---
title: Cut It Out of Colored Paper
url: https://oosioo.com/en/p/%EC%83%89%EB%8F%84%ED%99%94%EC%A7%80%EB%A1%9C-%EC%98%A4%EB%A0%A4-%EB%A7%8C%EB%93%A4%EA%B8%B0
date: 2026-10-15T10:26:51+00:00
author: SYSOP
summary: The line in my notebook read, "Cut it out of colored paper." At some point I stopped seeing the colored paper I used to fold as a kid. People who pushed origami all the way folded a rabbit from a single sheet, cut out a star with one snip, and built robots that fold themselves and walk away. After following that research, I borrowed the approach of Anatomy, a Claude skill that explains machines by building them as drawings, and made a skill for papercraft. Give it a shape and it produces a net for colored paper, then checks overlaps, glue tabs and refolding on its own. The fourteenth piece in A Page for Today.
---
# Cut It Out of Colored Paper

![A sheet of A4 printed with a red soccer-ball net lies at an angle on yellow colored paper, and beside it two half-folded pieces of the ball are coming together. On the left is the title "Cut It Out of Colored Paper"](/uploads/5132f0327a0f1f91.webp =720)

When I opened my idea notebook today, this is what was written there.

> Cut it out of colored paper.

As a kid I folded a lot of colored origami paper at school. Cranes, frogs, ddakji tiles, paper airplanes. There was always a half-used stack of colored paper in my desk drawer. But at some point I stopped seeing colored paper at all. I can't even remember the last time I folded something.

One of my friends with a young child folds toys for the kid out of paper. At first I thought it was just a knack with the hands, but as I followed the videos my friend showed me, I found that origami, pushed all the way, ends up in some rather strange places. There are people who fold an insect with six legs and antennae from a single sheet, people who cut out a star with one snip of the scissors, and people who use origami to design solar panels that unfold in space.

Then a thought came to me. If we turned the principles of origami into a program, could we make the shape of whatever we want out of paper? Could we go further and fold it so it moves like the real thing? If we also allow cutting and gluing, far more becomes possible. Today I follow that thought.

## Origami that began in kindergarten

There's a reason we folded colored paper as kids. Friedrich Fröbel (1782–1852), the German educator who created the first kindergarten, put paper folding on the list of "occupations" for children. This kindergarten education reached Japan in the late 19th century and blended with Japan's traditional folding ([Origami Heaven](https://origamiheaven.com/historyfreidrichfroebel.htm)), and that branch made its way into our classrooms.

The dashed and dash-dot lines that always appear in folding books are also a convention someone set. Akira Yoshizawa of Japan first introduced a notation in a 1954 book that drew valley folds as dashed lines and mountain folds as lines alternating dots and dashes, and Samuel Randlett of the United States refined it into a system in a 1961 book. This "Yoshizawa–Randlett system" is still the standard ([Wikipedia](https://en.wikipedia.org/wiki/Yoshizawa%E2%80%93Randlett_system)). The red dash-dot lines and blue dashed lines in the playground below follow this convention too.

## Folded paper has rules

When you start looking at origami through mathematics, the first thing you meet is the rule for a single vertex. Unfold a folded sheet and you'll see spots where creases branch out from a single point. For that spot to fold flat, two things have to hold.

- **The number of mountain folds and valley folds always differs by 2.** It's usually called Maekawa's theorem, but Murata published it first in 1966, and Justin found it independently in 1986.
- **Add up every other angle between the creases and you get 180 degrees.** This is called Kawasaki's theorem, or the Kawasaki–Justin theorem. Several people found it independently in the 1970s and 80s ([Hull 2013](https://arxiv.org/abs/1307.1065)).

There's one more. Paper can't pass through itself, so the order of the stacked layers has to work too. With a single vertex, checking these three in turn is all it takes. Drag the creases in tab ② of the playground below and you'll feel it right away.

The trouble starts when many vertices are tangled together. In 1996 Marshall Bern and Barry Hayes showed that, given a crease pattern, the question "does this fold flat?" is NP-hard ([Bern·Hayes 1996](https://ics.uci.edu/~eppstein/junkyard/origami.html)). It stays that way even if you're told which creases are mountains and which are valleys. The hard part, in the end, is the order of the layers: deciding which piece of paper goes on top of which so that nothing passes through anything else, and the number of possibilities explodes. This comes up again later.

