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> Folds are powerful. One can trisect or n-sect any angle for finite n. Does that mean folding allows you to construct (without trial-and-error) an accurate he
by Terr_ 1y ago
> Folds are powerful. One can trisect or n-sect any angle for finite n.
Does that mean folding allows you to construct (without trial-and-error) an accurate heptagon, even though you can't with a straight-edge and compass?
Intuitively, that seems wrong, I would expect many of the same limitations to apply.
- avhon1 1y agoSeems like you can https://origamiusa.org/thefold/article/diagrams-one-cut-heptagon-square https://origamiusa.org/thefold/article/diagrams-one-cut-hept... The one cut is to remove the perimeter of the square that lies outside the heptagon. Without the cut, you could make a crease, and fold the excess behind the heptagon.
- Terr_ 1y agoMy reading is that it's a convenient near-7 approximation someone developed, like using 22/7 for pi. Certainly good enough for practical handheld construction purposes, but not geometric-proof-y stuff.
- srean 1y agoCheckout Scimemi, Draw of a regular heptagon by folding. Proceedings of the 1st International Meeting of Origami Science and Technology. 1989 Simultaneous folding is mathematically a strictly more powerful primitive. Are you familiar with Lill's method of finding real roots of polynomials of any degree ? Simultaneous folds are a realization of the same idea https://en.m.wikipedia.org/wiki/Lill%27s_method#Finding_roots_using_paper_folding https://en.m.wikipedia.org/wiki/Lill%27s_method#Finding_root...
- jo-han 1y agoThis paper discusses constructing heptagons, with some history and the maths. http://origametry.net/papers/heptagon.pdf http://origametry.net/papers/heptagon.pdf It shows both a single sheet and a modular version.
- srean 1y agoYes. But remember one is dealing with idealized / axiomatized folding. The situation is similar with compass and straight edge geometry -- those physical lines and circles marked on paper are approximate but mathematically, in the world of axioms we assume the tools are capable of perfect constructions.