100 thoughts on “Fold and Cut Theorem – Numberphile

  1. Does this theorem also hold in higher dimensions? Like, is it possible to turn a cube into any abritrary shape by folding it in the 4th dimension?

  2. Maybe being an American there's something I'm missing. So cutting things like letters and turtles out of paper with one cut is certainly impressive. But this idea that an adult had no idea how to cut a square out of piece of paper is unfathomable. An American kindergartner could demonstrate this.
    Do they not make paper snowflakes where your from?

  3. could u try and line up the different bits of paper correctly and cut out the whole alphabet with literally one cut

  4. So does that mean if you were willing to do all the folding that you could do one cut and get every letter in the alphabet from that one cut?

  5. You need to make up for the less number of cuts with more number of folds so essentially the "work done" is same.
    For example, in a square, no. of folds + no. of cuts = 4 always.

  6. now someone needs to get the letters better so that they are more curved like pixels. no curved lines but just make it like 8 bit letters

  7. 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."

  8. If you can cut out any two dimtional shape with one cut by folding in 3 demension can you cut out 3 denensional shape with one cut by folding in the 4th deminsion and so on?

  9. 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.

  10. So we can basically create any polygon with a single straight cut. Now can we build any shape with a single curvy cut 🤔

  11. Can't you create any sort of piecewise continuous shape, given the condition that you don't have to cut a straight line, and so long as all the piecewise functions are similar? Seems a pretty trivial extension of the theorem that might allow G, S, O, and maybe Q to look nicer.

  12. it was stated that you can "one-cut" anything with straight lines. but how about a really big circle? i mean a really big collection of straight lines that it is a circle? im sorry i cant express this well in english.

  13. I've worked out for myself how to cut out all 26 letters of the alphabet in one cut! It was really fun (and really frustrating, especially for R and S)

  14. That's really impressive!
    Now do Å, Ä and Ö.
    Jah just kidding, but that's mighty interesting that a field like that exists in math! ^^

  15. 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!

  16. 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.

  17. Does that mean if you could fold 3 dimentional objects through the 4th dimension, you could carve them into any shape with a single slice

  18. Umm… Can't you just fold it along the diagonal twice, then cut once? That would be mathematically more efficient, correct?

  19. wtf? for a square just fold the thing diagonally twice and then make one cut… why would you claim that 3 folds and one cut is the optimum?

  20. More efficient? That is if you assume that making one cut is easier than making one fold. I don't see how making three folds and one cut is more efficient than making one fold a three cuts. It is still four actions in all.

  21. 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!

  22. You can cut a square with a single cut with only 2 folds. It will be rotated by 45 degrees, but still a square.

  23. Your square in three folds is not efficient. It take only two folds. Fold across the diagonals.

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