Now that it is true that the Parallel Postulate is false, we can return to the constant C, which is the proportionality constant between the difference between 180 and the angles of a hyperbolic triangle and the area of that triangle. The constant C can differ for different hyperbolic planes. This is because not all hyperbolic planes have the same curvature. To fully explain curvature, one must also look at the Euclidean plane.
The Euclidean plane is defined to have curvature 0. That means that the surface of Euclidean geometry is flat. Hyperbolic planes, however, have negative curvature. There are different hyperbolic planes because there are different possible values for the curvature of the hyperbolic plane. This is unlike Euclidean geometry, which only has one possible value for the curvature, i.e. 0. But what happens if there is positive curvature? If there is positive curvature another whole type of geometry is introduced, which is even more radical than hyperbolic geometry. This is elliptic geometry. This is the only one of the three types of geometry that has a finite plane, which is one of a sphere. In elliptic geometry, all structures are projected onto a sphere. The theorems resulting include the fact that the angles of a triangle add up to more than 180. A representation of a triangle projected on a sphere is shown below.
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This particular triangle is projected onto the Earth. The angles on the triangle add up to 230 degrees. Note that, in the close up image, the relative curvature of the area approximates zero, and the angles do add up to very close to 180 degrees. In elliptic geometry, Euclid's other postulates are violated. In this geometry, all but the fourth of Euclid's postulates are false (the fourth postulate is that all right angles are equal. The first postulate, that exactly one straight line can be drawn between any two points is false because, on a sphere, multiple lines can connect two points. Take longitude lines, for example. All longitude lines are drawn between the North and South poles, and in elliptical geometry, they all all straight lines. The second postulate, that a line can be continued indefinitely without ever intersecting itself is also false, because any latitude line on a globe eventually wraps its way around and intersects with itself. The third postulate (that a circle can be drawn with any center and radius) is false because if a circle is big enough, it will wrap around the sphere and intersect with itself, violating the very definition of a circle because the points on the edge of the circle would not all be the same distance from the center.
Just as the area of a triangle on a hyperbolic plane can be expressed with the equation 180-(x+y+z)=CΔ where C is a constant and Δ is the area of the triangle, the area of an elliptic triangle can also be expressed in terms of its angles, with the equation Δ=R^2(a+b+c-180). Again, Δ is the area of the triangle, while R is the radius of the sphere where the triangle is located. It is apparent that the quantity a+b+c-180 must be positive because the radius must be positive for the area of the triangle to be positive. If you combine these two equations and solve for C in terms of R, you obtain
C=-1/R^2
Going back to hyperbolic geometry, C, as the constant of variation for the area, must have a positive value. In hyperbolic geometry, therefore, for C to be positive, 1/R^2 must be negative (because then -1/R^2 would be positive). But this is impossible because all numbers square to a positive number or 0. Therefore, the solution to this equation for R must be imaginary, or the square root of a negative number. For information on imaginary numbers, see here. This seems impossible. How can a sphere have a radius that isn't positive or even a real number? It turns out an imaginary radius does have a geometrical meaning. The shape produced in this situation is called a pseudo-sphere. A representation of a pseudo-sphere is shown below.
In essence, this structure is kind of that of an inside-out sphere. The pseudo-sphere is wonderful, but what does it mean? It came about because the radius of a sphere that is a hyperbolic plane is a pseudo-sphere. It makes sense to say that hyperbolic geometry is actually projected onto this shape, just as an elliptic triangle is projected on a normal sphere. This may not seem like the case, but a pseudo-sphere is yet another Euclidean representation of hyperbolic geometry.
In conclusion, these are the only three types of geometry that fit the criteria. The criteria is that the geometries are homogenous and isotropic. Homogenous means that each point on the surface of the geometry is the same, and isotropic means that the perspective of the area around each point is the same as from any other point. There are other geometries that are combinations of the above three, and these usually include some regions with one geometry, and some regions with other types.
For the conclusion of this post, see Hyperbolic and Other Geometries (Part III)
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