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Why can a sphere not be flattened onto a plane without distortion?

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differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

In differential geometry, a sphere cannot be flattened onto a plane without distortion because the two surfaces possess different intrinsic geometries, specifically regarding their Gaussian curvature.

According to Gauss’s "Remarkable Theorem," if two surfaces are isometric—meaning one can be transformed into the other without stretching or distorting intrinsic distances—they must have the same Gaussian curvature at corresponding points. A plane has a constant Gaussian curvature of zero, making it a developable surface that can be flattened without distortion.12

In contrast, a sphere has a constant positive Gaussian curvature. Because the Gaussian curvature of a sphere is not zero, it is mathematically impossible to map it onto a flat plane while preserving the intrinsic distances between points. Consequently, any attempt to flatten a sphere necessarily introduces distortion, as the surface is fundamentally undevelopable.134