Area of Spherical Triangles Hahnemühle German Etching Print

In spherical geometry, a "triangle" has arcs of great circles as sides. The total interior angle of a spherical triangle exceeds \(\pi\) by an amount proportional to the triangle's area. On a sphere of radius \(R\), the constant of proportionality is \(1/R^2\), the "Gaussian curvature" of the sphere. This deceptively-simple fact, a special case of the "Gauss-Bonnet theorem" in differential geometry, has profound consequences: Beings in a two-dimensional universe can define and empirically detect curvature without making reference to a higher-dimensional space. In the same sense, our own universe is curved, and the deviations from Euclidean geometry manifest to us as the force of gravity.

Printed on archival quality Hahnemühle German etching paper (310gsm) with a velvety surface.