Preface

This book began with a photograph.

It showed a solar panel array folded into a cylinder small enough to fit inside a rocket nosecone, and the caption explained that the entire 8-square-meter panel had been collapsed by a single degree of freedom — one pull on one mechanism caused every crease to fold simultaneously, every panel to stack neatly against its neighbor, with a geometry so beautifully constrained that the only way it could fold was the way the designer intended. The origami pattern had been computed. The fold was optimal.

That photograph was taken aboard the ISAS Space Flyer Unit in 1995, and the fold pattern was the Miura-ori, invented by astrophysicist Koryo Miura in the 1970s as a solution to a very specific engineering problem: how do you package a large flat surface inside a small cylinder and deploy it reliably in zero gravity, with no hinges, no motors, and no chance of a second attempt?

Miura’s answer — a parallelogram tessellation that collapses along two axes simultaneously — turned out to be one of the most elegant intersections of geometry and engineering in the twentieth century. And it is, at its core, a solution to an optimization problem.

What this book is about

The argument of this book is that origami and numerical optimization are the same subject wearing different clothes. A valid flat-foldable crease pattern is a feasible point in a constrained geometric space. Designing an origami base is a circle-packing problem. Finding the equilibrium shape of a partially-folded structure is an energy minimization. And recovering the crease pattern that produces a desired deployed shape — the inverse problem — is a nonlinear program.

We develop all of these ideas from scratch, with Python code at every step.

Who this book is for

Engineers and scientists who are comfortable with linear algebra and calculus, and who have written some Python but may not have used scipy.optimize seriously. No prior origami knowledge is assumed. No prior optimization theory is required beyond the intuition that minimizing a function means finding its lowest point.

Each chapter ends with exercises that range from “verify this by hand” to “implement this extension.” The exercises are the book. Reading without working them is like reading a recipe without cooking.

A note on the code

All code runs inside the pixi environment defined in pixi.toml. To reproduce any result:

pixi install
pixi run kernel     # register the Jupyter kernel
pixi run render     # render HTML
pixi run preview    # live preview

The only dependencies are numpy, scipy, and matplotlib — all available from conda-forge with native ARM64 builds on Apple Silicon.

# Preface {.unnumbered} This book began with a photograph. It showed a solar panel array folded into a cylinder small enough to fit inside a rocket nosecone, and the caption explained that the entire 8-square-meter panel had been collapsed by a single degree of freedom — one pull on one mechanism caused every crease to fold simultaneously, every panel to stack neatly against its neighbor, with a geometry so beautifully constrained that the only way it *could* fold was the way the designer intended. The origami pattern had been computed. The fold was optimal. That photograph was taken aboard the ISAS Space Flyer Unit in 1995, and the fold pattern was the Miura-ori, invented by astrophysicist Koryo Miura in the 1970s as a solution to a very specific engineering problem: how do you package a large flat surface inside a small cylinder and deploy it reliably in zero gravity, with no hinges, no motors, and no chance of a second attempt? Miura's answer — a parallelogram tessellation that collapses along two axes simultaneously — turned out to be one of the most elegant intersections of geometry and engineering in the twentieth century. And it is, at its core, a solution to an optimization problem. ## What this book is about {.unnumbered} The argument of this book is that origami and numerical optimization are the same subject wearing different clothes. A valid flat-foldable crease pattern is a feasible point in a constrained geometric space. Designing an origami base is a circle-packing problem. Finding the equilibrium shape of a partially-folded structure is an energy minimization. And recovering the crease pattern that produces a desired deployed shape — the inverse problem — is a nonlinear program. We develop all of these ideas from scratch, with Python code at every step. ## Who this book is for {.unnumbered} Engineers and scientists who are comfortable with linear algebra and calculus, and who have written some Python but may not have used `scipy.optimize` seriously. No prior origami knowledge is assumed. No prior optimization theory is required beyond the intuition that minimizing a function means finding its lowest point. Each chapter ends with exercises that range from "verify this by hand" to "implement this extension." The exercises are the book. Reading without working them is like reading a recipe without cooking. ## A note on the code {.unnumbered} All code runs inside the `pixi` environment defined in `pixi.toml`. To reproduce any result: ```bash pixi install pixi run kernel # register the Jupyter kernel pixi run render # render HTML pixi run preview # live preview ``` The only dependencies are `numpy`, `scipy`, and `matplotlib` — all available from conda-forge with native ARM64 builds on Apple Silicon.