Lecture 3: When Do Single-Vertex Crease Patterns Fold Flat?

TL;DR

A single-vertex crease pattern folds flat when it can collapse its boundary circle onto a line, which requires at least one fold and holds when the alternating angles balance (Kawasaki's condition: the sum of the odd angles equals the sum of the even angles). Such a pattern is a disk around one vertex with n creases and n angles that sum to 360 degrees for flat paper, or 360 or less for a convex cone. Read on for how the boundary circle maps and when folding fails.

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: Is a single-vertex crease pattern always flat foldable?

No. Sometimes these patterns fold flat and sometimes they do not. Whether a given single-vertex pattern folds flat depends on the sum of the odd angles being equal to the sum of the even angles.

Q: What is a single-vertex crease pattern?

It is a disk-shaped or small region around a single vertex with n creases emanating from it. The pattern is defined by a sequence of angles, theta 1 up to theta n, measured between consecutive creases in clockwise order.

Q: What condition makes a single-vertex crease pattern fold flat?

The pattern must satisfy Kawasaki's condition: the sum of the odd angles equals the sum of the even angles. When this balance holds, the vertex can fold flat.

Q: Why must you make at least one fold to fold a circle onto a line?

A flat folding of a disk folds its boundary circle onto a portion of the circle, which you can then unroll onto a straight line. An unfolded full circle occupies the whole circle and cannot be unrolled, so at least one fold is needed to collapse it into a portion that unrolls onto a line.

Q: What does it mean that the crease pattern came from flat paper?

It means the sum of the angles around the vertex is 360 degrees, exactly as in a flat sheet of paper. This assumption is used when reasoning about a flat crease pattern one vertex at a time.

Q: What is a convex cone in single-vertex crease patterns?

A convex cone relaxes the flat-paper constraint so the angles sum to 360 degrees or less. For example, folding a crimp on two bottom creases can leave two angles of 135 degrees each, for a total of 270 degrees, which is less than 360.

Q: What are vertices and faces in a crease pattern?

Vertices are the corners where all the edges (creases) come together. Faces are the regions, such as triangles, that the creases divide the paper into.

Q: How does single-vertex flat folding relate to folding a circle onto a line?

A flat folding of the disk maps its boundary circle onto a circle, and unrolling that gives a folding of the circle onto a line. This is a step up from folding one-dimensional line segments onto a line, using the same mindset but with different topology, so some earlier one-dimensional results do not carry over.

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.