For example, suppose you are a disc of radius 1 living on the flat part of the surface, so your circumference is 2 = 6.28.. and your area is = 3.14.

Suppose part of the surface is curved like the graph of z= (x^{2}+y^{2}) (this graph has positive curvature). If you slide over there so that the middle of your body is at (0,0,0), your perimeter will fit exactly on the circle at height 1, but your area will have been stretched to 5.33.. units. 

On the other hand suppose part of your surface is curved like the graph of z=x^{2}y^{2} (this graph has negative curvature). If the middle of your body is at the point (0,0,0), your perimeter will fit the circle in the graph that lies over the circle r=.715.. in the plane; your insides will have to fit in the enclosed area which is only 2.26.. units. 

Geodesics: The geographers in your twodimensional universe need to be able to locate and measure areas of nonzero curvature without risking their insides. They may have discovered a theorem due to Gauss which permits these measurements. This theorem is stated in terms of geodesics. These are the paths on the surface which are as straight as possible: they turn neither to the left nor to the right, and their only bending is that which is forced on them by following the surface.
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