Comment by srean
14 hours 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.
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/Conic_section
https://en.wikipedia.org/wiki/Quadrature_(mathematics)
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
It can be used to trisecting an angle, an impossible construction with straightedge and compass:
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_t...
https://en.wikipedia.org/wiki/Dedekind_cut
So ultimately I don't know if it would have changed the course of history that much.
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
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.
> 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.
Seems like you can
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.
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.
Akira Yoshizawa actually used origami in a factory setting to communicate geometric and engineering concepts.