Concept

Spherical geometry

The geometry of the surface of a sphere, where the straight lines are great circles and every two of them meet. A triangle's angles add to more than two right angles by an amount equal to its area, so triangles of different sizes are never similar.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A right triangle on a sphere. A spherical triangle with a right angle where the equator meets a meridian and legs of 50 and 60 degrees; its hypotenuse is shorter than the flat theorem predicts.

The triangle that a globe gets wrong

On a sphere, a right triangle with legs of fifty and sixty degrees has a hypotenuse of seventy-two, not seventy-eight. The theorem is not approximately true there — it is false, and what replaces it says exactly how much room the surface has.

geometry · Pythagoras

Morley's construction on a sphere. A spherical triangle with angles 76.1, 97.3, 48.8 degrees, its angle trisectors as great-circle arcs, and the inner triangle they form, with sides 0.23902, 0.23881, 0.23902.

The trisectors on a sphere almost agree

Draw Morley's construction on a sphere and the triangle the trisectors cut out is not equilateral — but its sides agree to within a tenth of a percent even when the angles add to 222°. Every step of the rotation proof survives on the sphere except one, and that one is the angle sum.

geometry · Morley

Named alongside it

The objects these essays reach for when they reach for this one.

Hyperbolic geometry Angle sum Angle trisection Curvature Geodesic Morley theorem Parallel postulate Pythagorean theorem Rotation Spherical excess

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