Can the Same Net Fold into Two Shapes?
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- čas přidán 1. 12. 2022
- If you cut out the net from the card and solve a fold: send me a photo! Maybe hang it on your tree as a decoration.
To receive a physical card you officially need to be a "statistically significant" supporter (or higher) on Patreon by the end of 06 December 2022. And all supporters get emailed a digital copy! / standupmaths
This is the tweet from Jun Mitani in 2017: / 870414535034757120
And the recent retweet by Vincent Pantaloni I saw: / 1582175737213509633
Huge thanks to Ryuhei Uehara for did much of this research and was very helpful then I emailed. You can see their page (with all 2263 nets) here: www.jaist.ac.jp/~uehara/etc/or...
All the publications I waved around:
2007 - "Geometric Folding Algorithms" by Erik Demaine and Joseph O'Rourke
2008 - "Polygons Folding to Plural Incongruent Orthogonal Boxes" by Ryuhei Uehara
2011 - "Common Developments of Several Different Orthogonal Boxes" by Zachary Abel, Erik Demaine, Martin Demaine, Hiroaki Matsui, Günter Rote and Ryuhei Uehara
2013 - "Common Developments Of Three Incongruent Orthogonal Boxes" by Toshihiro Shirakawa and Ryuhei Uehara
2015 - "Common Developments of Three Incongruent Boxes of Area 30" by Dawei Xu, Takashi Horiyama, Toshihiro Shirakawa, and Ryuhei Uehara
Here is Geometric Folding Algorithms on Amazon if you want your own copy to wave around. But it costs 'text-book money'. amzn.to/3OYDXrF
Huge thanks to my Patreon supporters who funded those crazy nets I was sticking together. Support me and help me make more ridiculous shapes (also: videos). / standupmaths
CORRECTIONS
- At 07:36 I place the top shape in the wrong spot. But it totally works! Look at the diagram!
- Let me know if you spot anything else! My bad taping skills do not count as a mistake.
Filming and editing by Alex Genn-Bash
Props by Lisa Mather
3D plot in Geogebra by Ben Sparks
Written and performed by Matt Parker
Music by Howard Carter
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... - Zábava
I can’t be the only one that finds the non-orthogonal cube tremendously satisfying and elegant rather than awkward…..
I went from "Well, this is instantly a horror" to "that was damn clever and satisfying!".
Yeah, that was my favorite one in the video! I suddenly want to see more examples of strange folding angles being used to create cuboids.
You are not the only one.
Absolutely
I thought it was absolutely brilliant. So you are not the only one.
The non-grid √5 cube is more beautiful than trying to find a grid-only solution.
I love the actual cubic cuboid.
That's related to the Fibonacci number isn't it? Or at least involved in some sort of series that produces it?
@@krissp8712 well sorta, the golden ratio is (1+sqrt(5))/2 but i wouldn‘t say it has anything to do with it.
That one was my favourite too. Although the large 'zipper' had its attractions too.
@@krissp8712The golden ratio is related to root 5, but here it’s just that root 5 is the hypotenuse of a 1x2 right triangle.
Not only do I count the non-orthogonal fold option, I think it demonstrates an exceptional level of out of the box thinking.
Out of the box when it is a box. Hah. Love it.
Outside of the box, actually ;)
Can you give me the time stap?
In the box thinking but it definitely folds outside the lines.
@@brunojambeiro6776 16:45.
There needs to be a √5 non-orthogonal fold cube appreciation society! That little fella was my favorite by far. Matt you need to give this guy some love. Support the √5NOF Cube T-shirt is now right on top of my Christmas list!
I love how as I was watching the video, I was thinking "What are you on about, Matt, I absolutely LOVE the one with non-orthogonal folds and look, it even folded into a perfect cube!" and wondering if I'm weird or something and then I went to the comments and like 50% of all comments are from people specifically loving that oddball (eh, oddcube?). This is also like peak nerdiness to argue about and I'm not ashamed of it.
I'm less upset that the third cuboid had zero thickness, and more upset about the little half folds on the end
Couldn't you avoid the half folds by quadrupling the grid? Each of the original squares is now a supersquare of 4 little squares, and the half folds are now perfectly aligned with the smaller grid.
