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 :)
geometry optimization recreational-mathematics volume origami
asked Nov 18 at 18:36
Shaun
8,582113580
|
show 3 more comments
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 :)
geometry optimization recreational-mathematics volume origami
asked Nov 18 at 18:36
Shaun
8,582113580
1
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 at 19:17
1
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 at 19:31
1
yes, it is that book.
– achille hui
Nov 18 at 19:41
1
that result is for a unit square. i.e. $ell = 1$.
– achille hui
Nov 18 at 19:50
1
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 at 5:23
|
show 3 more comments
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 :)
geometry optimization recreational-mathematics volume origami
asked Nov 18 at 18:36
Shaun
8,582113580
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 :)
geometry optimization recreational-mathematics volume origami
geometry optimization recreational-mathematics volume origami
asked Nov 18 at 18:36
Shaun
8,582113580
asked Nov 18 at 18:36
Shaun
8,582113580
asked Nov 18 at 18:36
Shaun
8,582113580
asked Nov 18 at 18:36
Shaun
8,582113580
asked Nov 18 at 18:36
Shaun
8,582113580
8,582113580
1
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 at 19:17
1
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 at 19:31
1
yes, it is that book.
– achille hui
Nov 18 at 19:41
1
that result is for a unit square. i.e. $ell = 1$.
– achille hui
Nov 18 at 19:50
1
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 at 5:23
|
show 3 more comments
1
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 at 19:17
1
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 at 19:31
1
yes, it is that book.
– achille hui
Nov 18 at 19:41
1
that result is for a unit square. i.e. $ell = 1$.
– achille hui
Nov 18 at 19:50
1
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 at 5:23
1
1
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 at 19:17
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 at 19:17
1
1
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 at 19:31
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 at 19:31
1
1
yes, it is that book.
– achille hui
Nov 18 at 19:41
yes, it is that book.
– achille hui
Nov 18 at 19:41
1
1
that result is for a unit square. i.e. $ell = 1$.
– achille hui
Nov 18 at 19:50
that result is for a unit square. i.e. $ell = 1$.
– achille hui
Nov 18 at 19:50
1
1
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 at 5:23
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 at 5:23
|
show 3 more comments
1 Answer
1
active
oldest
votes
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 at 3:38
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 at 3:59
Well, until that day, @achillehui, this answer should suffice.
– Shaun
Nov 24 at 4:01
In fact, here's a tag for @JosephO'Rouke. I hope he doesn't mind this comment $ddotsmile$.
– Shaun
Nov 24 at 4:03
1
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 at 4:08
|
show 1 more comment
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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 at 3:38
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 at 3:59
Well, until that day, @achillehui, this answer should suffice.
– Shaun
Nov 24 at 4:01
In fact, here's a tag for @JosephO'Rouke. I hope he doesn't mind this comment $ddotsmile$.
– Shaun
Nov 24 at 4:03
1
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 at 4:08
|
show 1 more comment
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 at 3:38
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 at 3:59
Well, until that day, @achillehui, this answer should suffice.
– Shaun
Nov 24 at 4:01
In fact, here's a tag for @JosephO'Rouke. I hope he doesn't mind this comment $ddotsmile$.
– Shaun
Nov 24 at 4:03
1
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 at 4:08
|
show 1 more comment
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.
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 at 3:35
community wiki
Shaun
@achillehui, I'll leave this here until you post the answer yourself.
– Shaun
Nov 24 at 3:38
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 at 3:59
Well, until that day, @achillehui, this answer should suffice.
– Shaun
Nov 24 at 4:01
In fact, here's a tag for @JosephO'Rouke. I hope he doesn't mind this comment $ddotsmile$.
– Shaun
Nov 24 at 4:03
1
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 at 4:08
|
show 1 more comment
@achillehui, I'll leave this here until you post the answer yourself.
– Shaun
Nov 24 at 3:38
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 at 3:59
Well, until that day, @achillehui, this answer should suffice.
– Shaun
Nov 24 at 4:01
In fact, here's a tag for @JosephO'Rouke. I hope he doesn't mind this comment $ddotsmile$.
– Shaun
Nov 24 at 4:03
1
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 at 4:08
@achillehui, I'll leave this here until you post the answer yourself.
– Shaun
Nov 24 at 3:38
@achillehui, I'll leave this here until you post the answer yourself.
– Shaun
Nov 24 at 3:38
1
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 at 3:59
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 at 3:59
Well, until that day, @achillehui, this answer should suffice.
– Shaun
Nov 24 at 4:01
Well, until that day, @achillehui, this answer should suffice.
– Shaun
Nov 24 at 4:01
In fact, here's a tag for @JosephO'Rouke. I hope he doesn't mind this comment $ddotsmile$.
– Shaun
Nov 24 at 4:03
In fact, here's a tag for @JosephO'Rouke. I hope he doesn't mind this comment $ddotsmile$.
– Shaun
Nov 24 at 4:03
1
1
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 at 4:08
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 at 4:08
|
show 1 more comment
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1
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 at 19:17
1
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 at 19:31
1
yes, it is that book.
– achille hui
Nov 18 at 19:41
1
that result is for a unit square. i.e. $ell = 1$.
– achille hui
Nov 18 at 19:50
1
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 at 5:23