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Possible Unfolding of the Sphere
There is no point, no straight line, no circle, no sphere and no space in a manner how Euclide described them. There must be tiny elements, of which the plane is composed. These elements must have axes and radius to be able to slide and rotate around each other. If the particles -
i.e. atoms - of the material (what have been called "ideal plane" by Euclid) can rotate like a spiral string, the sphere can be unfold to the plane without any problem preserving its metrical measures along the string and the other distances and angles also can be calculated. The inner arms are the rotating string arms, it will be a half of the final sphere. They are becoming straight lines on the surface of the sphere. The outer parts of these "S"-like figures can be bent and twisted and creating another halfsphere on the top of the other one.
Enjoy this music:
http://www.youtube.com/watch?v=5T3FXFnoTzE&feature=quicklist&playnext=8&playnext_from=QL
The plane what bisects the sphere can be deformed similar way as it is shown on (partly) the figure above. The orange ribbons are showing the tangent planes with simultaneously equivalent Gaussian curvature.
i.e. atoms - of the material (what have been called "ideal plane" by Euclid) can rotate like a spiral string, the sphere can be unfold to the plane without any problem preserving its metrical measures along the string and the other distances and angles also can be calculated. The inner arms are the rotating string arms, it will be a half of the final sphere. They are becoming straight lines on the surface of the sphere. The outer parts of these "S"-like figures can be bent and twisted and creating another halfsphere on the top of the other one.
Enjoy this music:
http://www.youtube.com/watch?v=5T3FXFnoTzE&feature=quicklist&playnext=8&playnext_from=QL
The plane what bisects the sphere can be deformed similar way as it is shown on (partly) the figure above. The orange ribbons are showing the tangent planes with simultaneously equivalent Gaussian curvature.
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