3D modeling with paper
28 comments
·September 12, 2025cousin_it
Btw, 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.
vunderba
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.
amelius
Doesn't a paper cutter like the Cricut generate these parts out of the box?
coldfoundry
Oh wow, this brought me back! I used to be obsessed with papercraft back in the day as a kid, specifically “pepakura”. I used to print out halo 3 helmets and build them and wear them. It was like a puzzle on steroids in the cool department!
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.
srean
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
olooney
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.
srean
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.
null
WillAdams
Akira Yoshizawa actually used origami in a factory setting to communicate geometric and engineering concepts.
WillAdams
As 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.
Stevvo
"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.
RodgerTheGreat
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.
syntaxing
This is super cool. In theory, a lot of this could have been automated. Quad remesher would probably get you close enough to import to the paper software and Cricut like machine for the cutting and scoring(?).
RivieraKid
I remember paper models being very widespread when I was a kid in the Czech Republic, they were always included in a popular magazine for kids, no idea whether it has changed. Per ChatGPT this is unique for this region - Czech Republic, Poland, Slovakia.
rimunroe
If anyone's a fan of papercraft models and the game Homeworld, you might enjoy this collection of models from the games. I remember my sister put together several of these back in the early/mid 2000s.
https://www.homeworldaccess.net/infusions/downloads/download...
meindnoch
You 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.
mk_stjames
That type of shape constraint would be called having a ruled surface with a Gaussian curvature of 0 everywhere, otherwise known as a 'Developable Surface'.
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.
zaphar
He specifically set a constraint for now curved surfaces. Using cylindrical and conical surfaces would have violated that constraint.
turtlebits
Semi-related, but Canon has a great papercraft site, with varying difficulties. My kid especially loves the moving models.
https://creativepark.canon/en/categories/CAT-ST01-0071/top.h...
It is/was quite popular in Poland. 35 years ago, as a kid, I was assembling paper models. Planes were the easiest, usually it took about 2 days to do one. Couple of years ago I wanted to get back to it, so I bought a plane. Well, it turned out that fashion for paper models had changed and now 'reductionist' models are in full swing - being as close as possible to original. That plane has 160 pieces (a lot of them also subdivided), and every part that has size about 10cm in real life, has been modelled. In two weeks I was still in cockpit. Here is paper model of SR-71: https://www.sklep.model-kom.pl/sr-71-model-samolotu-rozpozna... From drawings it looks like it is more than 167+, not including subparts.