Lecture 3: Single-Vertex Crease Patterns

TL;DR

The local behavior of crease patterns is analyzed to determine if they can fold flat, with a focus on single vertex patterns.

Transcript

PROFESSOR: Today, we are talking about the local behavior of a crease pattern. So you take some crease pattern for some flat folding-- we're thinking about flat foldability. This is a foldability question. I give you a crease pattern like this. I want to know, does it fold flat, like this one does. And we're studying what happens locally right arou... Read More

Key Insights

• 🍹 The sum of the odd angles in a crease pattern must be equal to the sum of the even angles for it to fold flat.
• 🙏 Folding a circle onto a line requires at least one fold and assumes that the crease pattern came from a flat piece of paper.

Questions & Answers

Q: What is a single vertex crease pattern?

A single vertex crease pattern consists of a disk-shaped or small region around a vertex with n creases emanating from it.

Q: Does a single vertex crease pattern always fold flat?

No, sometimes a single vertex crease pattern does not fold flat. It depends on the sum of the odd angles being equal to the sum of the even angles.

Q: What is the condition for a crease pattern to be flat foldable?

The crease pattern must satisfy the condition that the sum of the odd angles is equal to the sum of the even angles.

Q: What is the significance of Kawasaki's theorem?

Kawasaki's theorem states that a crease pattern is flat foldable if the sum of the odd angles is equal to the sum of the even angles.

Q: What is the difference between convex and flat paper crease patterns?

Convex crease patterns allow for angles less than 360 degrees, while flat paper crease patterns have angles that must sum to 360 degrees.

Summary & Key Takeaways

• Crease patterns consist of vertices (corners) and faces (regions divided by creases).

• Single vertex crease patterns can fold flat if the sum of the odd angles is equal to the sum of the even angles.

• Folding a circle onto a line requires at least one fold and assumes that the crease pattern came from a flat piece of paper.