Why can't you flatten an orange peel without tearing it?
Gauss proved that a surface's curvature survives any bending without stretching. That is why every flat map of the Earth is wrong.
▶ Start the storyYou cannot flatten an orange peel without tearing it because of a remarkable result of Carl Friedrich Gauss, proved in 1827. The Theorema Egregium, "remarkable theorem", says that a surface's curvature can be determined entirely by measuring angles and distances on the surface itself. So a surface that is curved in this sense stays curved however you bend it, as long as you bend without stretching. A sphere, like an orange or the Earth, has positive curvature, a flat plane has zero, and so no amount of bending turns one into the other.
Concretely: a piece of paper cannot be bent onto a sphere without crumpling, and the surface of a sphere cannot be unfolded onto a flat plane without distorting distances. Step on an empty egg shell and its edges have to split before it will lie flat. This is also why no flat map of the Earth can be perfect, even for a small portion of the globe: every map projection distorts at least some distances.
Curvature zero
- Plane and cylinder
- A flat sheet rolls into a tube
- Pizza slice: bend one way, stays rigid the other
Curvature not zero
- Sphere: positive, like an orange or the Earth
- One-sheeted hyperboloid: negative, like a saddle
- Flattening means tearing or stretching
The counterpart is the cylinder. A cylindrical tube has Gaussian curvature zero, the same as the flat sheet it unrolls into, so a flat sheet can be rolled into a tube with no stretching or tearing. It also explains your pizza: a flat slice has curvature zero, and bending it must roughly keep it so. Curve it across, and the other direction has to stay straight, which stiffens the slice along the fold so it won't flop over.
Gauss's geodetic survey of Hanover fuelled his interest in the mathematics of curves and surfaces.
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Recap
Bending never changes a surface's Gaussian curvature, so a sphere cannot be flattened without stretching or tearing.
💡 A trick to remember it · Bend it, roll it, fold it: the curve that counts is the one you can't iron out.
Surprising fact · A pizza slice stays stiff when you curve it across because of Gauss's theorem.
Sources (3)
No source, no claim. Every fact in this lesson (16 claims) cites at least one of these.