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Is there a world where circle is square? (like when triangle can have sum of degrees more than 180 on sphere) What is the mathematical or at least common-sense proof?

roslav
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    What do you mean by "Is there a world"? Are you asking about physical reality, or just "another world" in a metaphorical sense? – Zev Chonoles Apr 16 '13 at 13:38
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    user54358's answer below is a really good answer for "Is there a metric space whose circles look like Euclidean squares?" Is that what you meant to ask? Or do you really mean to find a space whose circles are also squares? You would have to explain what a square is in that space. – rschwieb Apr 16 '13 at 13:59
  • "Is there a metric space whose circles look like Euclidean squares?" I have meant this. – roslav Apr 16 '13 at 16:24

3 Answers3

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If you define the norm to be the maximum norm, circles in that normed space would look like a square.

user54358
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Yes. If you paint a circle on a rubber sheet, and deform the sheet with a different deformation rate at each point until the circle becomes a square you got one. Now, you could define a bijection between both spaces.

All this assuming that what you meant is an isomorphism in between a topological space (euclidean 'rubber sheet') and another space of the same dimension.

On the other side, you can't flatten a rubber sphere into a plane.

Ambesh
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As Arya mentioned, in topological sense this is possible.

You can also define the distance (or "norm") in other forms, say 1-norm $|x|+|y|$ or $\infty$-norm $\max \{x,y\}$, then the circle will "look like" a square.

Jun Zhang
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