Research on folding along curves started early too. David Huffman, famous for his compression code, wrote just one paper on paper folding, in 1976, proposing a visual language that handles straight and curved creases together ([Huffman 1976](https://doi.org/10.1109/TC.1976.1674542)).

## Programs that fold the shape you want

Some people turned the idea of "folding the shape you want" into actual programs.

Robert Lang, a physicist turned origami artist, built TreeMaker in the 1990s. You draw the animal you want to fold as a "tree," like a stick figure, and it turns each flap, such as a leg, horn or tail, into a circle on the square. The longer the flap, the bigger the circle. Pack these circles in as large as possible without overlapping, and the places where creases run between them are set ([Lang 1996](https://doi.org/10.1145/237218.237249)).

Tomohiro Tachi's Origamizer goes a step further. Give it any polyhedron and it produces a crease pattern that folds it from a single square sheet ([Tachi 2010](https://doi.org/10.1109/TVCG.2009.67)). In 2017 Erik Demaine and Tachi proved that the method works for any polyhedron ([Demaine·Tachi 2017](https://doi.org/10.4230/LIPIcs.SoCG.2017.34)). This is also the program that folded the "Stanford bunny," the test model of computer graphics, from a single sheet. The catch is that the creases for tucking away the leftover paper are enormously numerous, so folding it by hand is very hard.

If you want to preview a fold before folding it, there's [Origami Simulator](https://origamisimulator.org) by Amanda Ghassaei at MIT. Feed it a crease pattern and it shows the paper folding in your browser ([Ghassaei et al. 2018](https://erikdemaine.org/papers/OrigamiSimulator_Origami7/)). There's even a common file format ([FOLD](https://github.com/edemaine/fold)) these programs use to exchange crease patterns.

## If you allow cutting

Bring scissors into origami and another world opens up.

*Houdini's Paper Magic*, published in 1922, explains how to fold a sheet a few times and cut out a five-pointed star with a single snip. The oldest recorded example is a Japanese book from 1721. In 1999 Erik Demaine, Martin Demaine and Anna Lubiw pushed this to the limit, proving that any shape made of straight lines can be cut out with one straight cut after folding the paper the right way ([Demaine·Demaine·Lubiw 1999](https://erikdemaine.org/foldcut/)). A star, a swan, letters — even several separate shapes at once.

Cutting and gluing, that is, papercraft built from nets, is older still. Albrecht Dürer drew nets of the regular polyhedra in his 1525 book *Instructions for Measuring*. Yet even this simple-looking task still has an open question. In 1975 Shephard asked: can every convex polyhedron be cut along its edges and unfolded into a single piece that doesn't overlap itself? Fifty years later, we still don't know ([Wikipedia](https://en.wikipedia.org/wiki/Net_(polyhedron))). For non-convex polyhedra, there are examples that can't be unfolded into one piece however you cut them ([Bern et al. 2003](https://arxiv.org/abs/cs/9908003)).

Research on making papercraft patterns by computer took off in the 2000s. Jun Mitani and Hiromasa Suzuki came up with a way to approximate rounded toy models with narrow strips and build them from paper ([Mitani·Suzuki 2004](https://doi.org/10.1145/1015706.1015711)), and Shigeo Takahashi's team used a genetic algorithm to unfold models into a single non-overlapping piece ([Takahashi et al. 2011](https://doi.org/10.1111/j.1467-8659.2011.02053.x)). There's also work that automatically designs pop-up cards, where a building rises as you open the card, from a 3D model ([Li et al. 2010](https://cg.cs.tsinghua.edu.cn/people/~xianying/Papers/Popup/index.html)).

## Paper that moves

Can we fold not just a shape but a movement? That's already underway on several fronts.

