Can the Same Net Fold into Two Shapes?

here I have two copies of the same cube I mean all cubes are the same but these are also exactly the same size which is quite nice and I am going to cut seven edges on each of them so I'm gonna... basically I'm uncutting the tape that I stuck on a moment ago there's that one [Music] and there's that one so the same shape the cube can be folded out into two different nets and I did a previous video all about this there are 11 different nets that fold into the same cube in that video we took it in a four dimensional direction next however we didn't stop to think if a single shape can have multiple nets can an individual net fold up into multiple shapes like could you fold the same net up different ways and get different shapes and whenever you mention this to a mathematician they always like no that can't be done Wham it can there it is this net same net two shapes forget nets of a cube this is where it's at so these are two identical nets you can see I can line one up perfectly on top of the other one they are precisely the same shape they're also based on a square grid and I'm going to fold them up but importantly I'm going to fold them on slightly different lines and we'll see well obviously we're going to get the same shape twice let's do it area [Music] okay and there it is so it turns out this net folds up into a one by one by five cuboid well I may as well do the second one just to just to double check that and there it is Isn't that amazing so the same net gives two different cuboids I'm gonna call these generalized version of a cube there are lots of different cuboids we have a one by one by five cuboid we have a one by two by three cuboid in exactly the same net if you fold it in different places gives you different cuboids I was recently at the MathsJam conference and I mentioned this idea to a few people and they're all like no like it feels so wrong that the same net can give more than one shape depending on when you fold it but yet it does work and how could we have seen this coming well it turns out by reading a book this is geometric folding algorithms by Demaine and O'Rourke yes Erik Demaine who's done a lot of amazing bits of mathematics including writing this incredible book and on page 425 you can see the picture of that net and how it folds into the two different shapes and actually they give more than one example of how this can work they've also got a one by one by eight and a five by two by one and this book came out in 2007 in the text they mentioned that these were found by a group at the University of Waterloo and the footnote has that from September 1999. so there you are if you look it up you can see that this idea of the same net giving more than one shape has been with us for pretty much a quarter of a century that's not right can someone check 1999 is a quarter century ago meanwhile should we have seen this coming let's say we didn't know these existed but we wanted to see if such a net was possible well by definition I guess the two shapes at the same net can fold into must have the same surface area because we're not letting the nets kind of overlap or anything like that and yeah this has the same surface area as this so your starting point would be are there two cuboids with the same surface area and that's easy enough you just search varying the three directions you've got how high it is this way how high it is that way and how high it is this way and then you add up all the squares and you get the surface area I'm very pleasingly if you wanted to represent that data you would put it on a 3D plot however the 3D plot is itself of the cuboids every number that's the surface area of one possible cuboid is at the corner of where that cuboid ends starting at the origin so nice now you don't want to look at ones that are duplicates because obviously this is the same shape as this is the same shape as this you want to have distinct ones and this is the smallest pair these both have an area of 22 and 22 is the smallest integer that there are two different cuboids with that area so this is the starting point and how many nets could we possibly have to check a lot there are so many nets oh my goodness and a lot of them work so this one was found in 1999 in 2008 someone went through and checked all of the nets that are made up of 22 squares to see which of them fold into both of these shapes and there are 2 263 such distinct nets they're all on our website I recommend checking it out the rules are the nets are allowed to touch in a corner but they can't touch along an edge so when you fold them up you never have to make like different cuts to get different shapes as long as the corners aren't considered attaching you you've got 2263 of these options absolutely incredible also in 2008 the Japanese mathematician Ryuhei Uehara who found those 2263 nets also published this incredible paper and in it they show not only did they find those over 2 000 nets that fold into these two specific cuboids but if you're just looking at grid nets that fold into more than one distinct cuboid there are infinitely many all right here is the net from the paper so I'm going to pop those out of the way for a second and you can see same same net twice there it is but fantastically not only does this fold into two different cuboids but you can extend it as far as you want this did a little sticky out there at the top is always one wide but then this bit here is not always two this is actually one plus k and I've just made the k equals one case so that is two but k could be a bigger number and this would be wider this would be wider this would be wider the whole thing just kind of stretches out and because there are infinitely many possible values of k you now got infinitely many nets and they all fold up into one of two different boxes fun side fact that gets mathematicians so excited these shapes tile the plane so they can repeat to completely cover 2D surface they lock in like that at and then that carries on in each direction and then these they wrap around I think they look in there yeah that's right because now you've got this bed here to fill and that's filled when you put that one there like that so there you are they told a plane let's put them together all right here we go [Music] okay and before I stuck them together this time I thought I'd just fold them and show you it's a little bit more intuitive this way you can see this version if you fold it this way it becomes this really really long tube and then you've got the caps on the end that just kind of seal it up and as you increase k you're stretching these out and so the tube just gets longer and longer and longer and longer whereas if you fold it the other way around this is kind of wrapping it up instead of rolling it down this way you're rolling it the other way k is actually this side oh actually it's this one here so that would fold off that caps the top that caps the bottom those don't change this length