Posted by joshuawootonn 18 hours ago
There used to be an entire finishing process with this yellow and blue bottled smooth-cast resin and sanding before painting, but they always stayed paper for me.
Was a cheap way for me to have fun, and definitely holds a special place in my heart forever. Great share and thank you for posting! Brought me through memory lane.
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
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.
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.
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.
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.
http://origametry.net/papers/heptagon.pdf
It shows both a single sheet and a modular version.
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.
Certainly good enough for practical handheld construction purposes, but not geometric-proof-y stuff.
Scimemi, Draw of a regular
heptagon by folding.
Proceedings of the 1st
International Meeting of
Origami Science and
Technology. 1989
https://creativepark.canon/en/categories/CAT-ST01-0071/top.h...
While there are a lot of models available for purchase/download, the classic tool for this sort of thing is
https://pepakura.tamasoft.co.jp/pepakura_designer/
as noted by coldfoundry --- that said, an unlikely tool which has this is PythonSCAD:
which allows one to use OpenSCAD or Python to create a 3D model and export it in a number of formats, including "Foldable PS" which automates this process.
https://www.homeworldaccess.net/infusions/downloads/download...
The Kushan Carrier looks exactly like the one I put together as a kid after playing Homeworld, right down to the readme file saying "if you've never done anything like this before, I'd suggest starting with something else"... a warning I ignored as a kid!
I wonder if there are algorithms for approximating arbitrary geometries with a combination of planar, cylindrical and conical faces? Sheet metal fabrication should be facing the same constraints.
Fitting a -single- such surface to a set of points is nearly trivial; finding a way to best fit -multiple- such surfaces together to approximate a non-trivial shape (cloud of points) where they share edges in a way that could be joined like this paper model.... feels very NP-hard to me. This is a subset of the problem in the 3d-scan-to-CAD industry where you have a point cloud/mesh and you need to detect flat planes, cylinders, fillets, etc of a 3d scan and best-fit primitive surfaces to those areas and then join them into a manifold while respecting a bunch of other geometric and tolerance constraints.
There is a reason why there are only a few software packages that even attempt to do this, and it is almost always human-guided in some way. It's a fascinating problem.