of the 3D nets do tile 3D space does that possibly point to a cycle that would translate up dimensions to where any cube in any of the possible dimensions will infact tile its own and every other dimension.?

[@redstonianemboar5711]

This is a very good question

[@protocol6]

Actually, what I was thinking was "are those Parker tilings?" Followed by "what's wrong with them?"

[@cj719521]

The orders of magnitude increases between net numbers in higher dimentions is interesting. (+1), +1, +1, +1, +2, +2, +2, +2, +2. I wonder when it starts jumping by 3 digits. I’m sure it does at some point, right? It’s fascinating.

[@abdulmuhaimin9780]

10 hours in and I cant find a single net without any pending submissions

[@ishoottheyscore8970]

153 and 154 currently showing as unsolved, possibly their pending solutions failed

[@Lofty1783]

Next Challenge: How many of the Hypercubes tile without rotation?

[@Rurdock]

Was thinking the same... I proved the 84 without rotation.

[@kmn1794]

Next Next challenge. How many combinations of hypercubes can tile together. Bonus points: also without rotation.

[@erkinalp]

@@kmn1794 Which hypercube though. Tesseracts?

[@kmn1794]

That depends how you want to look at it.

[@tomc.5704]

I rotated mine in 4D and now I can't get it back -- send help

[@ulysg]

7:49 Oh my god, I can smell these cubes

[@CtrlAnimate]

someone pointed out to me, when you see an object, you almost always know the taste and toungh feel of that object. this one checks out.

[@ncot_tech]

@@CtrlAnimate My inner 4 year old agrees. And the slightly inaccurate rubbery way they clip together but might pop apart again.

[@Manabender]

There's an old DSi-ware game called Boxlife. In it, you're given an irregularly-shaped sheet of paper, divided into squares, and your goal is to make boxes (cubes) out of it. You do this by cutting the paper into cube nets and folding them up into cubes. Because of that, I've known for a *long* time that there are 11 possible cube nets. I also mentally divided them into "the six nets made of a line of four and any two side squares" and "the other five". You might like that game, Matt. I'm not sure if DSi-ware that old is still even on the Nintendo store, but I'm sure it can be emulated.

[@pyrotempestwing]

I played the Dungeon Dice Monsters game for the GBA, which is how I know.

[@diligar]

Less than 24 hours after posting this, every single tiling has a pending confirmation! Congrats!

[@peterkelley6344]

ONLY YOU Matt Parker would ignite a global search like this! Keep us updated on this fantastic Quest.

[@johnchessant3012]

Yay for crowd-sourced math!!

[@MichaelMurraySting]

And we can show that they stack through.... the magic of buying (or making) two of them!

[@cadekachelmeier7251]

But first we have to learn about latent heat and the refrigeration cycle.

[@xymaryai8283]

@@cadekachelmeier7251 AHAHAHAHA yes, you see the compressor... awesome to see Technology Connections fans here, even if it isn't that suprising to know people who like smart maths stuff would also like smart engineering stuff

[@WindsorMason]

@@cadekachelmeier7251 I am now reading these comments in his voice and inflections.

[@juliankandlhofer7553]

or you could build them in minecraft :)

[@PentagramOfTemplars]

That went quick, but was really fun! Tried some myself but all are already solved (about half pending check but I have so far not seen any wrong solution, and each folding has a bunch of solutions sent in). This kind of recreational mathematics is why maths has always been my favorite among the aesthetic subjects.

[@mattshaw5179]

I've now got the Tetris music going round and round in my head!!! Thanks Matt!! Great video by the way!!!

[@Julio974]

7:15 It's interesting how the patterns with rotational symmetry correspond to the graphs with axial symmetry (including the pairing lines)

[@MatMan64]

I believe this phenomenon is similar to how stereo graphic projections preserve symmetry, which is super neat

[@GroovingPict]

maybe when all 261 are solved (or all the ones that are solvable, if it's not all of them), do a follow-up video on the second channel where you go through all of them? And importantly, why the ones that dont work dont work, if there are any

[@alexanderf8451]

Yes, the real question will be why some of them don't work if there are any like that.

[@feuermurmel]

Spoiler Alter: All tile space. I'm wondering whether thats a coincidence.

[@hammerth1421]

@@feuermurmel Probably not. It's not proven but I think it's reasonable to conjecture that all possible nets of n-cells tile (n-1)-space.

[@qsquared8833]

I noticed that the growth factor is diminishing in higher dimensions, seeing 10D really confirmed it in my mind's eye; That is VERY interesting!

[@Verrisin]

" What is your favorite 3D net of a 4D cube? " - Good question to break the awkward silence on a first date!

[@Mrsparky492]

There is something really cool about the fact that we could hand this to a 4-D being and they could fold it into a hypercube.

[@arowhite]

Is it something possible in our universe? Or is our universe a 3D space only?

