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











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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







share|cite|improve this question


















  • 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

















up vote
0
down vote

favorite












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







share|cite|improve this question


















  • 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















up vote
0
down vote

favorite









up vote
0
down vote

favorite











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







share|cite|improve this 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 :)






geometry optimization recreational-mathematics volume origami




geometry optimization recreational-mathematics volume origami






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 18 at 18:36









Shaun

8,145113577




8,145113577








  • 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




    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












1 Answer
1






active

oldest

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up vote
0
down vote



accepted










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.






share|cite|improve this answer























  • @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











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up vote
0
down vote



accepted










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.






share|cite|improve this answer























  • @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















up vote
0
down vote



accepted










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.






share|cite|improve this answer























  • @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













up vote
0
down vote



accepted







up vote
0
down vote



accepted






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.






share|cite|improve this 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.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








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


















  • @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


















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