Theorema Egregium: Why Every Flat Map Is Wrong
Gauss’s Theorema Egregium (“remarkable theorem”) states that the Gaussian curvature of a surface is intrinsic: it depends only on distances measured within the surface, so it is preserved by isometries. Equivalently, two surfaces with different Gaussian curvature cannot be isometric, even locally.
The Gaussian curvature of a sphere
By symmetry, a sphere of radius \(R\) has the same Gaussian curvature \(K\) at every point. Its surface area is \(4 \pi R^2\), so by the Gauss–Bonnet theorem,
\[\chi(\Sigma) = \frac{1}{2\pi}\int_{\Sigma} K \, dA =\frac{1}{2 \pi } \cdot 4 \pi R^2 \cdot K = 2R^2 K,\]where \(\chi(\Sigma)\) is the Euler characteristic. A sphere has genus zero (no handles), so \(\chi = 2\); this is the same \(2\) as in Euler’s formula \(V - E + F = 2\) for polyhedra. Hence
\[K=\frac{1}{R^2}.\]The Gaussian curvature of a plane
A plane is flat: both principal curvatures vanish, so \(K = 0\). A sphere and a plane therefore have different Gaussian curvatures.
Isometric surfaces
Two surfaces are isometric if one can be bent into the other without stretching or tearing, so that all lengths and angles measured within the surface are preserved:
Conclusion
This is why every flat world map is wrong. A plane and a sphere have different Gaussian curvatures, so by the Theorema Egregium no map from (any part of) the Earth to a flat sheet can preserve distances. Every projection must compromise: it can preserve angles, or areas, but not both, and never distances.
The familiar Mercator projection preserves angles (it is conformal) but badly distorts areas: regions far from the equator, such as Greenland and Europe, look far too big compared with equatorial regions. This is one reason the size of India and of Africa is often underestimated.
Other projections make different compromises. The gnomonic projection maps great circles (the shortest paths on the sphere) to straight lines, which is useful for navigation, but it distorts shapes and areas heavily and can only show less than a hemisphere, so it is rarely used for world maps.
Exercises
- The first sentence says that isometric surfaces have the same Gaussian curvature. Why is this equivalent to saying that surfaces with different Gaussian curvatures are not isometric? (Undergraduate, year 1)
- Show that the helicoid and the catenoid are locally isometric. (Undergraduate, years 2–3)
- If you deform a sphere, how does its Gaussian curvature change? What stays the same, and why? (Undergraduate, years 2–3)
- (***) Interpret the first and second fundamental forms geometrically. Show that the third fundamental form, defined by \(d\mathbf{n} \cdot d\mathbf{n}\) for the unit normal \(\mathbf{n}\), carries no new information. (Solution in my other post.)
Changes on 25 October 2026
What was wrong. Nothing in the mathematics. The Gauss–Bonnet calculation for the sphere stopped at \(\chi = 2R^2K\) without finishing.
What changed. It now concludes \(\chi = 2\) and connects this to \(V - E + F = 2\), and the statement of the theorem is clearer. Previously titled “Theorema Egregium — The reason why all maps are wrong”.