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3D modeling with paper
11–20 of 52 posts
Re: 3D modeling with paper
#12Btw, there's a pretty well known origami version of the SR-71 by Toshikazu Kawasaki. One square, no cuts, the usual. I folded it as a kid from diagrams in "Origami for the Connoisseur". It's not as detailed as the papercraft version, but I think it symbolizes the real airplane very well.
Re: 3D modeling with paper
#13As a person who wonders where the paper X-15 model he had vanished to after he joined the service, this resonates with me.
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.
Re: 3D modeling with paper
#14"3D Rendering with Paper" might have been a more accurate title. The modelling process is very similar to regular 3D modelling. In theory, with perfect paper and cutting and gluing skills you could print any UV map and cut, fold and glue it into a paper model using this method.
UV maps, especially for low-poly models, do not generally have a 1:1 geometric relationship with polygons in the original model. Areas with more significant detail will get more space on the UV map, mirrored or repeating areas will be overlapped, and of course UV maps will never include the tabs you'd need to physically glue parts together.
Re: 3D modeling with paper
#15I 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.
Akira Yoshizawa actually used origami in a factory setting to communicate geometric and engineering concepts.
Re: 3D modeling with paper
#16I 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.
Re: 3D modeling with paper
#17Btw, there's a pretty well known origami version of the SR-71 by Toshikazu Kawasaki. One square, no cuts, the usual. I folded it as a kid from diagrams in "Origami for the Connoisseur". It's not as detailed as the papercraft version, but I think it symbolizes the real airplane very well.
That's pretty awesome. I'd love to see the Lockheed F-117 Nighthawk get the same treatment. Seems like its angular design would lend itself well towards an origami version.
Re: 3D modeling with paper
#18You could have replaced a bunch of faces with larger cylindrical/conical faces (aka 3D developable surfaces) to get a more realistic look. Paper can bend!
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.
Re: 3D modeling with paper
#19You could have replaced a bunch of faces with larger cylindrical/conical faces (aka 3D developable surfaces) to get a more realistic look. Paper can bend! 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.
He specifically set a constraint for now curved surfaces. Using cylindrical and conical surfaces would have violated that constraint.
Re: 3D modeling with paper
#20I 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 o…
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.