> 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 3 days ago | parent | next [-]

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.

▲ Terr_ 3 days ago | parent [-]

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 3 days ago \| parent [-]
		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_root...

▲ srean 3 days ago | parent | prev | next [-]

Yes.

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.

▲ jo-han 3 days ago | parent | prev [-]

This paper discusses constructing heptagons, with some history and the maths.

http://origametry.net/papers/heptagon.pdf

It shows both a single sheet and a modular version.