Alternatives to Euclidean Geometry and Its Applications
Negations to Euclid’s fifth postulate, known as the parallel postulate, give rise to the emergence of other types of geometries. Its existence stands in the respective models which their originators have imagined and designed them to be. The development of these geometries and its eventual recognition give humans some mathematical systems as alternative to Euclidean geometry. The controversial Euclid’s fifth postulate is phrased in this manner, to wit:
“If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that ...view middle of the document...
So did the weather forecasters, in predicting paths of weather.
Another type of geometry exists with three-dimensional curved spaces as its model, is the hyperbolic (also called saddle or Lobachevskian) geometry being named after the Russian mathematician, Nicholas Lobachevsky, who proposes the following:
“If L is any line and P is any point not on L , then there exists at least two lines through P that are parallel to L.”
Euclid’s fifth postulate holds no water in hyperbolic geometry due to the presence of parallel lines upon construction. Hyperbolic geometry is used by astronomers in predicting behaviors of space objects’ orbits. It is also useful in space travel and computations involving gravitational pulls.
Each of these geometries has defined terms and own independent system that normally posits different and varying logical equivalents and consequences which responds and answers to problems in the models that each of them stands for. The geometries bear distinct properties as summarized in the following table:
Properties | Types of Geometry |
| Euclidean | Spherical | Hyperbolic |
Curvature | | | |
Given a line m and a point P not on m, the number of lines passing through P and parallel to m | 1 | 0 | many |
Sum of interior angles of a triangle | 180° | > 180° | < 180° |
Square of hypotenuse of a right triangle with sides a and b | a2 + b2 | < a2 + b2 | > a2 + b2 |
Circumference of a circle with diameter 1 | π | < π | > π |