The one he does at 10:33 brings that idea to a whole new level :D
@@eggsquishit It gets a pass in my books, cause its neat that the angles come together at 90 degrees
@@pierrefley5000 Thank you. I am now satisfied again.
I'm just glad I'm not the only one feeling this.
"How could we have seen this coming? By reading a book". This is my new favourite quote ever. And I already know my kids are going to hate me for this.
maybe also by trying things :)
Until one of them shows you a porn magazine. "How could we have seen him coming? By reading this book!"
2:36
It's like "the magic of buying two of them."
@@vigilantcosmicpenguin8721 Yeah, TC is brilliantly written as well, but these are simply words to live by. :-)
I like how this is something where the math community is like, "No! It can't be done!"
I can picture Erik Demaine barging into some kind of court of mathematicians, dramatically placing down this net and saying "BEHOLD!"
czcams.com/video/FfGZlGsnvDs/video.html
"FEAST YOUR EYES, YE NONBELIEVERS!"
Actually, someone else placed this net on his desk, and then he put it in the book. I don't think he's ever discovered anything by himself.
BEHOLD, A MAN
Sirr, that's a plucked chicken.
I wouldn’t classify myself as a mathematician (perhaps more of a math enthusiast) but I was very satisfied learning that the one pattern tiled the plane 😂
Also the non-othogonal one reminded me strangely of the strategy of superposing an angled grid to find such tessellating shapes....
that pattern also reminds me of ms paint on diagonal lines
I'm surprised this isn't an overtly gift wrapping themed episode
I thought the same thing!!
Missed the chance.
Matt's family and friends are going to roll their eyes so hard when he explains that to open their gifts they have to unfold the net of a cuboid.
Missed opportunity...
Matt Parker’s wrapping paper company, PLC
I actually think those off the grid folds are pretty interesting. you glossed over the other example from the book, folding one net into a cuboid AND a triangular pyramid; I was hoping to see more of that in this video. Break off from the grid, don't even limit yourself to cuboids!
It's essentially Dudeney's Dissection turned into a net
Yeah the off grid ones are so interesting in my opinion, would be interested to know how many of those there are for a cube (for different surface areas) or how many surface areas are possible to achieve for a cube using them ect. c:
Become ungovernable, maths edition
That one wasn't even off-grid. It was on a grid of triangles. And since zero-degree folds seem to be allowed, ALL the shapes in the video were also on a grid of triangles.
parker cuboid (square) power is too strong
I actually really liked the sqrt(5) sided cube from 2015, very clever
Imagine making a box for a christmas present with one net and then wrapping it with another net. The container itself is the perfect gift for a mathematician
I love that 30SA cube. The fact that it doesn't fold "on the grid" makes it more interesting to my damaged brain.
It’s so satisfying when folded too
It was by far the best.
I was surprised an example didn't come up earlier tbh
it uses the 3 4 5 pithagorean triplet, it's so cool
Also √5 as a side length factor is just amazing.
Especially considering that (√5+1)/2 is the golden ratio (and I have about 30 other reasons to like the 5).
I like that the color mismatch provided higher contrast. That made it very obvious that the nets still covered all regions.👍
There was no color mismatch, they are exactly the same shapes. That was the joke.
@@morosov4595 was there not? I thought the prints had lines printed on them for help when folding. That would dictate which color folded into which shape
@@morosov4595 late, but what do you mean joke? matt wanted to have the same cuboids be the same color BECAUSE they are the same cuboid. but he messed it up, hence the mismatch
@@morosov4595 I feel so stupid right now.
@@faland0069 He said he messed it up when he was sending the files.
The “don’t look at the time code” genuinely made my day
9.52
Same. He said it right as I was checking it.
Well this has got my holiday shipping problems sorted. No more having to buy a bunch of different-shaped boxes for all my different gifts, as long as they can all fit into boxes with the same surface area!
That's ... actually an ok idea wow
The root 5 cube is awesome! I love that it doesn't fold on the grid lines!
If only to follow the reasoning of the people coming up with it. Overlaying the grid with another over the 1x2 domino's and then realising there is a cube with area 30 and ribbon square root of five, and then finding one grid that works must've been soooooo satisfying!
