This video for me shows the definition of a mathematician mind: one day you come with a silly pointless idea, but requires an original intelligent reasoning behind it, and even though you realize how silly and pointless it is, you still invest a lot of thinking, work and passion. And when you finish you feel like you accomplished something. A mind in love with reasoning, that screams "feed me with interesting puzzles"
This theorem actually has a connectiom with Lang's Axiality theory and the packing technique. The cut line, in origami crease pattern, becomes axial creases. Explained in greater detail in Origami Design Secrets, 2nd edition.
+Tadashi Mori When I made my thesis about math in origami, I also saw the Between the Folds documentary explaining a bit about the fold and cut, Erik Demaine cut the paper to make a swan, and then was talking about some algorithms in the computer. He also talked about the curved folds and how the Hyperbolic Paraboloid worked in the curved folds, just amazing.
+Tadashi Mori I saw some of Demaine's lectures in the MIT open courses, it's all about math and foldings. You should take a look, totally worth it (Don't know why i'm writing in english, sou br)
Here is what I think we need a video on. How does someone start at "Isn't that a theorem" and go about finding the answer out. I think the greatest deficiency of most people is their lack of knowledge of how to look things up and I wish more STEM channels talked about that process more.
I picture the scene where she's at the pub talking to her math friend and messing with paper the friend being just as geeky because she already knows it's a thing really funny
+William Brall This is a surprisingly hard problem in math, actually. The same mathematical object sometimes shows up in completely different contexts, using completely different notation. How do you build a search engine for that?
As humorous as it is to watch all of you answer the question for me, it isn't one that I need the answer to. I was not asking for help learning to research, I was pointing out a curious deficiency in existing STEM channel's content. Those of you who are replying with how to do research should stop and instead make videos on the process yourself.
Fun Fact: Erik Demaine who worked on the 'single cut' problem was home schooled and joined college at 12, completed his PhD by 20 and joined the faculty of MIT. He was the youngest professor in the history of MIT.
When I was around 7 or 8 my dad had some missionaries over for dinner. After dinner they found out I was into origami so they decided to show us a trick (that of course turned into a religious lesson) and his first question to me was: "Can you fold a piece of paper, so if you make only one cut and unfold it, it'll unfold into a cross?" So I thought about it for a few seconds, and said "I think so" and proceeded to fold a piece of paper in half twice, and then on the diagonal (like in the video) and snip it in half. Unfolding it it was a cross and the missionary says "Wow, I've never had someone actually do it before."
I know right. the humor in messing around with paper while being a mathematician and the process she went by being a mathematician here is also cool because it's symbolic of mathematical thinking
+pindakaas42 A million times yes! Please start posting links directly to papers referenced in your videos (when freely avaible online). These are channels trying to promote science, make it easy for the viewers to get to the actual science!
+Senia Maybe, but it is the Ø and Å that is impossible. It is not just an O with a line in. The line is longer than cirkel. And the Å is two, not connected symbols. But here you can cheat and make a AA. A lot of danes do it is more internet friendly. Cheating with the Ø is OE.
The ability to cut out a complex shape in just one cut astonishes me. I had no idea that a theorem such as " The fold and cut theorem" existed. The only theorems I ever knew of existing included physics and geometry. Lynda's video inspires me to think out of the box more often and to search up easier ways to complete tasks.
That square cut optimisation is a bit off... The best optimisation is to fold diagonally once into a triangle, then fold that in half a second time and cut once. (2 fold 1 cut)
It is still 2 cuts though, because the 1 cut you are talking about is divided in 2 segments. But I see what you mean, it would probably take less time to do what you say.
+Parax77 True - it can be done more optimally in terms of folds - although it is optimal for number of cuts. In this case, I wanted to carry on the same pattern I'd established, to keep things simple. The question of optimal folding is another one entirely and also quite interesting!
I think this is the most fascinating Numberphile video I've watched. It's impressive enough to share with people who wouldn't typically like Numberphile videos.
The Betsy Ross story was made up. But it's good. Incidently, we chose five points, because six, which was standard in Western heraldry at the time, was associated with nobility, which we didn't want to be. Oh, and we started with 13 stars.