The best known is the Miura fold. Devised by Koryo Miura of Japan in 1970, it repeats a group of four identical parallelograms. Pull on just one corner and the sheet opens in both width and depth at once; push and it folds up together. Because it has only a single motion, one motor can deploy it. In 1985 Miura proposed the fold as a way to unfold large membranes in space, and a deployment experiment was actually carried out on Japan's SFU experimental satellite, launched in 1995 ([Wikipedia](https://en.wikipedia.org/wiki/Miura_fold)). In 2014 NASA JPL worked with origami artist Robert Lang on a prototype solar array that folds to 2.7 m across and unfolds to 25 m ([NASA JPL](https://www.jpl.nasa.gov/news/solar-power-origami-style/)).

There are robots that fold themselves, too. In 2014 a Harvard and MIT team made a laminate of paper and shape-memory polymer. When heated, it folds itself into a robot in about four minutes and crawls away ([Felton et al. 2014](https://doi.org/10.1126/science.1252610)). The following year MIT presented a 1.7 cm square weighing 0.31 g that folds itself, walks and swims, and can be dissolved in a solution when its job is done ([Miyashita et al. 2015](https://dspace.mit.edu/handle/1721.1/97147)). Origami stents that enter a blood vessel folded and open at body temperature have also been studied ([Kuribayashi et al. 2006](https://doi.org/10.1016/j.msea.2005.12.016)).

On the cutting side there's kirigami. You only put regular slits in a sheet, yet when you pull it, the flat sheet pops up into 3D ([Rafsanjani·Bertoldi 2017](https://doi.org/10.1103/PhysRevLett.118.084301)) or stretches in two directions at once. By varying the slit pattern from place to place, you can design a flat sheet to open into a curved surface of your choice ([Konaković et al. 2016](https://doi.org/10.1145/2897824.2925944), [Choi·Dudte·Mahadevan 2019](https://doi.org/10.1038/s41563-019-0452-y)). Research that gradually changes the cell shapes of a Miura fold to produce a desired curvature belongs to the same line ([Dudte et al. 2016](https://doi.org/10.1038/nmat4540)).

Folding a shape you want has algorithms. So does cutting one out. Movement isn't yet at the stage of "give it the motion you want and it hands back the creases," but patterns with simple motions, like the Miura fold and kirigami, are being designed one by one.

## Can AI fold paper in its head?

Research that has AI models do origami has grown sharply lately. The results are still underwhelming. ORIGAMISPACE, presented at NeurIPS 2025, tested models on 350 problems that bundle crease patterns, folding steps and photos of finished models, and included a task of writing crease patterns directly as code ([Xu et al. 2025](https://arxiv.org/abs/2511.18450)). GamiBench, released that December, mixed 186 patterns that fold with 186 that don't, and even models like GPT-5 and Gemini 2.5 Pro struggled with one-step spatial judgments ([Spencer et al. 2025](https://arxiv.org/abs/2512.22207)). A paper from March this year concluded that simply making models bigger doesn't steadily improve their ability to reason about physical deformations like these ([Agarwal et al. 2026](https://arxiv.org/abs/2603.13856)).

Asking a model that learned from words to fold paper in its head is asking it to do something hard even for people. So what if we don't make it fold in its head, but hand the folding math to a program and let the model use that program? As it happens, a skill that works exactly that way came out last week.

## Anatomy: a skill that builds machines to explain

On October 10, a developer named Ryan released a Claude skill called [Anatomy](https://github.com/wheresryan22/anatomy). Give it a concept in one line and it builds a machine that explains it, as an isometric drawing — one you can rotate, take apart and watch move. Within five days it passed 10,000 stars on GitHub (11,556 as of October 15).

Reading the skill's documentation, what stands out more than the drawing skill is the order in which it works.

1. **Collect real numbers first.** Sizes, counts, thresholds. Every number and movement in the drawing must come from the same model.
2. **Turn the concept into a real object.** The concept "if the cache key matches, skip the work" becomes a lock that turns when all the pins line up. In the documentation's words, "the object is the explanation."
3. **Write the parts list first.** A single drawing has 40 to 150 parts.
4. **Build the geometry in code.** Cylinders, solids of revolution, tubes, down to a single screw, all made with functions.
5. **Prove it with checks.** `audit` proves that "nothing passes through anything, every end is attached to something, and every overlap is drawn in depth order." Nothing is matched by eye.
6. **Zoom in and look.** It takes the drawing captured with headless Chrome, enlarges it four times, and inspects every joint.