@@MeriaDuck The only thing better would be a cuboid with a side that would be a cube root.
Sitting here at half past midnight chuckling away. Wife wakes up, sees what I'm watching, mumbles something about me being a nerd and falls back asleep...
But I'm a happy nerd. 😀
Sounds like Matt might like to attend 8OSME (if it happens)
The Eighth International Meeting on Origami in Science, Mathematics, and Education. The Demaines and MItani have been regular presenters at previous conferences, and Uehara is on the steering committee for 8OSME.
Origami maths is pretty incredible.
Ooooh
Yeah that would be the dream. Off to look it up and see if I can lend encouragement to the endeavor
Due to finances it's folded
@@stevesmith2044 damnit
22:28: "but if I know mathematicians, they definitely wouldn't have bothered to do that"
But if I know Erik Demaine, he *definitely* would've bothered to do that. He's freakishly good at everything origami and often folds large, complex models, and is a fan of doing things for no reason.
On a more interesting note, I'm very happy that there's finally a video on cuboid folding. There's also a bunch of interesting research on the half-grid model and polyomino-based cube folding by Erik and Martin Demaine - it turns out that there's a very nice way to fold a 3x3 square into a 1x1x1 cube if you can make half-grid folds, and the same for a 2x4 rectangle.
A 3x3 square or a 2x4 rectangle into a 1x1x1 cube are impossible- the surface area would go from 9 or 8 to 6.
@@jessehammer123 There are overlaps.
@@KingstonCzajkowski Oh, we’re working with a different rule set than standard maps. Got it.
What a great 11 minute and 26 second video that was! I wish there was more!
check again!
wham!
Find the 46 cuboid!
right? Now I’m wondering if there are nets that fold into three different cuboids too!
This video unfolded in several ways.
1999 being a quarter century ago was the most surprising thing in this video.
2025, 1999 is forever more than 25 years away.
Get ready to mark that.
it's only 23 years, not 25
@@NoNameAtAll2 A Parker quarter of a century.
@@NoNameAtAll2 24 in a month. Close enough to call it a quarter century ago
That terminology needs to be illegal.
It feels like the 46 area cube could be bruteforced for sure
I'll be surprised if the next A Problem Squared doesn't tell us that he received dozens of submissions of programs which compute the net(s) in question, and that they produce answers in times from 30 minutes to 30 milliseconds.
And if it doesn't, the next best thing is to try and engineer something with diagonal folds (perfect cube case, very unlikely) or half-folds (degenerate cuboid, more possible than the diagonal case)
I don’t think so. The easiest brute force imo would be “unfolding” each of the three shapes to get all of the nets that could possibly fold into those shapes, and then “folding” each one of those in turn in every possible orthogonal and non-orthogonal folding pattern to try to generate the other two shapes. This feels like a “more combinations than there are atoms in the universe” type of thing
@@hoebare And it turns out there are like 317 answers.
presenting: BoxFolding@home
The fact that they have different volumes is tripping me up 😂
In the Domain book, the figure 25.51 (folding into a cuboid and a tetrahedron) could scale vertically to fold into a much more satisfying christmas tree (and a present.)
"By reading a book. (Long Pause)" Matt is on his A-Game with snarky remarks again!
Most of your videos are at the very edge of my understanding or beyond it. But there are moments when you say something like "this is currently humankind's best effort" and I get swept up in the excitement of seeing these paper boxes as the physical embodiment of the border between "all human knowledge" and "what lies beyond, yet to be discovered." Thanks for making those moments happen.
The "Wham! It can be done!" Made me lol
This branch of geometry should be addressed as standup geometry
because it is basically geometrical analog of a pun.
So satisfying seeing the nets fold into the different shapes
bro spoilers
Finding SemiHypercube on as many channels as possible should be a game by now! I just realised your name, so I should have probably expected this.
The flat "cuboid" and the diagonal folded cube made me laugh. Brilliant answers! Beautiful!
The non-orthogonal one is my favourite! Such cleverness to fold it like that with no overlaps! :o
My flatland mind is blown.
Edit: I know want to start a business offering three different shapes of gift boxes using the same 532 net - one more posters and other long objects, one for clothes, and one for knickknacks. Since they are all built from the same net, makes ordering supplies easier.