I like how any straight-lined shape can be created with just one cut and one sheet of paper, & how the number and form of folds of the paper sheet are possibly endless.
As a side note, the square requires 1 fewer folds than was displayed to make a square hole in one cut. If you just fold across the diagonal. Then, fold along the other diagonal.
Katie, you can actually improve the one cut square even further, you can do a single cut with only two folds. If you fold it in half diagonally, you get a triangle, and then you fold it in half, and cut on the line, unfold, and you have a square!
Just to say when I saw this video i got really intrested. I didnt no you could make a cut through paper without poking a hole with a pen or pointy thing. And after just learning that seeing you be able to cut out the whole alphabat one by one with one cut I was amazed. Thanks for showing us this awsome video!
I used to learn how to do Origami once. Sometimes it was so complicated that I couldn't understand how people figured it out. I would love to see more videos about the subject.
Okay, I know the mythbusters did an episode a while back on how many times you can fold a piece of paper. I turned out, of course, that it had something to do with the size of the paper (bigger meaning you could fold it more times) but that at some point the thickness became prohibitively stiff to bend. Of course, that is straight folding in half, and your folds don't necessarily overlap entirely so you might not double the total thickness with each fold, but taking that into consideration, what do you suspect the upper threshold for shape complexity is?
Counting cuts and folds as operations, then there is a set of n possible shapes that can be made with m operations. For uniqueness shapes should only belong to the set with the lowest m the can be made with, otherwise the set of shapes that can be made with m operations would contain all sets of shapes that can be made with m - i operations for i = {1,...,m-1}.
Watching this channel reminds me of my dream to become a mathematician. I still want to but I think it's too late and that I let too much time pass. I am 29 and dropped out of college due to work during my first year. I used to be the guy that Aces every test and people copied off (I didn't even try hard). I used to love math because it was the only subject that made sense to me (and physics); and helped me with my work as an analyst for fortune companies. I want to go back to school and study it, but sometimes I feel like I'm too old for anyone to take me serious at college. I've been wanting to for the last couple years, but scared. I sometimes also think that with a decade passing since I went to school (for math) made me forget some of the basics. I don't know what to do to build up the confidence and go for it... Is it too late for me?
The accurate part about Betsy Ross is that whatever seamstress(es) made those flags probably did use this easy trick to make a reliable symmetrical star, because it was a common thing known among such folk. No secret, no mystery, no advantage over seamstress competition in the next town - just an age-old reliable way to do the thing. It's terrible for mass production, though, and not likely used when flags were mass-produced. It would waste wayyyy too much cloth!
Interesting thinking! Do you have any knowledge of the actual history around this, or is this just speculation? Either way, it's interesting to think about, I just think it would be nice to know which this is. :)
Why isn't re-discovering something rewarded in our society? It's people like this, figuring things like this out, that may make the fundamental discoveries in 30 years. Great work discovering something about Math(s). I personally don't care that someone already published something about it. Good work.
it makes me wonder if there's mathematically interesting shapes that can be obtained from strategic cuts on well known oragami patterns, perhaps at particular times during the folds
If you get the star (something I'm struggling with somehow... I _think_ I'm following correctly from 2:51 through 3:14 (hah! -- hmm, what's at 6:28? Oh look, my last name initial! ;) ), and then I'm unsure about the "fold that back" at 3:15 -- it _seems_ like I'm doing the same thing, but then 3:17 comes along, and... mine are different sizes. Is it something to do with the size of the paper I'm starting with, somehow?? (I'm using A4; should I be using US-letter? Would that make the difference??)
@@mustafak.2101 actually, it's not square... see my top-level comment for the math I did on the correct ratio (20:17, roughly). Square fails, too, just differently.
awesome, its an interesting idea that we could technically (if paper thickness weren't a factor) make any shape with one cut and lots of folding. cool vid!
now I wonder... Does the theorem take into account this other theorem that proved that a piece of paper cannot be folded in half more than 8 times (I think that was the number)
+Ciroluiro "The polynomial running time is a function of the number n of vertices in the input polygon, even though the number of required simple folds can be arbitrary large for a fixed n;". I don't know much about this but I really think that the ultimate goal is not about paper folding, but about understanding folding in a very general way (ultimately, understanding protein folding if I remember well).
