Lotusland of Mathematical Dreams

What happens when sum of all angles of a triangle is not . Is it possible? Can the area of a triangle be the sum of all its angles? Can a straight line be curved?

If the mind says, no. Its wrong. It is possible. Beyond the Euclidean geometry that we are used to though. The case is totally different with Hyperbolic spaces.

Think of a line drawn on a sphere, a straight line. There is no doubt that the line is perfectly straight. But when you see the line after drawing it, you can feel the curve in the line. The curvature is attributed to the spherical surface. So if you accept the curvature, or rather think of such a line on earth, the curvature is so big that we feel the plane as prefectly flat. So the line becomes perfectly straight. Thus it is the preception that makes the line straight or curved. Hyperbolic geometry is on a spherical universe. So when we try to see it in the Euclidean space, we get confused. We start thinking if the straight line is actually straight or not.

Now think of a sphere. A triangle is drawn on it. The projection is taken onto 2-d plane. The bounding circle will be the projection of the equator (actually this is one of the formal representation method for Hyperbolic geometry, called Poincaré Hyperbolic Disk). The lines under this representation will be straight only if it is a diameter to the circle. With such a representaion, the angle doesnt always sum upto , as it normally does. It falls short, and by how much depends on the area of the triangle.

Taking the constant C=1, we have the formula for the angle.

Yet again, this is not inside the universe of Hyperbolic geometry. Taking a triangle in a hyperbolic space, and trying to represent it in Euclidean one gives us this pain. There are alternate representations available, which keeps the lines straight, and angles add to . But these add in other types of complications. It is even said that an equation for the distance between two points gets more complex this way.