Turning the square into a circle

[Observeraren]

Hello Forum,

Does topology reckon the art of turning a square into a circle? I am quite new to topology and maths in general, I have only dabbled and eyed on my collection of mathbooks. I have come to a conclusion of how to turn the Square into A Circle without cutting.
I wonder if I am cheating and not following the rules of the problem of turning the square into a circle? Am I trying to make a fool out of math?
Think of a paper, and when you fold it, it can fold into itself. Is this cutting? I call it penetrating, but as for me, making a circle from the square is easy business in the aforementioned "penetrating" manner and I thought it was a question of the century, but the answer is so easy.

If I am correct, and you Topology Wizards of the forum do not know how to make the square into a circle in the aforementioned manner, I can show you.
I bet I have solved nothing.
Best wishes,
Observeraren

 

[Orodruin]

Folding is not how topology works. Rather, you should imagine that you have a perfectly malleable sheet in the shape of a circle. A different shape is homeomorphic to the circle if you can drag the sheet out into that shape. By "folding" your function between the circle and the square would not be injective and therefore not a homeomorphism. Also, you would be better off by looking at the actual mathematical properties that are required than "thinking in words" and flimsy descriptions such as "deforming without cutting".

 

[Observeraren]

Folding is not how topology works. Rather, you should imagine that you have a perfectly malleable sheet in the shape of a circle. A different shape is homeomorphic to the circle if you can drag the sheet out into that shape. By "folding" your function between the circle and the square would not be injective and therefore not a homeomorphism. Also, you would be better off by looking at the actual mathematical properties that are required than "thinking in words" and flimsy descriptions such as "deforming without cutting".

Is there a problem if I "squish" the square into a circle? Try to stand out with my laymans terms, thank you.

 

[Orodruin]

No.

 

[PeroK]

Hello Forum,

Does topology reckon the art of turning a square into a circle? I am quite new to topology and maths in general, I have only dabbled and eyed on my collection of mathbooks. I have come to a conclusion of how to turn the Square into A Circle without cutting.
I wonder if I am cheating and not following the rules of the problem of turning the square into a circle? Am I trying to make a fool out of math?
Think of a paper, and when you fold it, it can fold into itself. Is this cutting? I call it penetrating, but as for me, making a circle from the square is easy business in the aforementioned "penetrating" manner and I thought it was a question of the century, but the answer is so easy.

If I am correct, and you Topology Wizards of the forum do not know how to make the square into a circle in the aforementioned manner, I can show you.
I bet I have solved nothing.
Best wishes,
Observeraren

I thought "squaring the circle" was about ruler-and-compass geometry; not topology:

https://en.wikipedia.org/wiki/Squaring_the_circle

 

[Orodruin]

I thought "squaring the circle" was about ruler-and-compass geometry; not topology:

https://en.wikipedia.org/wiki/Squaring_the_circle

Well, that is a different problem than the problem of showing that the circle and the square are homeomorphic (which they are).

 

[PeroK]

Well, that is a different problem than the problem of showing that the circle and the square are homeomorphic (which they are).

Yes, but I've never heard of a problem to show that a circle and a square are homeomorphic, since it's fairly obvious that they are.

 

[Orodruin]

Yes, but I've never heard of a problem to show that a circle and a square are homeomorphic, since it's fairly obvious that they are.

True. But it is one thing to think it is "obvious" and another to actually construct the homeomorphism. It also works as a good basic example when learning topology.

 

[Observeraren]

I thought "squaring the circle" was about ruler-and-compass geometry; not topology:

https://en.wikipedia.org/wiki/Squaring_the_circle

Whoopsie. I should take my solution to the appropiate part of the forum. What part of the forum would you suggest?

True. But it is one thing to think it is "obvious" and another to actually construct the homeomorphism. It also works as a good basic example when learning topology.

True.

Thanks guys for your time.

 

[Mark44]

Whoopsie. I should take my solution to the appropiate part of the forum. What part of the forum would you suggest?

Your thread is fine, here. The thing that PeroK was talking about was a very old problem of how to construct a square with the same area as a given circle, using only a compass (the dividers kind, not the device that shows directions) and a straightedge. This has been proven mathematically to be impossible. See https://en.wikipedia.org/wiki/Squaring_the_circle.