Non-Euclidean Geometry

Drop the parallel postulate and the angle sum changes.

The idea

Non-Euclidean geometry is the geometry obtained by replacing Euclid's parallel postulate while keeping his other postulates. Euclid derived plane geometry from five postulates, and the fifth, the parallel postulate, states that through a point not on a given line there passes exactly one line parallel to it. Geometers spent two thousand years trying to derive it from the other four, and failed because it is independent of them: replace it with something else and the remaining postulates still describe a consistent geometry that is no longer flat.

There are two replacements. In spherical geometry the lines are the great circles of a sphere; any two great circles meet, so no line has a parallel. In hyperbolic geometry, infinitely many lines through the point never meet the given line.

The angle sum of a triangle separates the three geometries. That a Euclidean triangle's angles total $180^{\circ}$ is proved from the parallel postulate, so it changes with the postulate: the angles $\alpha$, $\beta$, $\gamma$ of a spherical triangle total more than $180^{\circ}$, and the surplus is its excess; a hyperbolic triangle's total falls short of $180^{\circ}$, and the shortfall is its defect.

Ways to work on it

• Walkthrough. The parallel postulate and spherical excess.
• Practice. Match a property to its geometry.
• Hardest. Hyperbolic parallels and the angle defect.

Not sure where to start? Take the ten-question placement test.