What is the minimum square that circumscribes Jordan curve?

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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 Related Link:

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

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

[Guest]

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]

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!

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

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

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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.

[Guest]

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.

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]

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.

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Remark: Our simple answer is not so simple!

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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.

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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]

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]

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]

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]

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]

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.

[Guest]

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!

Relevant Reference Link:

'Fractal',

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

[Guest]

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!

Relevant Reference Link:

'Fractal',

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