geometry

A)     Neutral geometry, which is more often referred to as absolute geometry, is based on the first four of Euclid’s postulates and ignores the fifth postulate. One of the more general ways to state the fifth postulate is “for a two-dimensional geometry when two straight lines are both intersected by a third line, if there are two interior angles formed on either side of the third line such that their sum is less than 180 degrees then the first two original lines will eventually intersect on the same side as the interior angles aforementioned.”

The major difference between these geometry types is the behavior of parallel lines, or lines for which a third line is perpendicular to each. In Euclidean geometry, the parallel lines can be extended for an infinite distance and will always remain the same distance from one another. These are the traditional parallel lines we are used to dealing with and they do not intersect. In neutral geometry such lines will curve away from each other (hyperbolic geometry) and not intersect, or they will curve toward each other (spherical geometry) and intersect one another. Many types of these non-Euclidean geometries exist, including elliptical geometry, saddle geometry, and Lobachevskian geometry.

B)      The development of non-Euclidean geometry is important because it further solidified the relationship between science, mathematics, and experiment. An excellent example of this is that relationships that hold true on a flat desktop or sheet of paper will not necessarily hold true on the surface of the Earth (and vice versa). A necessary conclusion from this development is that Euclidean geometry is merely a special case of geometry in the actual universe.

Mathematicians have spent centuries trying to show that Euclid’s fifth postulate can be inferred from the first four. During the course of these attempts, several equivalent statements have been proposed. An offshoot of this is dealing with geometries where no assumptions are made about parallelism, of which hyperbolic and spherical geometries are examples. A few examples of equivalent statements for Euclid’s fifth postulate are:

1) There exists a triangle whose angles add up to 180 degrees
2) There exists a quadrilateral where all the angles are 90 degrees
3) The upper limit for the area of a triangle is unbounded

Part 1.

The great circle containing points B and C can be thought of as the equator of the Earth. This means that point A can be considered one of the poles, for simplicity we will call it the North Pole. With these analogies, it can be seen that the line segment BC runs east/west while the line segment AB runs north/south. This requires that angle ABC measures 90 degrees. A similar argument shows that angle ACB also measures 90 degrees. It is obvious by inspection that angle BAC is greater than zero degrees, making the sum of the three angles for triangle ABC greater than 180 degrees.

Part 2.

This is a proof that the angles in a triangle equal 180°:

The top line (that touches the top of the triangle) is
running parallel to the base of the triangle.

So:

  • angles A are the same
  • angles B are the same

And you can easily see that A + C + B does a complete rotation from one side of the straight line to the other, or 180°

The above is from http://www.mathsisfun.com/proof180deg.html

Construct a quadrilateral with all four angles equal to 90 degrees. This means that the quadrilateral will consist of two sets of parallel lines. Consider the top line in the above diagram to be a portion of one side of the quadrilateral and the bottom line in the above diagram to be a portion of the opposite side of the quadrilateral. This means that the top and bottom lines of the above diagram are parallel.

The two angles marked A are opposite interior angles, and thus have the same value.

The two angles B are also opposite interior angles, and thus have the same value as well.

Now look at the top of the triangle. It is obvious that A + B + C = 180 degrees since the three angles added together make a straight line.

This means that the three angles within the triangle add to 180 degrees.

Part 3.
The above diagram is drawn in a hyperbolic geometry. This means that the sum of the angles for any triangle will be less than 180 degrees.

Notice that triangle ABC and triangle BCD share a common side, namely line segment BC.

This means that the three angles in triangle ABC (A, ABC, and ACB) will add to less than 180 degrees, and the three angles in triangle BCD (D, BCD, CBD) will add to less than 180 degrees.

Now look at the diagram as being a quadrilateral with four angles (A, ABD, ACD, and D). Notice that angle ABD is the sum of angles ABC and CBD. Notice also that angle ACD is the sum of angles ACB and BCD.

This means that the sum of the four angles of the quadrilateral will equal the sum of the six angles of the two triangles, which is less than 360 degrees.

If the sum of the angles of the quadrilateral must be less than 360 degrees, then it is not possible to have each of the four angles of the quadrilateral equal to 90 degrees.

This means that there are no rectangles in hyperbolic geometry.