Lotusland of Mathematical Dreams

1. There is a unique straight line segment connecting any two points

2. Unlimited (continuous) extendability of any straight line segment .

3. Existence of a circle with any center and any value for radius .

4. Equality of right angles.

5. If two straight line segments, a and b in a plane intersect another straight line c such that the sum of the interior angles on the same side of c is less than two right angles, then a and b, extended far enough on that side of c, will intersect somewhere.

Playfair’s Axiom is an alternate variation of the 5th Euclidean postulate (parallel postulate). It states that for any line and any point that is not on the line, there is a unique straight line through the point which is parallel to the line.

Parallel postulate has a lot of story attached to it. History says of a lot of people who tried to make the fifth postulate a theorem, derivable from the rest four. But finally Beltrami proved the independence of 5th postulate from others.

Remarkable is the effort by Saccheri, which speaks of the origins of elliptic geometry and hyperbolic geometry.

This is an alternate representaion for Hyperbolic geometry. Wikipedia says that this done by projecting a point on a sphere onto a plane tangential to the sphere at the point antinodal to the center of projection (that is the point diametrically opposite to the center of projection).

With this projection, any circle that crosses through the center of projection becomes a straight line.Any other circle (that does not touch center of projection), can become circles (possibly ellipses in the case of inclined circles).

For an example, think that we are projecting Earth. The center of projection is absolute North Pole. The plane of projection is parallel to the equitorial circle. Now consider the projection of a longitude. It will be a straight line.

Consider the case of a latitude. This becomes a circle. An extreme case will be the equator. It will be a bounding circle. It is called primitive of the projection. Any other latitude becomes a circle concentric to the primitive and inside it. Infact those latitudes in southern hemisphere will have projection with radius greater than the primitive (Still haven’t figured out how).

A beautiful description is also available from International Union of Crystallography. Stereography is used in crystallography it explains.

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.