## Can Anatomy design origami?

Not as is. Anatomy draws machines made of rigid parts. Its documentation explicitly says it doesn't do people or maps. It could probably make "a picture of folded paper," like an origami diagram, but that's a picture, not a design.

Yet Anatomy's skeleton carries over to paper almost unchanged. Build a model from real numbers, build the geometry in code, check the result yourself, and capture it to confirm. Check number 5 is especially interesting. The "depth order of overlapping parts" that Anatomy proves is the same question as the layer order in origami we saw earlier. The difference is that machine parts have 3D positions, so casting a ray tells you right away what's in front. In flat-folded paper, every layer lies in the same plane. The geometry doesn't tell you front from back, so you have to choose an order, and that choice is the NP-hard problem. That's why designing single-sheet folds is where Anatomy-style checking gets most expensive.

Papercraft that is cut and glued is a different story. A net is a set of polygons laid out on one plane, so the "doesn't overlap" check is just a matter of whether polygons collide. Whether it becomes the right shape when folded can also be traced back by calculation. The checks are cheap and certain. I figured this is where an Anatomy-style skill could prove useful first. So I built one.

## Paper Anatomy: a skill for cut-and-glue papercraft

I borrowed the name from Anatomy and called it "Paper Anatomy (papercut)." It consists of a skill document, a calculation kit, a command-line tool and the playground below. Its working order follows Anatomy.

1. **Real numbers.** Finished size, A4 paper (210×297 mm, 10 mm margins), a 7 mm glue tab height, and 4 mm as the shortest edge you can cut with scissors.
2. **Build the model in code.** The mushroom is a solid of revolution made by rotating its profile line, the fish is built by joining elliptical cross-sections, and the house by extruding a pentagon. Fewer faces are easier to cut; more faces look more like the real thing.
3. **Unfold.** Lay one face on the paper and attach neighboring faces one at a time. Each time, check whether it overlaps faces already placed and whether it goes past one A4 sheet; if there's a problem, cut that edge instead of folding it. When there's nowhere left to attach, start a new piece. Try dozens of times with different starting faces and orders, and pick the one with the fewest pieces and the shortest total glue length.
4. **Tabs and numbers.** Give each cut edge a pair number, and put a glue tab on only one side of the pair. If a tab covers another face or tab, put it on the other side; if that doesn't work either, fall back to a half tab.
5. **Prove it with checks.** It's done only when all six rows pass.

| Check | What it asks |
|---|---|
| Overlap | Do no two faces overlap within a piece? |
| Tabs | Does no tab cover a face or another tab? |
| Pairs | Does every cut edge have exactly one number and one tab? |
| Refold | Folding by the computed angles, does every vertex return to its original place (within 0.01 mm)? |
| Paper | Does every piece fit on one A4 sheet? |
| Scissors | Is the shortest edge at least 4 mm? |

The refold check is what I learned most from Anatomy. A net that looks plausible and a net that really becomes the original shape when folded are different things. So it rotates every face by the computed angle at each hinge to fold it back up in 3D, and measures whether each vertex lands where it belongs. The folding animation in the playground shows exactly this calculation, so if it fits on screen, it fits on paper.

```map /embed/papercut-2026-en.html 3:4
Paper Anatomy playground. In tab ①, pick a cube, house, soccer ball, mushroom or fish, or upload your own model (.obj), and it makes a non-overlapping A4 net with a check table and shows the cut paper folding into shape. You can also play with ② the mountain-valley rules at a single vertex, ③ the Miura fold and ④ rotating-squares kirigami.
```

Printing is blocked inside the frame. To print on colored paper, press "Print on A4" or "Download SVG" in the [playground opened in a new window](/embed/papercut-2026-en.html). If you use a cutting plotter, you can feed it the SVG as is. Uploaded model files never leave your browser.

## Fewer pieces, or tabs on every seam?

What I wrestled with longest while building this was the glue tabs. On a soccer ball, the angles at a vertex where one pentagon and two hexagons meet add up to 348 degrees. That's 12 degrees short of the 360 degrees of flat paper. So on the net, the two cut edges next to that vertex sit almost touching, with a 12-degree gap. There's no room to fit a tab between them. In the first version, fifteen or so of the soccer ball's 60 seams couldn't get a tab. Once I allowed half tabs, which cover only half of an edge, all 60 seams got tabs in 2 pieces.

