Differences in Geometry…
Geometry is the branch of mathematics that deals with the properties of space. Geometry is classified between two separate branches, Euclidean and Non-Euclidean Geometry. Being based off different postulates, theorems, and proofs, Euclidean Geometry deals mostly with two-dimensional figures, while Demonstrative, Analytic, Descriptive, Conic, Spherical, Hyperbolic, are Non-Euclidean, dealing with figures containing more than two-dimensions. The main difference between Euclidean, and Non-Euclidean Geometry is the assumption of how many lines are parallel to another. In Euclidean Geometry it is stated that there is one unique parallel line to a point not on that line.
Euclidean Geometry has been around for over thousands of years, and is studied the most in high school as well as college courses. In it’s simplest form, Euclidean geometry, is concerned with problems such as determining the areas and diameters of two-dimensional figures and the surface areas and volumes of solids. Euclidean Geometry is based off of the parallel postulate, Postulate V in Euclid’s elements, which states that, “If a straight line meets two other straight lines so as to make the two interior angles on one side of it together less than two right angles, the other straight lines, if extended indefinitely, will meet on that side on which the angles are less than two right angles.”
For centuries, mathematicians tried to contradict Euclid’s Postulate V, and determine that there was more than one line parallel to that of another. It was declared impossible until the 19th century when Non-Euclidean Geometry was developed. Non-Euclidean geometry was classified as any geometry that differed from the standards of Euclidean geometry. One of the most useful Non-Euclidean Geometry is the Spherical Geometry, which describes the surface of the sphere. Spherical Geometry is also the most commonly used Non-Euclidean geometry, being used by astronomers, pilots, and ship captains. In Euclidean geometry it is stated that the sum of the angles in a triangle are equal to 180. As for Spherical geometry it is stated that the sum of the angles in a triangle are always greater than 180. When most people try and visualize a triangle containing angle sums greater than 180 they say it’s impossible. They’re right, in Euclidean geometry it is impossible, but as for Spherical geometry, it is possible. Think of the triangle on a sphere, and then try and visualize it. See Appendix 1-1.
When thinking of the Non-Euclidean Spherical Geometry, we start of with a basic sphere. A sphere is a set of points in three-dimensional space equidistant from a point called the center of the sphere. The distance from the center to the points on the sphere is called the radius. See Appendix 1-2 to visualize tangents, lines, and centers between the sphere, lines, and planes.
Unlike standard Euclidean Geometry, in Spherical Geometry, radians are used to replace degree measures. It is usual for most people to measure angles and such with degrees, as for scientists, engineers, and mathematicians, radians are used to substitute degree measures. The size of a radian is determined by the requirement that there are 2pi radians in a circle. Thus 2pi radians equals 360 degrees. This means that 1 radian = 180/pi degrees, and 1 degree = pi/180 radians. See Appendix 1-3.
In Euclidean Geometry, the simplest polygon is known as the triangle, containing three sides. There is no such thing as a two-sided polygon in Euclidean Geometry, as for Spherical Geometry, there is, being referred to as a lune, or biangle. See Appendix 1-4. A two-sided polygon is closely related to part of the moon that is commonly seemed. In lunes there are two noticeable things which should be noted. One, the two vertices do not lye on the poles of the sphere. Second, the two angles of the lune are congruent.
The area of a sphere is closely related to finding the area of a circle in Euclidean geometry. To find the area of a circle, the formula, piR? is used in Euclidean Geometry. In Spherical geometry, a similar formula is used, 4piR?.
Besides Euclidean Geometry and the Non-Euclidean Spherical Geometry, there is also the commonly known Non-Euclidean Hyperbolic Geometry. Hyperbolic Geometry is closely related to Einstein’s General theory of Relativity. Hyperbolic Geometry is a “curved” space, related to some Spherical Geometry. Einstein’s General Theory of Relativity can be understood by saying that matter and energy distort space, and the distortions of space affect the motions of matter and energy, being related to, “curved space.” Cosmologists today believe that “curved space” is the fourth dimension, although it hasn’t yet been proven by means of postulates, theorems, or proofs.
It is very difficult to draw a mental picture of four-dimensional space. Although, it has been drawn in a way excepted by crazed mathematicians and cosmologists. Four-dimensional space is derived from flatland that contains sliding figures within. See Appendix 2-1. This flatland is then manipulated into a curved plane. See Appendix 2-2. Both the flatland and the manipulated curved flatland, is related to that of Mercury’s orbit, replacing Euclidean geometry with Hyperbolic.