[@JBroMCMXCI]

@@arowhite Some planets are 4 dimensional but we can’t detect them because we can only see 3 dimensional light waves

[@d.l.7416]

@JBroMCMXCI no

[@Jehannum2000]

@@JBroMCMXCI wHAT?!?!

[@raymitchell9736]

Imagine a world where all possible unfoldings of hyper-dimensional cubes are dangling above your head... Thanks to this video I don't have to anymore! LOL 😁

[@eideticex]

I too am grateful for the same reason. I have actually calculated all the unfoldings of hypercubes as hashing function. It's essentially how you handle mapping coordinates into voxel geometry in a non-recursive manner. It's also a very interesting thing to apply to seam smoothing in level of detail systems, really tricks the eye into not noticing things are changing rapidly in front of it. Seen a very similar approach used in a game, their solution was much more elegant than anything I had going and gave me the impression of imaging them all. Easy to do as weighted indices into space (numbers), hard to visualize until now as shapes.

[@mariojoia1337]

Thanks for the nightmare... ... ...

[@TerryLigocki]

What a wonderful channel! I have been enjoying it for some time. Doing a “crowd source” to find (and verify) unfoldings of the hypercube that tile 3D was fantastic. I imagine you were a bit surprised by how quickly all 261 cases were submitted (I was)! I look forward to your next video AND any future mass participation problems...

[@bsharpmajorscale]

"The rest of the tilings are trivial, and left as an exercise to the viewer."

[@artful1967]

I am beginning to feel that my Maths Degree from 35 years ago is about as relevant now as how fast I used to run when I was 22. The subject has gone so far beyond what I knew I struggle to follow the questions, let alone the answers. sad times.

[@calpolman5791]

What kind of stuff was in the syllabus back then? Or if you've looked at a more modern syllabus, how much has it changed?

[@thaumaTurtles]

Matt: so, how many such nets are there? Me: uh, idk... these look kinda like pentominoes, and there are 12 of those, so... 12? Matt: 11! Me: *surprised pikachu face*

[@thesharpestknife]

They are hexominoes though

[@thaumaTurtles]

@@thesharpestknife I know, but in the second after he asked the question, my brain made an immediate association

[@samuelthecamel]

@sixequalszero haha it's an exclamation point not a factorial

[@SimonBuchanNz]

@sixequalszero one of these days I'm going to define a mathematical operator for ?, That'll show those mathematicians.

[@dirm12]

@@SimonBuchanNz You could get Matt to announce it on this channel as the first place its published (cos if we're honest, isn't Matt using youtube as just another medium to publish maths discoveries), and play around with it.

[@pXnTilde]

I got a windows notification while watching this and it almost blew out my eardrums 😂

[@MegaMisch]

This is why i love your content. You put in so much effort and the subjects are so awesome!!! Thanks for the great vid.

[@dirkh8335]

Matt showing his school teacher influences through his choice of prop

[@vigilantcosmicpenguin8721]

I think he robbed a primary school for all the blocks.

[@blankblank4949]

Audio is super quiet just a heads up matt, thanks for all the awesome content

[@standupmaths]

Is it 're-upload' bad? UPDATE: The consensus seems to be that it is not great, but not impossible. Re-uploading is such a mega-faff (which angers The Algorithm) that as long as it is not causing accessibility issues I'm going to leave it as is. And promise that I'd do a better job balancing the audio in all future videos. I'm sorry. I also get annoyed having to ride the volume up and down on different videos, so I feel your pain.

[@AKAAndreas]

@@standupmaths Probably not, but it's up to you. It is a bit low., but not too low.

[@roseroserose588]

@@standupmaths i just turned it up a bit wasn't a bother

[@cooldude8475]

@@standupmaths i can easily watch with my volume on 90%, so i think it's not that bad :)

[@culwin]

@@standupmaths I have volume cranked up to the max and I can mostly hear it.

[@sebumpostmortem]

Not a single rotation from any perspective fitted my brain like unfolding the Dalí ' s one. Literally in tears. *THANK YOU*

[@mrakoslav7057]

"It's not gonna be quick"-12h later it's done lol

[@davekuder1590]

Heinlein’s “And He Built A Crooked House” comes to mind

[@cholten99]

Thanks for reminding me of a great story I've not read in probably more than 30 years

[@z-beeblebrox]

@einstein9073 although it would also depend on Earth's gravitational center of mass in 4 dimensional space. Otherwise, you might pick the right one, only to find out everything falls sideways in 4D and your orientation's all wrong

[@gmalivuk]

That was a Dalí cross shape, right?

[@alexinator1288]

Why am I so interested in mathematics? It's crazy how very few people I know are interested in this and I'm here watching these all day.

