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February 25[edit]

Help me to understand some mathematical picture?[edit]

Help me to understand something.

Wikipedia has this picture.

toroidal picture

If we imagine a toroidal chess board, A1 would be conected to A8 and A8 conected to A1, B1 would be conected to b8, A1 would be conected to H1,........

Now, how this picture would works (the sperical one)?
I am not fully getting the idea.
http://upload.wikimedia.org/wikipedia/commons/thumb/c/c1/SphereAsSquare.svg/240px-SphereAsSquare.svg.png
PS: The chess thing is just to make easier to me see it and explain the question.201.78.165.137 (talk) 11:24, 25 February 2015 (UTC)[reply]

Along the red lines, a7 is connected to b8, a6 to c8, a5 to d8, all the way to a1 connected to h8. Along the blue lines, a1 is connected to h8, b1 to h7, c1 to h6, etc.

Looked at another way, if you take the left-top half of the depicted square, you can fold it to create a cone, with a seam along the red A line. Likewise you can fold the bootom-right half into a cone. Putting the two cones together gives you a surface homeomorphic to a sphere. -- Meni Rosenfeld (talk) 18:51, 25 February 2015 (UTC)[reply]

Or, more graphically, take a square sheet of thin, stretchy rubber, fold over diagonally (fold runs from top left to bottom right in the diagram) to make a triangle, seal the open edges – then simply inflate like a balloon into a spherical shape. —Quondum 19:36, 25 February 2015 (UTC)[reply]

Right, and for the first picture of a torus, that's like taking your sheet of rubber, rolling it into a tube, then gluing the circular edges together. This leaves us with the (shell of) a donut, the typical picture of a torus. Also it's worth pointing out that the sphere-like thing you construct out of rubber is topologically a sphere, but of course not geometrically a sphere, because your balloon will have pointy bits, and a geometric sphere does not. SemanticMantis (talk) 22:26, 25 February 2015 (UTC)[reply]

One can get rid of the pointy bits through a suitable continuous deformation during inflation. One can say that it is homeomorphically (as noted by Meni, and equivalent to topologically) but not diffeomorphically equivalent to a sphere. (But then, if one chose, one could pre-shape the sheet so that the sealed shape has no pointy bits and is diffeomorphically equivalent; it just wouldn't start as a polygon.) —Quondum 22:50, 27 February 2015 (UTC)[reply]