+Ciroluiro The thesis states any bazillion-edged-thingmashape can be one-cut into a folded surface. I think the maximum individual edges we can cut through any physical surface folded in halfs is 2^8. The thesis is not constrained by number of edges, only that all the edges must be straight, so it cannot accomodate physical limitations on half folding.
Ya it is pretty cool but to make it more intuitive use jordan curve theorem you have 2 regions an inner and outer region. So when lining up the lines your essentially separating the regions into to distinct pieces when you cut.
From the title i thought this was dedicated to cutting Hasse diagrams of anisotropic G-varieties to obtain motivic decompositions or to folding Dynkin diagrams. You should a video about this (or linear algebraic groups in general). Maybe Alexander Vishik from Nottingham can help Brady :)
this video needs to be retitled to SUPER AMAZING AWESOME!!! cause the title you have now made me not watch it for a few days and now i cant belive what i missed out on!
1. optimizing cutting process 2. optimizing FOLDING process So you see the square can be cut with ONE cut and just TWO folds if you fold diagonally. Now optimize the number of folds needed for all the letters :D
Hypothetically it is possible to make a circular shape. First, you fold a large square piece of paper into a 90 degree triangle and then you fold it into a 45 degree triangle. After doing this, repeat folding the triangle an infinite number of times until the the triangle is infinitely thin. Finally cut the infinitely small triangle corner off and open to see a perfectly curved circle. This method is of course limited by physical limitations. The most obvious being that you can not fold paper infinitely. And even if you could then you would never get to cutting the corner because you could always fold the paper one more time!
There is a physical limit of ~8 in-half folds (256 individual layers) unless you have an extra dramatic difference between the sizes and the depth of the paper
I was wondering how critical the paper size was... or how accurate it is, even then. I get a pretty big fail re 3:17 when I try with A4 paper. It gets _better_ with US/Letter (8.5x11") paper, though it's still far from perfect rotational symmetry. I guess that's what the "ish" at 3:08 is about? ;) Edit: oh, and there's another ish at 3:20! So, yeah. OK, I feel better about my imperfection, then. :)
+pindakaas42 the problem is that this sort of math is actually pretty hard. you can easily convey the punchline in a funny internet video, but just listing punchlines makes for bad content for math education, you'd want to enable kids with tools to tackle these sorta ideas. I do think it would be great if we could integrate these sorta things into math education, but it's difficult to do in a way that's not just disconnected math fact you need to memorize for test or something.
Either string or surface. Folding is strings. Straight lines are parallel to planes that's why it is possible. You can do the same only with sin curves.
Francis Hopkinson (1737 - 1791) designed the first official American flag. He was an author, a composer, and one of the signers of the Declaration of Independence as a delegate from New Jersey.
1:10 is technically not max efficient. yes you only have to cut one cut. but u have to fold it 3 times. u can actually do the 1 cut thing also by only doin 2 folds.
I love this and things like this. Where do I study math to be able to work doing things like this. Do I need a math school for all of this at all? I did take a semester of math at university but that was almost no fun at all... Anyway what a great video, this seems like a lot of fun!!
If we're talking about optimization, then you should minimize the number of folds as well as the number of cuts. The square can be reduced from 3 folds to 2 folds. Make the first fold line to span from one corner of the square to its opposite corner. Then make the second fold to collapse the two remaining line segments onto a single line segment. Voila!
+user255 Well, if you're passionate enough to comment about it, maybe you're passionate enough to try it yourself. I'm not being snide here, try it, it's fun! Imagine how cool it would be to beat a math PhD at her own game!
This is really cool. I had no idea. But I guess it makes sense. It seems like all shapes with straight lines are made up of triangles which squares are also made up of
I wonder how far you can take this, like for example, can you say a circle is a infinite set of infnitely small straight lines, therefore would it (theoretically) be possible to cut a circle in one cut, and we could extrapolate this for literally any shape as long as we allow infinitely foldable paper to cut out shapes made out of infinitely small lines
My immediate thought about the square is that you don't have to fold thrice to do it with one cut: Two diagonal folds should work nicely. Of course the cut will be twice as long, but it still seems more efficient.