Hyperbolic geometry has an interesting “twisted” figure to it, the pseudosphere. The pseudosphere is the two-dimensional object, which is related to the normal sphere used in Spherical Geometry. However, the pseudosphere is smaller then the plane it lies on and tends to “bend back” on itself. Since the pseudosphere is bigger than the plane it lies on, it is hard to be drawn out and to be visualized. In Appendix 2-3, the fourth dimension is shown, for us to better visualize parts of the pseudosphere.
In Euclidean Geometry, the theorem, “If two lines are parallel to a third line, then the two lines are parallel to each other,” is disproved in Hyperbolic Geometry. Appendix 2-4 shows how it is proved false in hyperbolic geometry by means of three lines being contained on the pseudosphere. Euclidean lines and hyperbolic lines are different in only one way. It is said that lines in Euclidean Geometry are straight and endless, although in hyperbolic geometry, lines are contained on the sphere and curve. Although the lines curve, they are still parallel to the sphere in hyperbolic geometry. Even though the lines do have some similarities, there are three theorems in Euclidean Geometry, which are false in Hyperbolic Geometry. These theorems are as follows:
1) If two lines are parallel to a third line, then the two lines are parallel to each other.
2) If two lines are parallel, then the two lines are equidistant.
3) Lines that do not have an end (infinite), also do not have a boundary.
One postulate used in Hyperbolic Geometry is the parallel postulate, which states, “Given a line and a point not on that line, there are at least two lines which contain that point, which are in the same plane as the line and are parallel to that line.”
In Euclidean Geometry, the area of a triangle is calculated by multiplying the length of any side times the corresponding height, and dividing the product by two. However, in hyperbolic geometry, if a triangle is scribed onto a sphere, the area formula for Euclidean geometry will not work out properly.
In Euclidean Geometry the area of a square is a side cubed. However, in Hyperbolic Geometry it is possible for a square on the sphere to contain three right angles and one acute angle. In this case the formula for a square in Euclidean Geometry will not pertain to that of Hyperbolic Geometry. In essence, area formulas for polygons in Euclidean Geometry are, under some circumstances, invalid in Hyperbolic Geometry.
As described within the preceding text, there are many different variations in Geometry. The term “Geometry” is very broad and can be quite speculated into other means. Mainly, there are two breeds of Geometry. The standardized Euclidean Geometry, which is mostly accepted and used by all, and there is the Non-Euclidean Geometry, anything other than the standardized Euclidean methods. Non-Euclidean geometry are not often studied and utilized as much as the Euclidean means of calculations. Most High School courses only study the areas of Euclidean Geometry and don’t get into the sophisticated area of any Non-Euclidean Geometry. The Non-Euclidean Geometry mainly focused on in the preceding text was that of Spherical and Hyperbolic. There are many other Non-Euclidean Geometries which can be studied, some being Demonstrative, Analytic, Descriptive and Conic, to name a few. Because of their extensive comprehending language, most people try to isolate themselves from studying any means of Non-Euclidean Geometry to avoid confusion between that and Euclidean Geometry.
The sphere on the left is a perfect example of a triangle containing angles which sum is greater than 180?.
The picture on the left demonstrates how in Spherical geometry, many things occur. In the uninteresting case the plane and the sphere miss each other. If they do meet each other there are two possibilities. First they can meet in a single point. In this case the plane is tangent to the sphere at the point of intersection. In the other case the sphere and the plane meet in a circle.
If A and B are two points on the sphere, then the distance between them is the distance along the great circle connecting them. Since this circle lies totally in a plane, we can figure this distance using the plane figure to our left. If the angle ACB is a, and if a is measured in radians, then the distance between A and B is given by
d(A,B) = R a,
where R is the radius of the sphere.
If two lines are parallel to a third line, then the two lines are parallel to each other. This is a theorem in Euclidean Geometry, yet in Hyperbolic Geometry it is proved false by the above counter example (Both BA and BC are parallel to DE, yet BA is not parallel to BC). However, you may not be convinced that BA and DE are parallel.
O’Reilly, Geometry in a Nutshell, O’Reilly & Associates, Inc. California, ?1996.
Leon L. Bram, Funk & Wagnalls New Encyclopedia, Funk & Wagnalls, Inc. ?1990.
The Geometry of a Sphere: http://math.rice.edu/~pcmi/sphere/