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.
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.
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.
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):
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:
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
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.
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.
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.
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.
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!
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.
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.
Hey, I'm the original author! I should have elaborated more on this constraint. First, many papercraft models do use cylindrical/conical faces - it's just something I prefer not to do stylistically. Part of the art here is the approximation, rather than aiming for perfect realism. There's also the fact that not all paper bends the same. Papers and cardboards come in various weights and textures, so they each can curve differently. Keeping only flat faces removes these variables from the assembly.
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.
Human problem? It's probably already solved by one of the many recent machine learning papers, often there is source on GitHub and Transformer models on HuggingFace or some random Google Drive or Biadu drive. So one such human problem is finding how to ask aXiv Assistant what the best SOTA papers for it are and searching for if they finally released code or not (hoping researchers have a real repo not a GitHub site without code).
I recall that Nvidia have some clean solutions. I wish it was a more pure principled solver though with some clean code. Probably OpenEvolve could iterate on a solution to it like the circle packing problem example but 3D.
Sometimes it's funny to think that there are human problems left, which itself really is a human problem.
But that's an arbitrary constraint choice that didn't need to be there. It's not inherent to the medium. He has a justification for it (curves are "flimsy and introduce variances") but that is easy to get around with perpendicular reinforcing pieces inside that constrain the curve.
"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.
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.
Whether this article would get more or less attention with this changed title depends a lot on the ratio "viewers from the USA"/"viewers from other countries".
While that would certainly be a factor, I think I'd argue it's less about where you're from and more about what your interests or experience are.
I actually think the title as it is now has more mass appeal; it's very general and might pique your curiosity if you're interested in either 3-D modelling or paper crafting.
On the other hand if it had the "SR-71 Blackbird" in the title, some readers might shy away due to either not knowing what that is, or thinking "well, I'm not really interested in planes".
Which would be kind of a shame, since I think the post has some nice points to make regardless of whether you're into the SR-71 Blackbird or planes; that's just the example chosen to paint the broader picture.
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(?).
I do some cardboard / papercraft, but mine is completely unplanned and without this high level of precision. So mine is not suitable for accurate scale model building, but rather for building random houses / castles / vehicles.
This is ridiculous. I’ll tell you why. Here I quote:
“All parts in the assembled model must be made of paper.
Each part must be a single, solid color. The parts must not use any printed textures or designs.
The model must be represented as a simple polyhedron.”
Must. Must. Must.
This is a game. Or an art school exercise.
Modeling is concerned only with attaining the necessary accuracy. Not conforming to a methodology.
> Modeling is concerned only with attaining the necessary accuracy. Not conforming to a methodology.
Maybe to you. More in general, your claim is simply wrong.
This is actually answered in TFA:
> Constraints: Let's set some constraints for how we're allowed to model our creation. These are self-imposed limitations that fit my preferred-style for model design:
> Why constraints? It may feel weird to impose constraints on an art. However, I find that these constraints encourage a better designed model that can be assembled easily and predictably, including by others.
It's ok if you disagree with this because you enjoy your model-making in a different way. The author explained why they chose this path, and it makes sense: a lot of art is about constrains ("don't do digital", "use only 2 colors", "origami without any cuts", etc).
"These are self-imposed limitations that fit my preferred-style for model design... I find that these constraints encourage a better designed model that can be assembled easily and predictably, including by others."
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.
Direct link: https://www.giladorigami.com/BO_Conn.html
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.
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.
Source for "origami CAD" Pepakura:
https://pepakura.tamasoft.co.jp/pepakura_designer/
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.
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.
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.
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.
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.
Checkout
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...
Akira Yoshizawa actually used origami in a factory setting to communicate geometric and engineering concepts.
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...
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:
https://pythonscad.org/
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.
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...
Hey, thanks for posting this!
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!
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.
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.
Hey, I'm the original author! I should have elaborated more on this constraint. First, many papercraft models do use cylindrical/conical faces - it's just something I prefer not to do stylistically. Part of the art here is the approximation, rather than aiming for perfect realism. There's also the fact that not all paper bends the same. Papers and cardboards come in various weights and textures, so they each can curve differently. Keeping only flat faces removes these variables from the assembly.
