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I always wonder what the Elements would have looked like had Euclid had included paper folding as a primitive. Folds are powerful. One can trisect or n-sect an

by srean 1y ago

I always wonder what the Elements would have looked like had Euclid had included paper folding as a primitive. Folds are powerful. One can trisect or n-sect any angle for finite n. One still needs the compass though for circle. Straight edge Compass Nuesis Paper folding Makes for a very powerful tool set.

WillAdams 1y ago
Akira Yoshizawa actually used origami in a factory setting to communicate geometric and engineering concepts.
olooney 1y ago
The Greeks were not adverse to studying topics outside of the classic axioms, for example neusis, conic sections, or Archimedes work on quadrature (which presaged calculus): https://en.wikipedia.org/wiki/Neusis_construction https://en.wikipedia.org/wiki/Neusis_construction https://en.wikipedia.org/wiki/Conic_section https://en.wikipedia.org/wiki/Conic_section https://en.wikipedia.org/wiki/Quadrature_(mathematics) https://en.wikipedia.org/wiki/Quadrature_(mathematics) https://en.wikipedia.org/wiki/Quadrature_of_the_Parabola https://en.wikipedia.org/wiki/Quadrature_of_the_Parabola They just preferred the simpler axioms on grounds of aesthetic parsimony. As far as I know, the ancient Greeks never thought to fold the paper. It has, however, been studied since the 1980's by modern mathematicians: https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms It can be used to trisecting an angle, an impossible construction with straightedge and compass: https://www.youtube.com/watch?v=SL2lYcggGpc&t=185s https://www.youtube.com/watch?v=SL2lYcggGpc&t=185s It's more powerful than compass and straight-edge constructions, but not by much. It essentially gives you cube roots in addition to square roots. You still need a completely different point of view to make the quantum leap the the real numbers, calculus, and limits: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... https://en.wikipedia.org/wiki/Dedekind_cut https://en.wikipedia.org/wiki/Dedekind_cut So ultimately I don't know if it would have changed the course of history that much.
- srean 1y ago
  Sure, it makes sense to isolate the minimal sets of primitives needed for an operation. Greeks experimented quite a bit with nuesis before focusing on straight edge and compass. Folding, as you noted, was not part of their mix. BTW nuesis can also trisect angles, so they could do it without origami. Origami folding is more powerful than the closure of rationale by square and cube roots. They were extended to the quintic roots by Robert Lang using a type of folding called multifold. Now it's known that with multifolds all of the algebraic numbers can be constructed with origami https://arxiv.org/abs/0808.1517 https://arxiv.org/abs/0808.1517 Yes one would not reach the reals (that's not the ultimate goal) but the geometry would certainly would have been richer. By no means is the area of folding a mathematical dead end as new theorems still get discovered.
- deleted 1y ago
  [deleted]
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 ago
  Seems 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 ago
    My 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 ago
      Checkout 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 ago
  This 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.
- 1y ago

by srean 1y ago

WillAdams 1y ago

Akira Yoshizawa actually used origami in a factory setting to communicate geometric and engineering concepts.

olooney 1y ago

The Greeks were not adverse to studying topics outside of the classic axioms, for example neusis, conic sections, or Archimedes work on quadrature (which presaged calculus): https://en.wikipedia.org/wiki/Neusis_construction https://en.wikipedia.org/wiki/Neusis_construction https://en.wikipedia.org/wiki/Conic_section https://en.wikipedia.org/wiki/Conic_section https://en.wikipedia.org/wiki/Quadrature_(mathematics) https://en.wikipedia.org/wiki/Quadrature_(mathematics) https://en.wikipedia.org/wiki/Quadrature_of_the_Parabola https://en.wikipedia.org/wiki/Quadrature_of_the_Parabola They just preferred the simpler axioms on grounds of aesthetic parsimony. As far as I know, the ancient Greeks never thought to fold the paper. It has, however, been studied since the 1980's by modern mathematicians: https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms It can be used to trisecting an angle, an impossible construction with straightedge and compass: https://www.youtube.com/watch?v=SL2lYcggGpc&t=185s https://www.youtube.com/watch?v=SL2lYcggGpc&t=185s It's more powerful than compass and straight-edge constructions, but not by much. It essentially gives you cube roots in addition to square roots. You still need a completely different point of view to make the quantum leap the the real numbers, calculus, and limits: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... https://en.wikipedia.org/wiki/Dedekind_cut https://en.wikipedia.org/wiki/Dedekind_cut So ultimately I don't know if it would have changed the course of history that much.

srean 1y ago
Sure, it makes sense to isolate the minimal sets of primitives needed for an operation. Greeks experimented quite a bit with nuesis before focusing on straight edge and compass. Folding, as you noted, was not part of their mix. BTW nuesis can also trisect angles, so they could do it without origami. Origami folding is more powerful than the closure of rationale by square and cube roots. They were extended to the quintic roots by Robert Lang using a type of folding called multifold. Now it's known that with multifolds all of the algebraic numbers can be constructed with origami https://arxiv.org/abs/0808.1517 https://arxiv.org/abs/0808.1517 Yes one would not reach the reals (that's not the ultimate goal) but the geometry would certainly would have been richer. By no means is the area of folding a mathematical dead end as new theorems still get discovered.
deleted 1y ago
[deleted]

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 ago
Seems 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 ago
  My 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 ago
    Checkout 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 ago
This 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.
1y ago