What is the minimum square that circumscribes Jordan curve?

Algebra

What is the minimum square that circumscribes Jordan curve?

Postby Guest » Tue Apr 14, 2020 1:34 pm

What is the minimum square that circumscribes a Jordan curve?

Moreover, how does one quantify (via parameters) such a square when given any Jordan curve?

Hmm. What circumscribes can be inscribed too... Think of the inscribed square problem.
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A miminum square that circumscribes a Jordan Curve..jpg
What is the minimum square that circumscribes a Jordan curve?
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Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 1:44 pm

A Related Link:

'Need Help on a program to draw Jordan Curves',

https://www.math10.com/forum/viewtopic.php?f=33&t=8696.
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 2:07 pm

Guest wrote:What is the minimum square that circumscribes a Jordan curve?

Moreover, how does one quantify (via parameters) such a square when given any Jordan curve?

Hmm. What circumscribes can be inscribed too... Think of the inscribed square problem.


A Simple Answer: Start with any square and draw any Jordan curve within the square that touches the boundary of the square at least two points.

Solving the inscribed square problem is much more difficult.
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 2:49 pm

Guest wrote:
Guest wrote:What is the minimum square that circumscribes a Jordan curve?

Moreover, how does one quantify (via parameters) such a square when given any Jordan curve?

Hmm. What circumscribes can be inscribed too... Think of the inscribed square problem.


A Simple Answer: Start with any square and draw any Jordan curve within the square that touches the boundary of the square at least two points.

Solving the inscribed square problem is much more difficult.


Important hint for solving the inscribed square problem:

The circumscribed square by design (the shape of the square) generates/induces inscribed squares within the inscribed Jordan curve. Go figure! :)
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 3:10 pm

Another hint: The circumscribed square is free to rotate/orientate about its center axis.
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 3:16 pm

Final hint: The shape of the circumscribed square is scaled-invariant.
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 3:57 pm

Our three hints for solving the inscribed square problem along with some basic calculus (first derivatives/tangents of piecewise smooth curves of the Jordan curve) and some basic analytic geometry are almost enough to solve the inscribed square problem.

And the ideas of contours, open cover (basic topology), and neighborhood (basic topology) are useful too.

To clinch the solution to the inscribed square problem, we need to introduce the idea of convergence/limit (basic calculus) with simple probability theory. :D
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 4:12 pm

Guest wrote:Our three hints for solving the inscribed square problem along with some basic calculus (first derivatives/tangents of piecewise smooth curves of the Jordan curve) and some basic analytic geometry are almost enough to solve the inscribed square problem.

And the ideas of contours, open cover (basic topology), and neighborhood (basic topology) are useful too.

To clinch the solution to the inscribed square problem, we need to introduce the idea of convergence/limit (basic calculus) with simple probability theory. :D


Relevant Reference Link:

'What is the solution to the inscribed square problem (Toeplitz square peg problem)?'

https://www.researchgate.net/post/What_is_the_solution_to_the_inscribed_square_problem_Toeplitz_square_peg_problem
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 4:32 pm

Guest wrote:
Guest wrote:
Guest wrote:What is the minimum square that circumscribes a Jordan curve?

Moreover, how does one quantify (via parameters) such a square when given any Jordan curve?

Hmm. What circumscribes can be inscribed too... Think of the inscribed square problem.


A Simple Answer: Start with any square and draw any Jordan curve within the square that touches the boundary of the square at least two points.

Solving the inscribed square problem is much more difficult.


Important hint for solving the inscribed square problem:

The circumscribed square by design (the shape of the square) generates/induces inscribed squares within the inscribed Jordan curve. Go figure! :)


Hmm. Our simple answer is wrong!

A Revised Simple Answer: Start with any square and draw any Jordan curve within the square that touches the boundary of the square at least one point...

The details (...) indicate complications.
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 4:48 pm

Remark: Our simple answer is not so simple! :o
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 5:06 pm

Another Remark: "It takes three to tangle here..."

The shape of the square and the shapes of various Jordan curves along with the definition of the inscribed square of a Jordan curve allow us to solve the inscribed square problem. :D
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 5:43 pm

According to the contents of the relevant reference link ( https://www.researchgate.net/post/What_is_the_solution_to_the_inscribed_square_problem_Toeplitz_square_peg_problem ), the following system of differential equations is not generally true!