The mushroom still wasn't solved by that. It has many spots where the area around a vertex is nearly flat, like the top of an umbrella. Here you have to choose.

![On the left is the mushroom net in "fewer pieces" mode: two large pieces with 18 seams that have no tabs. On the right is "tabs on every seam" mode: eight pieces, including petal-like pieces spreading from the decagon at the top of the cap, with tabs on all 77 pairs](/uploads/c1c70130e0fbac78.webp =720)

With fewer pieces it's done in 2 pieces, but 18 seams have to be butted together without tabs and taped. With tabs on everything, it grows to 8 pieces. If I were making it with a child, I'd choose the tabs. With more pieces you just glue them in number order, but seams without tabs are hard to line up even for adult hands. The playground includes both modes.

## I ordered a cat in one line

To see whether the skill really works as a skill, I gave a fresh Claude agent, with no connection to this conversation, only the skill document and the kit, and placed a one-line order: "/papercut a sitting cat, longest side 140 mm, A4 colored paper."

The agent split the cat into seven closed solids. It built the body and head as elliptical solids of revolution, the two ears as triangular pyramids, the tail as a square tube lying on the ground and wrapping around the body, and the two front paws as small boxes. The solids only touch each other; none pass through another. Then it unfolded, checked and fixed. After four rounds of fixes, out came a 102-face cat, and all six check rows passed. It took 9 minutes.

![At the top, a sitting low-poly cat seen from the front three-quarter, the side and the back three-quarter. An octagonal head with two pointed ears sits on a cone-like body, and the tail wraps around the body on the ground. Below are two A4 sheets printed with the nets for the body and front paws, and for the head, ears and tail](/uploads/ff2280fd0505329a.webp =720)

Unfolding each part separately scattered 10 pieces across eight A4 sheets. The agent tried grouping the parts in different ways and cut it down to one sheet for body + front paws and one for head + ears + tail, 11 pieces in all. That was a judgment call not in the skill document.

The agent's self-review was cool-headed. It wrote that from the front three-quarter and the side it reads as a cat, but the silhouette is closer to "a ball on a bowling pin," that without front legs it looks like two boxes set down, and that the face would have to be drawn by hand. It also said a human designer would have made the ears as bottomless triangular pyramids attached directly to the head, and printed eye and nose markings and part names on each piece.

More useful was the list of twelve gaps in the skill that it brought back. The revolution function couldn't be used outside the kit, there was no feature for gathering several parts onto one sheet, and the checks missed numbers overlapping on small faces, among others. Just as Anatomy keeps fixing until its checks reach zero, this list becomes the check items for the next version.

## What it can't do yet

This skill can't design single-sheet folds. Folding a rabbit from one square sheet is the domain of TreeMaker and Origamizer, and because of the layer-order problem we saw earlier, the checks are far more expensive. Curved surfaces are only approximated with flat pieces, so models like a smooth face need a different approach, such as Mitani and Suzuki's strips. The unfolding method is also still on the simple side. It greedily attaches neighboring faces and cuts when it gets stuck, so it lacks a human designer's sense for dividing things up — "this is an ear, so it goes separately; this is the belly, so white paper."

And to be honest, as of writing this, I haven't actually printed one on colored paper and cut it out. It passed the refold check on screen, but the thickness of the paper and misaligned glue spots you only learn by cutting.

## Back in front of the colored paper

Folding colored paper as a kid, I had no idea these rules existed. That at the center point of the "square base" you always pass through when folding a crane, the counts of mountain and valley folds differ by two; or that mathematicians wrestled for decades because of the obvious fact that paper can't pass through itself.

What Anatomy showed is that if you have the model build an object instead of drawing a picture, and then check it itself, the results change. It was the same with paper. It's fine if the model can't fold paper in its head. The code does the folding math, and the checks see whether anything overlaps. The model decides what to make, builds the model and fixes it until the checks pass.

The next time I see that friend, I plan to send this playground's address. So that when the child says in one line what they love, a few sheets of colored paper become a toy that evening.