Non-Euclidean geometry

What just happened?

From school you know one “law”: the angle sum of a triangle is always $180^\circ$. That is true — but only on the plane. When you draw a triangle on a curved surface, its angle sum stops being constant: it depends on the curvature of that surface.

Three geometries — three angle sums:

On a sphere (positive curvature) a triangle with one vertex at the pole and two on the equator has three right angles: $90^\circ+90^\circ+90^\circ=270^\circ$. On a saddle surface (negative curvature, hyperbolic geometry) the sum is less than $180^\circ$.

The invariant — angular excess and deficit:

The deviation of the angle sum from $180^\circ$ — the excess on a sphere or the deficit on a saddle — is directly proportional to the curvature $K$ of the surface and the area $P$ of the triangle. The larger the triangle on the sphere, the more its angle sum exceeds $180^\circ$.

The 19th-century revolution:

For over two thousand years mathematicians believed that Euclid’s geometry (with an angle sum $=180^\circ$) was the only possible, “true” geometry of space. They tried to derive his fifth postulate (about parallel lines) from the others — in vain. The breakthrough came when Nikolai Lobachevsky (1829), János Bolyai (1832) and Bernhard Riemann (1854) independently discovered consistent non-Euclidean geometries. These were not “wrong” geometries, but just as logically consistent as the Euclidean one — simply dependent on the curvature of space.

Why it matters:

The discovery revolutionised mathematics and philosophy: space need not be “flat”. Riemann’s differential geometry became the mathematical language of Einstein’s general theory of relativity, in which gravity is the curvature of spacetime — a triangle with angles other than $180^\circ$ is physics today, not just abstraction.

Why this matters for the exam:

The rule “the angle sum of a triangle = $180^\circ$” holds in Euclidean geometry — and that is what the exam uses. But it is worth knowing that this is an axiom of the plane, not an absolute truth: changing the fifth postulate gives other, equally correct geometries.

Remember: the angle sum of a triangle depends on the curvature of the surface — $=180^\circ$ on a flat one, $>180^\circ$ on a sphere (positive curvature), $<180^\circ$ on a saddle (negative, hyperbolic geometry).