[@NabeelFarooqui]

Same. I hardly understand anything

[@alexinator1288]

@Kuoltakoon imagine using the 3D nets from the hypercubes for houses or buildings?

[@saechaol]

@Kuoltakoon That’s computer science described in a single sentence lol

[@pkeric2626]

I think that a lot of people overlook the appeal of „real mathematics“ because of the way math is taught in schools. Students often get the feeling that not understanding something = being stupid and they get discouraged, not wanting to have anything to do with math.

[@DJL3G3ND]

I always sucked at most parts of maths, and failed my final maths exam 3 times (still never passed) but I can really understand some things like this and things related to geometry. I guess learning blender helped with that lol

[@ImrePolik]

Awesome video. The amount of work you put into these videos is mind-boggling.

[@Danicker]

Ah.. watching this about 12 hours after it was posted and it seems like WHUTS is already full! Well done MP fans!

[@wojciechszmyt3360]

Once in high school I was bored in class and I figured out how to tile a 2D plane with cross-shaped cube meshes, BUT with a requirement that meshes with the same orientation cannot touch! Funny exercise, I recommend ;)

[@blackcat5771]

what do you do now

[@wojciechszmyt3360]

@@blackcat5771 I'm a researcher full time, nanoscience :)

[@magnetron2.049]

After the Parker Square , we now have Parker audio !

[@redapplefour6223]

oh good, i'm not the only one. everything's been super quiet for me lately due to some reasons

[@rewrose2838]

@@redapplefour6223 same! 😂 only the ads play on proper volume

[@uncletony3264]

Thanks. One of the best (visualized) discussions on 3D vs. 4D I've seen.

[@badmath9099]

Thank you, Matt. Your videos are always enjoyable to watch. I love the moments like 5:53. I know you can edit out the thinking or start a new recording, but I'm glad you keep those moments in. :)

[@fussyboy2000]

Qoute from the mathgear site "Please note that without very specialist equipment: these nets cannot be folded back into a 4D hypercube."

[@revenevan11]

Lol

[@ulysg]

7:29 Speaking about that, what about a video about the Penrose tiling?

[@JanSenCheng]

"It's not going to be quick" Solved in the span of a day

[@skaryzgik]

"How many can there be?" *cut* *looks around, being repeatedly startled* Me: Oh wow did you *make* all those? Also, nice use of 3-d space when displaying the amount of 3-d nets. Very nice touch. :-)

[@TheyCallMePhinq]

video seem extra quiet for anyone else?

[@woutervanr]

Yeah, I had to turn up my phone volume to 90%

[@alexinator1288]

Yeah it seems a little quieter for me too I've just turned my headphones up a little more

[@rayhan_rac]

Yes

[@Cleath78]

Yeah my volume is up super high

[@captaingavin2662]

Except for when the music played over the presentation of the calculation for choosing pairs. That was nice and loud 😬.

[@whoeveriam0iam14222]

I'm constantly asking the question "which ones have you proven to not tile space"

[@fejfo6559]

yeah same, but from the way he phrased it at the end, I think we don't know any that don't tile space

[@Septimus_ii]

Proving that they don't tile space is more difficult. It will be interesting to see whether or not they all do

[@douira]

@@Septimus_ii I can imagine such a proof would be pretty difficult since the possibly repeating components can be large

[@gilroitto]

My thought regarding the 2d plane. Are there any 6 size figure that doesn’t tile the plane? @7:25

[@reddragon3132]

I would love to see more on the idea of breaking the 3d nets down into 2d nets. How many of these are there? How many tile a 2d plane? How do you present these as graphs while considering that you could get overlapping squares (are we allowing these or not?)? So many questions!

[@henrikoldcorn]

Matt’s patience, shown by this and other videos, is astounding.

[@Alex-rw9nn]

Thank you Moritz (sorry memory loss I do care about spelling it right but I literally can’t remember) I really thought your maths stuff was so cool!! Thank you for sharing

[@reidflemingworldstoughestm1394]

Hay Matt, how is that cylindrical 3 sided dice thing coming along? Really love to see how it turned out.

[@user-od2ld3jg9z]

It's been only 18 hours and there's already 105/261 tilings!

[@TeachAManToPhish]

My unfolded hypercube just came in! Number 178! Looks like it's solved too! Shoutout from California

[@coryman125]

I remember playing with blocks like those back in kindergarten. Was I really trying to figure out four-dimensional geometry? Was I just enjoying the little noise when they click together? Could have been either really

[@etepeteseat7424]

This Parker video really wasn't a Parker Video! :D Seriously, exciting stuff, and congrats on doing cool math stuff!

[@samkerchner6251]

I found that the unfolding #231 has at least 3 solutions

[@andriypredmyrskyy7791]

I live for stuff like the 3d nets. I'm always looking for nifty little math and STEM related knick knacks to put around the house, leave in my office, twiddle with, start conversations. Love your work.