here is k so this one's always two this one's always one and then this one here K this cuboid just kind of expands out this way but for completeness I will tape them together now foreign this one is always one by one by six k plus two the k equals one is the smallest case that works since that's why it's eight units tall and exactly the same surface area but this one's always one by actually I got this wrong before didn't I it's always one by five and then this is 2k which is why it's two when k equals one and this actually gets longer and longer this way like it kind of stretches out that way whereas this one gets taller that way or it's the same surface area from thinking that it's not possible to realizing there are infinitely many nets that fold into two different cuboids incredible now I first came across these in a tweet from 2017 that had an animation very nice it was retweeted a couple weeks ago by a math author I know Vincent who also expressed their thought that they didn't think it was possible to have these convex solutions classic no one thinks it can be done infinitely many mathematicians mathematicians versus maths in this case I thought this would be a great video and and that's it I mean I guess that's kind of that's the end of the video the only other don't look at the time code the only other thing I could do would be to go through every single one of those 2263 nets um and find the one that looks the most like a Christmas tree there it is so the Stand-up Maths Christmas card for 2022 I found look at that this pretty Christmas tree-y come on that folds into two different nets it will fold into this and it will fold into this however on that website they don't actually show you how they fold you've got to work it out yourself and so if you support me on Patreon you get emailed one of these everyone gets emailed one so then you've got to print it out yourself if you support me at a statistically significant level literally or higher I will post you a physical one of these on the inside it says I hope your holidays are net fun good joke and and then I write like you know thanks for everything in your name and I sign it and all that jazz and then what I want you to do is to cut out the Christmas tree from the front of this and see if you can work out how to fold it into these two different ones send me if Twitter still exists tweeted at me maybe I'll be on Instagram by then who knows um so yeah there you are and by the way you've got until very early December if you're at the correct level on Patreon you're guaranteed to get one of these in the post so do check it out and that's it that's the end of the video thank you so much for watching all right and that's it good job everybody another fantastic video of course we didn't discuss the possibility of a net that falls into three different shapes but that is absolutely not possible WAM you can have the same net that folds into three different cuboids in more than one way this is actually the first one that was discovered in 2011 in this fantastic paper and this one technically kind of works but it's very much let's say thinking outside the cuboid because this as you may have noticed is actually not that much bigger than the previous ones we were looking at because this is actually one of the 2263 nets that fold in to these two shapes and we'll do that in a second so once again this one will fold into that shape this one will fold into that shape but you're like hang on what's the third one we know there's not another cuboid that has an area of 22. or is there yeah I guess that one in a minute [Music] first up my friend Lisa Mather very kindly cut all of these out using one of those automatic kind of plotter machines with a scalpel energy the scoring and everything amazing but when I sent to these files I've just realized I swapped these two around so actually the purple one is this shape so if I put I can put that in there and you can see I've folded all the lines there it is that wraps perfectly around there so that one there folds up into the one by two by three cuboid just the colors are the the wrong way around and then this one here is the one that was purple last time so that wraps around like that and there there we go look at that a little more a bit of a crazy way of folding around but there you are I'm going to take some time by not trying to tape those together the remaining one though is this one you think what is that going to fold into so well let's just do it let's follow the folds that one goes there that one goes there that one oh hang on that goes there like that and there's like a little half flap there on the end that folds over there and then that falls like that that falls like that like that and like that and there it is so it folds into well it's flat it folds into a one by eleven by zero cuboid such a mathematician solution to a problem this is the degenerate case where one of the thicknesses of the cuboid is zero so technically yeah I mean it's a double covered rectangle it's got the same surface area it's technically a cuboid it just has zero length in one of the directions and so that was the first attempt to get the same net folding into three different shapes but rightfully you're allowed to be a bit upset at this the year is 2015 and this paper drops look at that this is common developments so the word development is used as a synonym for net because the word net already has too many synonyms I think and it doesn't translate well I believe so anyway this means common nets of three incongruent so different boxes of area 30 yeah someone did it they found three boxes that can all be folded from the same net so here I've got three copies of exactly the same net however they're still doing something a little bit sneaky I'll fold the first two which are a bit more normal and then we'll have a look at the Third right [Music] there it is the first one it's a one by three by three cuboid so it's got an area of nine on the front nine on the back and then four lots of three so that's 12 and 18 30. area of 30 One cuboid down two to go we'll do the this one next And there it is box number two this time it's one two three four five six seven one by one by seven so that means there's four times seven there's 28 around the edges plus one there plus one there Thirty so now we've got two cuboids from the same net both with an area of 30. we've drawn equal to our previous one that they both had an area of 22. what about this next one well this also folds up into a cuboid but it does bend the rules in so much as it bends where there's no rules it actually folds not in orthogonal folds to the grid that this is cut out of but the folds are on jaunty angles so this one here folds up like like bats how displeasing is that all right so I'm going to fold all these ridiculous angled folds and we'll see what we get okay there's the folding done so all the folds are at right angles to all the other folds they're just not at right angles to