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Sorry in advance for lacking the appropriate terminology, please help me edit it in below.

Take thease basic shapes:

triangle, square, ..., octigon
pyramid, cube
simplex, hypercube

Each flat surface of a cube is a square. The same applies to others in the rest of the table above.

{circle, sphere, hypersphere} also have a great deal in common with thease series.

Therefore, is it true to say that a surface of a sphere either is, or is in some respects, or can be thought of as being a circle, or more accurately a disc?

My thoughts as to how this could be:

A segment or slice isn't a surface.

Thanks Qiaochu Yuan, in topologically two discs can be used to form a sphere. Although they are not flat discs.

Perhaps as the number of sides approaches infinity, the shape of each side aproaches a circle, although it's apparent that circular surfaces do not fit together to form a sphere.

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So the question is?.. –  Ilya Feb 7 '12 at 16:46
    
I've marked the question. –  alan2here Feb 7 '12 at 17:03
    
Everything below the bolded question is my thoughts of how this could be. –  alan2here Feb 7 '12 at 17:10
    
Well, you would be certainly interested in Algebraic topology which in part deals with such 'common things'. I don't know if there is a good introductory book on this topic. –  Ilya Feb 7 '12 at 17:22
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1 Answer

Certainly it is possible to think of a sphere as being glued together from several disks (in mathematics "circle" refers to the object $x^2 + y^2 = 1$ rather than the object $x^2 + y^2 \le 1$, which is a disk). In fact it suffices to use two: the upper hemisphere and the lower hemisphere of a sphere are topologically disks, and gluing them together at their boundaries gives a sphere.

The study of topological spaces from this point of view used to be known as combinatorial topology, but nowadays it is subsumed under algebraic topology.

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This is probably as good an answer as there is, but I suspect OP is thinking of geometry, not topology. In geometry, things go wrong way before you get to spheres; there's already nothing in the second row of the table to correspond to hexagons and heptagons in the first row. –  Gerry Myerson Feb 7 '12 at 23:32
    
The disks would have to be distorted or stretched for that to work, try doing this with rubber without streching the rubber or creating a cut to reduce the length of the perimeter. Perhaps this stretching is expected in topologically. I'm more interested in the way it can seemingly be proven that a sphere can be constructed of flat disks when it's also apparently not true. –  alan2here Feb 8 '12 at 14:09
    
@alan2here: yes, some distortion is inevitable. As a geometric object, the sphere (with the usual metric) has nonzero Gaussian curvature (en.wikipedia.org/wiki/Gaussian_curvature) everywhere, while disks (with the usual metric) have zero Gaussian curvature everywhere. –  Qiaochu Yuan Feb 8 '12 at 17:41
    
So as I suspected, appart from my proof above proving it's possible it's definitely impossible :¬P –  alan2here Feb 8 '12 at 17:56
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