"In addition, we have the following system of differential equations:
dy0(x0)/dx = dyn(x0)/dx;
dy0(x1)/dx = dy1(x1)/dx;
dy1(x2)/dx = dy2(x2)/dx;
dy2(x3)/dx = dy3(x3)/dx;
...
dyn-1(xn)/dx = dyn(xn)/dx. "

Remark: The left side of each of the above equations may equal to the negative of the corresponding right side of those equations.
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 14, 2020 5:49 pm

Guest wrote:According to the contents of the relevant reference link ( https://www.researchgate.net/post/What_is_the_solution_to_the_inscribed_square_problem_Toeplitz_square_peg_problem ), the following system of differential equations is not generally true!

"In addition, we have the following system of differential equations:
dy0(x0)/dx = dyn(x0)/dx;
dy0(x1)/dx = dy1(x1)/dx;
dy1(x2)/dx = dy2(x2)/dx;
dy2(x3)/dx = dy3(x3)/dx;
...
dyn-1(xn)/dx = dyn(xn)/dx. "

Remark: The left side of each of the above equations may not equal to the corresponding right side of those equations.
Guest
 

What is the minimum square that circumscribes Jordan curve?

Postby Guest » Fri Apr 17, 2020 5:45 pm

Guest wrote:Another hint: The circumscribed square is free to rotate/orientate about its center axis.


We also include the appropriate (relative to our inscribed Jordan curve) horizon and vertical translations of our circumscribed square away from its center axis.
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Fri Apr 17, 2020 6:22 pm

Guest wrote:
Guest wrote:Another hint: The circumscribed square is free to rotate/orientate about its center axis.


An Update:

We also include the appropriate (relative to our inscribed Jordan curve) horizon and vertical translations of our circumscribed square.
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Fri Apr 17, 2020 6:24 pm

Guest wrote:
Guest wrote:Another hint: The circumscribed square is free to rotate/orientate about its center axis.


An Update:

We also include the appropriate (relative to our inscribed Jordan curve) horizontal and vertical translations of our circumscribed square.
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Tue Apr 21, 2020 6:31 pm

FYI: A Jordan curve inscribes at least one cell of a simple or centered square lattice.

Relevant Reference Links:

'Square lattice',

https://en.wikipedia.org/wiki/Square_lattice#cite_note-1;

'Inscribed Square Problem',

https://en.wikipedia.org/wiki/Inscribed_square_problem.
Attachments
Cells of a Simple or  Centered Square Lattice.png
The Inscribed Square Problem has an affirmative solution!
Cells of a Simple or Centered Square Lattice.png (8.83 KiB) Viewed 544 times
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Sat May 02, 2020 8:35 pm

Guest wrote:FYI: A Jordan curve inscribes at least one cell of a simple or centered square lattice.

Relevant Reference Links:

'Square lattice',

https://en.wikipedia.org/wiki/Square_lattice#cite_note-1;

'Inscribed Square Problem',

...


Our square lattice acts as a dynamic fractal that adjusts its scale to inscribe cells in any Jordan curve. Go figure! 8)

Relevant Reference Link:

'Fractal',

https://www.britannica.com/science/fractal.
Attachments
Examples of Fractals.png
The Inscribed Square Problem has an affirmative solution!
Examples of Fractals.png (4.1 KiB) Viewed 438 times
Guest
 

Re: What is the minimum square that circumscribes Jordan cur

Postby Guest » Sat May 02, 2020 10:33 pm

Guest wrote:
Guest wrote:FYI: A Jordan curve inscribes at least one cell of a simple or centered square lattice.

Relevant Reference Links:

'Square lattice',

https://en.wikipedia.org/wiki/Square_lattice#cite_note-1;

'Inscribed Square Problem',

...


[b]An Update[/b]:

Our square lattice acts as a dynamic fractal that adjusts its scale and orientation so that one or more of its cells are inscribed by any Jordan curve. Go figure! 8)

Relevant Reference Link:

'Fractal',

https://www.britannica.com/science/fractal.
Guest
 


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