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The course Differential Geometry told me that developable surfaces, of which the Gaussian curvature is $0$, can be flattened onto a plane without distortion.

Some says this is because a developable surface can have the same metric as a plane. But I still think metric here is a little abstract, since I can't figure out the connection between metric and the physical nature or the structure of a paper.

I cannot connect well between the physical phenomenon of paper folding and its mathematical description of Gaussian curvature or metric.

Is there a good physical explanation of this? Thanks.

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migrated from Oct 2 '12 at 14:17

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To learn something about developable surfaces in an historical context you can look at:… – Joseph Malkevitch Jan 12 '12 at 17:53
@JosephMalkevitch: Uh.. I'm sorry that I cannot get access to that article. – Roun Jan 13 '12 at 3:43

Too long for a comment, but definitely not an answer

I am not sure if that statement is 100% correct. Isn't a cylinder developable? Maybe you want to make the statement local?

Anyway, try the following at home. Take a piece of paper, and glue two sides together to get a cylinder. Try to glue a piece of paper into a sphere. You will notice that the latter will not work. you always have cringes if you try to do that. The problem is curvature. Draw any small triangle on a sphere, and you will notice that the angles do not add up to 180 degrees. In the plane this is always so. You cannot make the paper into a sphere, because however you do this, if you do not stretch, distort the paper you cannot have violate this property. If you do not like triangles, you can do this with circles. Then study the relation of the area and the circumference.

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