Fedex is taking notes furiously in the corner
i could see these types of boxes being used in tech products as an inner decorative box. maybe in the case of headphones; one box could hold the actual headphones, one could hold the cords, and another could hold accessories or the manual
Indeed, this is so much easier in Flatland - the 3×3, 2×4 and 1×5 rectangles all have identical nets.
I've got a feeling that the people working in the warehouse aren't going to be as excited about geometric nets.
Putting a USPS Priority Mail box into all of its 3-D glory isn't already time-consuming enough?
Imagine working in an Amazon warehouse and trying to keep up with the productivity requirements! 🙄
That net for the infinite family that tiles the plane looks like a worm-on-a-string, especially when it's purple
When Matt said "can someone check if 99 was actually a quarter of a century ago" I felt that
I love that small cube. Folding diagonally was the way i originally thought he was going to create 2 cuboids from the same net and it looks so good too!
0:59 lol didnt see that comming , I love how excited he is
The one that folds into a pyramid that was in the paper was cool.
19:35 "just going to very gently put them down here" *flagrantly cascades them off the table*
3:05 I'd love to see more about nets like the one that folds into a regular tetrahedron AND a rectangular box.
You know it's a good Stand-Up Maths video when the question in the title is answered in the first 2 minutes.
"You don't get more oidy than an actual cube." Put that on a shirt
I actually find it more fascinating that the same nets turn out to be different volumes!
SAME!
Volume and surface area have always had a weird relationship. Any of these paper cuboids you can crush and get something with less volume and the same surface area. In other words, a single surface can be realized in many different ways of similar surface area but nonsimilar volume. Cutting that surface up and folding it into a new surface is unlikely to share the same volume since you could have imagined it starting with any of the different crushed volumes. Of course the restriction to folding on a grid could have magically enforced similar volume since you no longer have these crushed examples, but it'd still be less likely since there's far more solutions to SA/2=xy+xz+yz than xyz=V
It seems strange at first, but it's also sort of obvious. Maybe another way to think about it that's more obvious is to drop down a dimension. Take a piece of string and lay it out in a circle. Then find two opposite points and pull them apart. You've got two shapes with the same circumference, but one has an area of 0 and the other of C²/(4π).
In any dimension, the shape that is the most circly is the one that minimises surface area or maximises volume.
@@SilverLining1 you cannot "crush a shape" without the shape losing its integrity
That's actually logical if you think about it. What you're essentially doing is construcing shapes of the same surface area but different dimensions. You can do this in 2D to create a 1x3 rectangle or a 2x2 square. Their perimeter is the same, so it's possible to make them with the same pieces, but their dimensions are different so the area changes.
origami time with matt is just great, I love seeing him struggling to tape them all together lol (technically this is kirigami but it's not as well known as the other word)
The most satisfying was the non-orthogonal folds! Everything else felt a bit simple by contrast.
The folding action definitely looks more complex to our orthogonally-minded brains, but the discovery of the net itself is evidently more complex for certain orthogonally-folding examples and I find that equally satisfying.
A very enjoyable 11 minutes, thank you!
yeah he honestly speaks way too slowly
I enjoyed all 26 minutes
@@beartankoperator7950 ⚙> Playback Speed >1.75
11:27 bookmarking this
“WHAM! You CAN have the same net that folds into 3 different cubboooiiiidsssss”
I almost quit the video at the flat cuboid... glad i stayed tbough. Its amazing how much effort you put into your videos. Every video of yours is a blast to watch.
Matt's arts and crafts videos are always good. 👍
Matt's videos are always good. 👍
The matt and adam savange episode on the tested channel was one of my favourite
That non orthogonal cube would probably make a pretty cool football/soccer ball
there has to be an industrial application to this. only printing out one shape that can be folded into different ones is a huge time saver.
21:00
"Is that even all in the frame?"
My thought, "Run it by at light speed, you'll get it in the frame."
I'd love to see the analytics on how many people stopped watching once that outro music started.
Me as well! I’ll wait until there has been enough views and then take a look at the data.
14:10 its a Parker cuboid!
The origami gang more pleased by the non-orthogonal folds
Origami people starting with the same square every time: Am I a joke to you?
