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

Question

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. — , Nov 18 '18 at 19:17
Well, a priori, at least, there's nothing to say that the shape is convex. But thank you :) — Shaun, Nov 18 '18 at 19:21
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
Okay. Thank you. It's assuming $\ell=1$, right, @achillehui ? — Shaun, Nov 18 '18 at 19:48
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. — , Nov 19 '18 at 05:23

score 0 · Accepted Answer · answered Nov 24 '18 at 03:35

0

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.

answered Nov 24 '18 at 03:35

Shaun

47,747

@achillehui, I'll leave this here until you post the answer yourself. – Shaun Nov 24 '18 at 03:38
1

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 03:59
Well, until that day, @achillehui, this answer should suffice. – Shaun Nov 24 '18 at 04:01
In fact, here's a tag for @JosephO'Rouke. I hope he doesn't mind this comment $\ddot\smile$. – Shaun Nov 24 '18 at 04:03
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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 04:08
Oh, okay. Thank you, @achillehui :) – Shaun Nov 24 '18 at 04:09

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

Motivation:

The Question:

Thoughts:

1 Answers1