[@ysquaredyobozo]

If you're interested, Matt, i made a program for doing slice-by-slice renders of 4D objects, its not particularly fast, but it runs on my computer at a stable 60FPS when only rendering only one object it has capability for doing rotations and translations across any axis, its made in Unity, but it is almost exclusively written in HLSL so it can run on GPU

[@loedje]

every single hypercube unfolding has a submission right now they just need to be verified

[@squibble311]

mine got verified! :D

[@t71024]

CONGRATULATIONS on your newly earned silver medal on the quest to 1M subscribers!

[@chihuahuajedi]

I just want to take a second to appreciate the pride you have in your discoveries. It's an intersection of the pursuit of knowledge with art, emotion, and passion. That state of you being so thrilled for something most of the world has never thought about. That pure joy that comes with discovery and again with sharing your discovery with the world, that's what I want to be the end state of the human condition. This story is not just about little toy cubes, it's about the human drive to accomplish things just for the sake of contributing to humanity. Never stop Matt Parker.

[@michaelharrison1093]

Two more questions to find answers to: 1) How many of these 3D nets are able to be unfolded into 2D nets that are able to tile 2D space? 2) For each 3D net that can unfold into a 2D net that can tile 2D space, how many different 2D nets exist that can tile 2D space?

[@Osmosium2507]

I wonder if there is a mathematician in a 2D parallel world making a video on YouCircle about unfolding of 3D squares.

[@kelly4187]

1, 11, 261. Ah yes, the familiar sequence A091159. I'm just impressed you and your team made them all!

[@Capslok23342]

i just love how every one of them tile 3d space. so pleasing

[@alexhaywood3139]

Somewhere out there a PhD student who has been working on these for the last 2 years has just given up as the internet overtook their progress within a single day.

[@bwayagnes2452]

LMAOOO TRUE

[@AnoopKhetani]

I looked at the title, didnt understand it so I thought: 1 point makes a point, 2 endpoints make a line, 4 lines makes a square, 6 squares make a cube, So 8 cubes make a 4D cube?

[@professorcube5104]

No no it could be 10

[@jsytac]

So 10x 4D cubes make a 5D cube??

[@AnoopKhetani]

@@jsytac I Guess? Hehe

[@SimonBuchanNz]

For the curious, yep, n-cubes have 2n facets of (n-1)-cubes. They also have 2^n vertices.

[@AnoopKhetani]

@@SimonBuchanNz Hmmm thanks for the info!

[@bryanbischof4351]

This is a really impressive video and a great way to use your community. Thank you

[@erinhopper6568]

the plane tiling fact was so satisfying because that’s the first thing i wondered looking at all the nets :)

[@camyron]

the purple hypercube net at the front of the screen will tile in exactly the same way as the "dali cross" that Giovanna and Joseph found. (I may be wrongabout which net it is because some of it is hidden - I'm talking about whichever net has a central stem of four cubes, and four cubes around the topmost cube of the stem.) Moving the topmost cube to the bottom of the stem doesn't change how one layer of these nets are grouped together (like at 14:25) when you have an infinite amount of them, and then you can stack the layers together in exactly the same way. Edit: Net 102 in Moritz Firsching's system

[@DanielVCOliveira]

Mathematicians: We've found two 4D-cube unfoldings that tile 3D space, but stacking cubes in random shapes proved too cumbersome to do so we dropped it. Minecraft players: hold my diamonds

[@officialEricBG]

i'm a mod on the website, legit half of the submissions are minecraft :)

[@rowgesage936]

seems like a very convenient way to do it thats very beginner friendly. minecraft saves the day again

[@sebastianjost]

For a moment I was thinking about coding my own visualization but Minecraft really seems good. Though some texture changes may help make it look better

[@officialEricBG]

my favourite submission method so far has been kerbal space program LMFAO

[@1kain3]

Dangit, here I nearly cut my fingers off trying to supplement my dice with some cork cubes so that I could have some physical objects to work with, never occurred to me to load up minecraft :(

[@yaminijoshi3740]

Thank you for such amazing content.. It was really so much effort into one video.. Keep doing the great work..

[@boiledham]

Great video! Exciting mathematical crowd-sourcing project!! Can't wait to verify some tilable unfoldings of the hypercube on whuts.org today!

[@realcolby]

“It’s not going to be quick.” Its been less than a day and WHUTS has proven that every 3D tiling of a hypercube is capable of perfectly filling space

[@1Herobrine1]

This is Mad. The video is like 12h old and every single net is just waiting for the confirmation. And by the looks of it, they are all very promising

[@rhysmanley3217]

Great work with this video, Matt! And great work everyone on WHUTS for solving the problem in less than 48 hours! I only have one question: WHUT'S next?

[@DamianReloaded]

Kudos for the production! Wow!