Wikipedia:Reference desk/Archives/Mathematics/2016 September 6
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September 6
editGeometry problem
editIs it possible to fold a 4-inch square piece of paper into a regular convex polygon with 16 sides or more and a circumcircle diameter of 1.38 inches or slightly less without cutting or tearing the paper? (An additional condition is that all the flaps must be folded to one face of the paper -- the other face must remain flat and smooth.) 2601:646:8E01:7E0B:F88D:DE34:7772:8E5B (talk) 06:44, 6 September 2016 (UTC)
- It depends on how thick your paper is (also, how strong your fingers are). --CiaPan (talk) 06:58, 6 September 2016 (UTC)
- Geometrical paper is infinitely thin and can be folded. Physical paper is thicker and can only be bended. Bo Jacoby (talk) 07:26, 6 September 2016 (UTC).
- Actually, I lied -- the material in my case is not paper, but a very thin stainless steel mesh. 2601:646:8E01:7E0B:F88D:DE34:7772:8E5B (talk) 09:53, 6 September 2016 (UTC)
- Geometrical paper is infinitely thin and can be folded. Physical paper is thicker and can only be bended. Bo Jacoby (talk) 07:26, 6 September 2016 (UTC).
- Yes you can do that quite easily. Draw a sixteen sided figure on the square which has four of its sides on the opposite sides of the square. Then just fold over the sides. The diameter of the circumcircle will be a little more than two inches. Or am I missing something in the question? Dmcq (talk) 09:29, 6 September 2016 (UTC)
- Yes -- the question was whether you can get it down to 1.38 inches or less. 2601:646:8E01:7E0B:F88D:DE34:7772:8E5B (talk) 09:49, 6 September 2016 (UTC)
- Do you allow to fold it more than once, i.e. fold on fold? If you do, then that's possible (provided that the thickness of paper is negligible). If you don't, then what do you mean by "a 4-inch square"? Do you mean that the perimeter is 4 inches (in which case the square can be folded as you wanted it), or you meant that - every side is 4 inches long - or the area is 4 square inches (in which case the square cannot be folded as you wanted it)? HOTmag (talk) 10:09, 6 September 2016 (UTC)
- Yes, fold on fold is allowed, as long as all the folds are on one face of the resulting polygon. And a 4-inch square means every side is 4 inches long. 2601:646:8E01:7E0B:F88D:DE34:7772:8E5B (talk) 22:50, 6 September 2016 (UTC)
- Do you allow to fold it more than once, i.e. fold on fold? If you do, then that's possible (provided that the thickness of paper is negligible). If you don't, then what do you mean by "a 4-inch square"? Do you mean that the perimeter is 4 inches (in which case the square can be folded as you wanted it), or you meant that - every side is 4 inches long - or the area is 4 square inches (in which case the square cannot be folded as you wanted it)? HOTmag (talk) 10:09, 6 September 2016 (UTC)
- Yes -- the question was whether you can get it down to 1.38 inches or less. 2601:646:8E01:7E0B:F88D:DE34:7772:8E5B (talk) 09:49, 6 September 2016 (UTC)
- (ec) It depends, partially, on how much force you're able to apply. You can roll a sheet of paper into a paper straw and (with appropriate force) flatten it into a stripe. Then fold it again and press once more to form a small square...
But the minimum size is actually limited by the radius of bending layers at edges of the figure (hence also by the total length of those parts of layers in the bends). See e.g. How Many Times Can You Fold a Piece of Paper? at 'mental_floss'. And here you can find some relevant maths by Britney Gallivan: Folding Paper in Half 12 Times at 'Pomona historical'. There is even Wikipedia article about it – see Britney Gallivan#Paper folding theorem --CiaPan (talk) 10:32, 6 September 2016 (UTC)
- (ec) It depends, partially, on how much force you're able to apply. You can roll a sheet of paper into a paper straw and (with appropriate force) flatten it into a stripe. Then fold it again and press once more to form a small square...
- Just fold down the original square to a smaller square so a side is a bit less than 1.38 inches and go on from there. Where does this 1.38 inches come from anyway that you think it is some problem? Dmcq (talk) 12:54, 6 September 2016 (UTC)
- @Dmcq: Note a specific material: 'a very thin stainless steel mesh' and a reference to Nominal pipe size#NPS tables for selected sizes – I suppose it is a practical problem to make a filter to fit inside the pipe junction in the absence of a cutting tool... --CiaPan (talk) 13:07, 6 September 2016 (UTC)
- Cutting it would be just as easy then, a hacksaw or a fretwork saw to cut steel would make short work of that. Dmcq (talk) 13:21, 6 September 2016 (UTC)
- I can in principle cut out a circle of the required size, but I'd rather not do it if I can get the same result by folding the square. 2601:646:8E01:7E0B:F88D:DE34:7772:8E5B (talk) 22:50, 6 September 2016 (UTC)
- Since (4/1.38)^2 is approximately 8.4, any folding will end up with at least 9 thicknesses of mesh in some places. That sounds very impractical to me. I think cutting will be much easier. Gandalf61 (talk) 10:49, 7 September 2016 (UTC)
- 9 thicknesses? That does sound pretty hard to do with steel mesh, even for a strong person like me. 2601:646:8E01:7E0B:F88D:DE34:7772:8E5B (talk) 01:44, 8 September 2016 (UTC)
- Worse: the area of the circle is 1/10.7 of the parent square, so 11 layers. —Tamfang (talk) 10:28, 8 September 2016 (UTC)
- Since (4/1.38)^2 is approximately 8.4, any folding will end up with at least 9 thicknesses of mesh in some places. That sounds very impractical to me. I think cutting will be much easier. Gandalf61 (talk) 10:49, 7 September 2016 (UTC)
- I can in principle cut out a circle of the required size, but I'd rather not do it if I can get the same result by folding the square. 2601:646:8E01:7E0B:F88D:DE34:7772:8E5B (talk) 22:50, 6 September 2016 (UTC)
- Cutting it would be just as easy then, a hacksaw or a fretwork saw to cut steel would make short work of that. Dmcq (talk) 13:21, 6 September 2016 (UTC)
- @Dmcq: Note a specific material: 'a very thin stainless steel mesh' and a reference to Nominal pipe size#NPS tables for selected sizes – I suppose it is a practical problem to make a filter to fit inside the pipe junction in the absence of a cutting tool... --CiaPan (talk) 13:07, 6 September 2016 (UTC)
- Just fold down the original square to a smaller square so a side is a bit less than 1.38 inches and go on from there. Where does this 1.38 inches come from anyway that you think it is some problem? Dmcq (talk) 12:54, 6 September 2016 (UTC)