Weird geometry

The area of a hyperbolic triangle, call it ABC, strangely enough doesn't depend on the lengths of the sides. It is totally determined by the angle measures. So for any two hyperbolic triangles ABC and DEF, if angles A=D, B=E, and C=F, then ABC and DEF are congruent. Weird!

The area of a hyperbolic triangle is given by

Area = π-A-B-C

where A, B, C are the measures of the named angles in radians. (In the special case of a triply-asymptotic or ideal triangle, where all three vertices lie on the boundary "at infinity", the area is simply pi. Weirder still! The area of a TRIANGLE is the ratio of the circumference of a circle to its diameter? (The picture shows two ideal triangles in the Poincare upper half plane model

Perhaps weirdest yet? In spherical geometry, the area of a triangle ABC is r²(A+B+C-π). r² here is the square of the radius of the sphere. Usually one chooses the unit sphere, so the term disappears. But what happens if you use i, the imaginary unit, instead? Well, you get

Area = i²(A+B+C-π)
= -1(A+B+C-π)
= π-A-B-C

which is the formula for the area of a hyperbolic triangle. Consequently, hyperbolic geometry "is just" spherical geometry on a sphere of radius i.


:O