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.
Human problem? It's probably already solved by one of the many recent machine learning papers, often there is source on GitHub and Transformer models on HuggingFace or some random Google Drive or Biadu drive. So one such human problem is finding how to ask aXiv Assistant what the best SOTA papers for it are and searching for if they finally released code or not (hoping researchers have a real repo not a GitHub site without code). I recall that Nvidia have some clean solutions. I wish it was a more pure principled solver though with some clean code. Probably OpenEvolve could iterate on a solution to it like the circle packing problem example but 3D. Sometimes it's funny to think that there are human problems left, which itself really is a human problem.
He specifically set a constraint for now curved surfaces. Using cylindrical and conical surfaces would have violated that constraint.
But that's an arbitrary constraint choice that didn't need to be there. It's not inherent to the medium. He has a justification for it (curves are "flimsy and introduce variances") but that is easy to get around with perpendicular reinforcing pieces inside that constrain the curve.
"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.
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.
These were popular in Soviet Union as well. At least in seventies in Baltic states where I grew up.
Given that the article's author used Blender to create their model, I'm surprised they didn't use it's built-in paper model exporter.
https://docs.blender.org/manual/en/4.1//addons/import_export...
They mentioned it, seem to prefer pepakura.
Very cool. Would probably get even more attention with the title "3D Modeling the SR-71 Blackbird with Paper".
Whether this article would get more or less attention with this changed title depends a lot on the ratio "viewers from the USA"/"viewers from other countries".
While that would certainly be a factor, I think I'd argue it's less about where you're from and more about what your interests or experience are.
I actually think the title as it is now has more mass appeal; it's very general and might pique your curiosity if you're interested in either 3-D modelling or paper crafting.
On the other hand if it had the "SR-71 Blackbird" in the title, some readers might shy away due to either not knowing what that is, or thinking "well, I'm not really interested in planes".
Which would be kind of a shame, since I think the post has some nice points to make regardless of whether you're into the SR-71 Blackbird or planes; that's just the example chosen to paint the broader picture.
I used to have books of models like these. Space stations, trains, wild west diorama sets, cars, etc. I wish I could find copies of the older ones.
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(?).
The final build looks great, I thought I was looking at a 3D render.
I like software: open source and only for linux
Doesn't a paper cutter like the Cricut generate these parts out of the box?
Interesting.
Optimization idea: Make your 3D modellers make their models first out of paper. Bet they'll be more cognizant about extra triangles!
I love this!
I do some cardboard / papercraft, but mine is completely unplanned and without this high level of precision. So mine is not suitable for accurate scale model building, but rather for building random houses / castles / vehicles.
This is ridiculous. I’ll tell you why. Here I quote:
“All parts in the assembled model must be made of paper. Each part must be a single, solid color. The parts must not use any printed textures or designs. The model must be represented as a simple polyhedron.”
Must. Must. Must. This is a game. Or an art school exercise.
Modeling is concerned only with attaining the necessary accuracy. Not conforming to a methodology.
What a bizarre objection.
> Modeling is concerned only with attaining the necessary accuracy. Not conforming to a methodology.
Maybe to you. More in general, your claim is simply wrong.
This is actually answered in TFA:
> Constraints: Let's set some constraints for how we're allowed to model our creation. These are self-imposed limitations that fit my preferred-style for model design:
> Why constraints? It may feel weird to impose constraints on an art. However, I find that these constraints encourage a better designed model that can be assembled easily and predictably, including by others.
It's ok if you disagree with this because you enjoy your model-making in a different way. The author explained why they chose this path, and it makes sense: a lot of art is about constrains ("don't do digital", "use only 2 colors", "origami without any cuts", etc).
Did you read the sentence above this quote?
> "These are self-imposed limitations that fit my preferred-style for model design"
If you have a different preferred style, then write your own article and how-to, stop complaining and touting nonsense yourself.
"These are self-imposed limitations that fit my preferred-style for model design... I find that these constraints encourage a better designed model that can be assembled easily and predictably, including by others."
Seems reasonable.