Turning the square into a circle

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Discussion Overview

The discussion revolves around the concept of transforming a square into a circle within the framework of topology. Participants explore the implications of this transformation, questioning whether it can be achieved without cutting and discussing the nature of homeomorphism in relation to these shapes.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant suggests that folding a square into a circle is a valid method, questioning if this constitutes "cutting" or if it can be considered a form of "penetrating."
  • Another participant argues that folding does not align with topological principles, emphasizing that a homeomorphism requires a function that is injective and continuous, and that one should focus on mathematical properties rather than informal descriptions.
  • There is a mention of "squaring the circle" as a concept traditionally associated with ruler-and-compass geometry, which some participants clarify is distinct from the topological discussion at hand.
  • Several participants agree that while it may seem obvious that a circle and a square are homeomorphic, constructing the homeomorphism is a different matter that serves as a useful example in topology.
  • One participant expresses uncertainty about the appropriate forum section for their original question, indicating a potential misunderstanding of the topic's classification.

Areas of Agreement / Disagreement

Participants express differing views on the validity of folding as a method in topology, with some asserting that it does not meet the criteria for homeomorphism. There is no consensus on the best approach to the problem, and the discussion remains unresolved regarding the implications of folding versus other methods.

Contextual Notes

Participants highlight the need for clarity in definitions and the mathematical properties involved in homeomorphism, indicating that assumptions about the nature of transformations may vary. The distinction between topological and geometric interpretations of "squaring the circle" is also noted.

Observeraren
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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
 
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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".
 
Orodruin said:
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.
 
No.
 
Observeraren said:
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
 
PeroK said:
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).
 
Orodruin said:
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.
 
PeroK said:
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.
 
PeroK said:
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?

Orodruin said:
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.
 
  • #10
Observeraren said:
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.
 
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