Christmas present idea... give someone the 1x2x3 box AS their present. Tell them to be careful when unwrapping it (just cut the tape and unfold it). They'll open up their present and see it's EMPTY! Tell them, "That's strange, I totally put a 1x1x5 box in your present, let's look around for it." Take the "wrapping" paper and refold it into the 1x1x5 box and say, "Ah see there it is."
A 5 inch long present in a 3 inch long box hahaha
How good is your slight of hand?
Or do they not get a present out of this box...
@@David-gk2ml the present is the box
there used to be a game from the DS store where you had to cut up nets from an endlessly scrolling grid and then fold them into boxes before they fell off screen. I remember that was how I learned about the 11 different nets for a cube and which ones tile the plane. Someone should remake that game into an app, I would play ut all day
Is it the game called "Boxlife"?
@@iteragami5078 yes! that's the name!
Parker color coordination 😂 what a fun video! So glad you brought up the four shapes question, and so disappointed that we don't have an answer yet 😭
@1:13 Matt, I do actually really appreciate you subtly flipping the order when showing them lined up, so we can see clearly that the line up works both ways. Saved me as I was in the middle of trying to study the bottom one to see if I could pre-emptively catch any sneaky tricks about it having an extra hole missing from it.
This video's title didn't specify that the shapes should be cuboids, so I read the title it and immediately came to the conclusion that it should obviously be possible. My even more obvious example of a cube 3x3x3 where one side has a 1x1x1 part either sticking out or sticking in.
I also should have mentioned I was only talking about convex shapes.
I would absolutely love to see cracks at this problem which are more flexible! Only rule, it has to be convex. How small can you get 3? Can you get 4? I want to know! The one with the weird folds was already absolutely wondrous in how it fit together
edit: also no self-intersection you hecks
Not to be *that guy* but why even restrict them to convex? As long as it's nonintersecting you can still realize them by folding. I think convexity is best reserved for when there are physical constraints or when you want to limit infinite sets to a finite subset (eg johnson solids), neither of which applies here, I think.
If that is your only rule, you can get all integer (trivial) solutions: 1 = 1 x 0.5 x 0 (using a half fold), 2 = 1 x 1 x 0, etc.
For me I have to say, that the "dark-blue-diagonal" folding was the most satisfying for me.
The christmas tree on the card at around 10:10 is so bad I love it
“The Parker Tree” is what I’ll call it.
The first three-option net (the one with the flat cuboid) would have made a really great string of lights for the christmas tree
What a wonderful video. My favorite since the last net one!
After rewatching a dozen of your videos, I wonder if 3D nets of 4D shapes can fold into different 4D shapes. And beyond that, if 2D nets of the 3D nets of 4D shapes can fold into new 3D shapes which are also nets of a different 4D shape (or even the same 4D shape, I guess that'd be cool too)
What would be really interesting is to send the Transcendental supporters two Christmas cards so that they can simultaneously have both folded cuboids next to each other.
That's a Charlie Brown Christmas Tree net that makes 2 polygons.
The Parker Christmas Tree
I think most mathematicians would think this is not possible because they would assume you wouldn't cut faces into different parts
Releasing this on a Saturday morning is perfect for watching with my kids. One of your best videos, enjoyed it a lot
Amazing production quality and learning this time! Love it!
Was waiting for a non-orthogonal example. That one is my favorite!
I absolutely love the off-angle-folded cube
I am so happy about the effort that you put into the videos
In a "How many" math question, usually the answer is undefined, 0, 1, 2, 3, or Infinity.
When Matt starts alluding to something being too big, all I think is that has never stopped him before. You're the best.
Absolutely amazing in a fun kind of way.
“WHAM!”
-Matt Parker, 2022
2:35 Erin Domain has great range! 🤣
I love the non-orthogonal one! I wanna laser cut one out of plywood with "living hinges" for the folds.
What about TRIANGLES? The net of a octahedron and three tetrahedra stuck together both have 8 triangle faces! I just haven't checked for possible solutions yet.
Bonus: if there's a solution, the faces aren't stitched together from multiple polygons.
After a bit of trial and error I found a solution.