the original grid of 33 squares I don't like this at all let's tape it together [Music] I'm not happy about it but there it is I mean that's definitely a cuboid I mean it's an actual cube you don't get more oily and an actual cube and everything is at right angles it's just not relative to the original grid and this length now this new length we've got that is the square root of 5 compared to the unit length of the grid that the net was cut out of it's just so ridiculous and because that's root 5 long and this is a cube each of the faces has an area of five and there are six of them area of 30. so it works there you are area of 30 area of 30 area of 30 they're all cuboids and they were all folded from exactly the same grid I mean you cannot argue people are going to argue with that so if this paper from 2015 is making you sad don't worry I skipped over one from 2013 was this paper and it also found a net that gives you three different cuboids and none of this folding on weird angle it follows all the rules it's got a straight up Square grid net fold it three different ways along the straight angles and you get three different cuboids the problem is it's absolutely freaking massive now they do the same trick as before where they show there's an infinite family so we now know there are infinitely many nets that are grids and fold on the grid lines and become three different cuboids because of the family the issue is the smallest one is this one and it has an area of 532. it's just ridiculously big and when Lisa very kindly made all of these other ones for me I was like ah there is one more but it's just stupidly big I don't think we should try and we did so this literally arrived in the post from Lisa this morning right before we started filming I'm gonna pop it open and against my better judgment we're going to build three boxes with an area of 532. here we go oh my goodness no Okay so I mean it's very well packed oh no okay sorry everything else is going off these will come back later don't worry I'm just gonna very gently put them down here all right here we go okay there's the first one look at it oh that's so big why did I think this was a good idea oh it's gonna oh they zipper together look at that that's organizer like that lovely oh okay yeah let's get that crease done first [Music] okay that all goes there that falls around there that folds in there [Music] okay I mean that's far from perfect definitely the hardest one to put together and you can literally see a lot of this a lot of the actual seams but there it is that's that's the shape so that is now the squares are a lot smaller this time so that's two by one two three four two by four by a lot so that must be the two by four by 43. so on the upside that's that's the longest is that even all is just all in the shot right so that's the longest one I'm gonna put that like there and then let's reveal the next two which a lot a lot more boxy so they won't be quite I mean obviously the Net's the same size that's the whole point so there's another there it is identical net but this time the fold should make it more of a box let's find out [Music] [Music] thank you that was ridiculous... ly good fun it was annoying but I'm so glad I did it you know what I don't know if the people who wrote the paper would have ever bothered actually making these so I very easily could be the first human who ever folded the same net into three distinct shapes only with orthogonal folds I mean conceivably someone else might have done it the author's working on the paper or someone who read it since then but that was a lot of faff and if I know mathematicians they would not have bothered actually trying to do that but anyway we can now see we got all three this one here that one is the seven by eight by fourteen we've now got the 2 by 13 by 16 more of a cereal box size and our original ridiculous 2x4 by 43 and so this is currently humankind's best effort at the same net that folds into three different cuboids orthogonal folds only and I think that's ridiculous we can do so much better I mean look at the case for two it's tiny an area of 22 but we need an area of 532 for the smallest one we've found now technically there could be a solution with an area of 46 that would give you a one by one by eleven and give you a 1 by 2 by 7 and a one by three by five however no one has found one they did say at the end of this is their 2015 paper now all right and then they point out that this is the paper where they did the the silly one here where they found the one with 30 which is the case with the non-orthogonal fold still very clever but not what we're after and the end in the concluding remarks they say here we conjecture that there exists an orthogonal polygon of 46 unit squares that admits to fold these three boxes referring to the the smallest possible ones however as of now this is the limit of humankind's understanding no one's found one with 46 squares that fold into three different cuboids no one's managed to prove it doesn't exist so if you're bored give it a go although I it's probably going to need a lot more computing power than we have at the moment but who knows people that watch my videos have written more ridiculous code than that in the past and the other thing we don't know is if there's a net that folds into four different cuboids or five different cuboids or is there a limit even so from the same concluding remarks they point out that if from there let me get this here from there we remind that theorem 2 says that we have no upper Bound by the constraint of the surface areas So in theory you could pick an arbitrary large number and there's one net that folds into that many different cuboids or they go on to say but it is hard to imagine that one polyomino which just means a bunch of squares stuck together a net can fold into say 10 000 different boxes but just because it's hard to imagine doesn't mean it doesn't exist so anyway that's what we don't know we don't know if there's a better way to do three than this and we don't know if four or five or however many above that work if you want to give it a go please do let me know if you have any success the one thing we do know though as a species is that you've only got until early December to make sure you're signed up to get the Stand-up Maths Patreon Christmas cards so I say early December I leave it as late as I can make sure you're supporting me statistically significant or higher if you want the physical card thank you so much for your support it not only means I can justify an entire day folding these things but I can get friends like Lisa involved to use to use their skills and resources to make possibly the first ever set of identical nets that fold into three different cuboids thank you for your support have a wonderful festive season

Channel: Stand-up Maths

Views: 315,925

Keywords: maths, math, mathematics, comedy, stand-up

Id: jOTTZtVPrgo

Length: 25min 59sec (1559 seconds)

Published: Fri Dec 02 2022