# Using (rigid) Origami moves only, what is the maximum volume that can be enclosed by a square piece of paper?

## Motivation:

This is inspired by this question.

## The Question:

What is the maximum volume that can be enclosed by folding a square piece of paper (with side length $$\ell$$) using only (rigid) Origami moves?

## Thoughts:

I found this, but it's not very helpful because it doesn't give a specific volume and I can't find the paper it references.

It's not a question I think I can answer myself. I have no formal training in Origami and know very little about it.

I'm guessing the shape is just a cube but I'm not sure how to prove that.

Please help :)

• It seems you want a closed (convex?) polyhedron whereas the linked question concerns an open "dish". If so, see Alexander, Dyson, and O'Rourke, "The Foldings of a Square to Convex Polyhedra, 2002. – Rahul Nov 18 '18 at 19:17
• The ref it pointed to is a book (I've a hard copy of that). In the book, it say the volume is about $0.056$ which is about $60$% of the volume of a unit-area sphere. – achille hui Nov 18 '18 at 19:31
• yes, it is that book. – achille hui Nov 18 '18 at 19:41
• that result is for a unit square. i.e. $\ell = 1$. – achille hui Nov 18 '18 at 19:50
• The same result is included in the paper I linked to (Joseph O'Rourke is a coauthor of both), and the paper is freely available via O'Rourke's webpage. – Rahul Nov 19 '18 at 5:23

## 1 Answer

In order to close the question, here is a community wiki answer from the comments.

The ref it pointed to is a book (I've a hard copy of that). In the book, it say the volume is about $$0.056$$ which is about $$60\%$$ of the volume of a unit-area sphere.

This is by achille hui, Nov 18 at 19:31.

• @achillehui, I'll leave this here until you post the answer yourself. – Shaun Nov 24 '18 at 3:38
• I won't post any answer. Joseph O'Rourke is a user on math.SE. If he ever see this question, he can provide an answer with much more details... – achille hui Nov 24 '18 at 3:59
• Well, until that day, @achillehui, this answer should suffice. – Shaun Nov 24 '18 at 4:01
• In fact, here's a tag for @JosephO'Rouke. I hope he doesn't mind this comment $\ddot\smile$. – Shaun Nov 24 '18 at 4:03
• Tagging a general user won't work. The system will only notify the user if he/she has interacted with a page before. – achille hui Nov 24 '18 at 4:08