Not sure what would be the best way to describe it, but here is a possible set of xy-coordinates of the vertices:
(0,0), (sqrt(3),1), (2sqrt(3),0), (2sqrt(3),2), (2sqrt(3), 4), (3sqrt(3), 5), (2sqrt(3), 6), (sqrt(3),5), (sqrt(3),3), (0,2), (0,0)
Amazing! So glad you actually made those 3 big boxes.
One of my favorite video in a while from stand up math. Merry Christmas to everyone.
The way the word net is used in this video differs in several ways from how I thought about nets until now. Here's my version: You optain a net of an n-dimensional shape by breaking up most of the (n-2)-dimensional "edges" such that the (n-1)-dimensional "surfaces" can be folded along those "edges" in such a way that they lie in a (n-1)-dimensional space without overlapping and while still being connected. If you consider the "surfaces" as vertices of a graph that are connected with an edge iff the "surfaces" share an "edge" then a net is basically a special spanning tree of the graph. So the folds are an inherent property of the net, which makes it a lot harder (if not impossible?) to find a net that folds into multipe different shapes, as only the angle of the folds can be different. I'm unsure whether angles of 0 should be allowed here as that feels kinda cheaty to me. If those angles are allowed and you also allow "edges" to cross through other "edges" you kinda end up at what this video is about. I also don't really get this fixation on gridlines. That concept falls apart very quickly as soon as you're not dealing with cuboids or at least shapes that are composed of cuboids or even just edges with irrational ratios. In my opinion it also makes more sense to say that e.g. a 1x1x2 cuboid as well as its nets consist of 6 surfaces rather than 8 surfaces 2 pairs of which meet at an angle of 0. Regardless of the fact that I disagree with the definition of nets here some of those constructions were still quite pleasing to look at.
I had the same thoughts. My gut tells me that if you want to find multiple 3D shapes from folding any of these nets that only have seams on edges, then they won't be convex polygons, but at least of of the 3D shapes will have a concave portion.
According to OEIS sequence A000104, there are on the order of 10^24 polyominoes with 46 squares (without holes, and up to symmetry). So brute force is not an option, since even if we could check trillions of them per second it would still take thousands of years to run through them all. We have to find some clever way to characterize nets that can fold into those three cuboids, 1x1x11, 1x2x7, 1x3x5.
Though repurposing folding@home might be able to brute force this perhaps?
Maybe enumerate non-self-intersecting paths over the cuboids (that visit each of the corners) and see whether they actually give nets, and find some fast way to compare them?
I thought you would start with all cuboids area 46, then find all unfolding nets for them, then compare if any nets are the same?
Oh, wait, nets are not necessarily paths. So enumerate all trees on the cuboids with the corners as leaves (not sure whether any way of cutting would not give a net, so check whether folding out actually gives a net) and then try to compare those nets in a fast way
The first step in comparing is probably binning them by their width and height. And there are probably more metrics.
Merry Christmas to you too Matt!
First one of the best episodes from mathologer, now one of most satisfying episode from you. What a day!
I think there's a mistake in 7:36. The blueish piece of paper should be moved one step to the left (and up of course) to fully cover the correct the surface.
yeah, I noticed that too
It's okay, we'll just call what he did a Parker Plane.
I wonder what the relationship is between the volumes of the different cuboids birthed from the same net
😎👌Great video Matt ! Thanks for the mention and for showing this surprising result to a wide audience.
I'm surprised this is surprising, anyone who has folded a towel should know it's possible.
I've been wondering since the last video... Can all 2D nets of a single 3D net of a 4D hypercube perfectly tile 2D space?
Wow lots of uploads recently, how spoiled we are!
I never have any clue what you're talking about but you have fun crafts and sound excited so I watch all your vids :)
I guess you can say that the root 5 non orthogonal cube is a parker square of a solution
Lisa is definitely the star of this episode! Thank you for enabling Matt with your wonderfully precise craft.
The title made me think we would be looking at arbitrary shapes, but this went in a different direction than I thought... Is there a reason why the problem is restricted to only nets of cuboids?
There are non-cuboid polyhedra in the papers. You can see a pyramid pictured as an example.
I absolutely anticipated different polyhedra
All that input of cutting and folding.
Thanks a lot.
10:44 - everyone knows the rule of a good comedian: inform your audience when a joke is a good one.