Geometry is one of the most diverse topics in Mathematics. The discovery of geometry can be traced as early as 2900-2000 B.C. wherein the Egyptians learned how to create pyramids out of blocks of stones and the earliest method of computing for the area of a triangle was formulated. 2

In the following years, the Greeks also had their contributions in the development of geometry resulting to more studies regarding similar triangles and the formulation of proofs of the postulates and theorems involved to it. Pythagoras also had his contribution, particularly the “Pythagorean Theorem”. He was also considered as the first pure mathematician to logically deduced geometry facts. However, the turning point for this period was accredited to Euclid and his book called “Elements”, a collection of definitions, axioms, and his well-known five postulates from which he constructed different theorems that were then included in the Euclidean geometry. 2

For several years, the Euclidean geometry was thought to be the only existing mathematics that could explain matters concerning space but not until the 1900s wherein new kinds of geometry were introduced. This period marked the modern age of geometry. 4

Modern geometry basically involves non-Euclidean geometries. These types of geometry are based on axioms, postulates, and theorems that are somehow or entirely different from the postulates that Euclid introduced. Some examples of this type of geometry are the Hyperbolic and Elliptic geometry.

Comparing the latter two periods of geometry, it could be deduced that the first four postulates of Euclid are contained to what is known as neutral geometry, a geometry known for not having an assumption of a parallel postulate. Relating this to non-Euclidean geometry, though the neutral geometry may seem very ideal there are still some valid geometric models that it may not be enough. Some mathematicians argued that the world we live exist in a surface of a sphere and this do not follow Euclid’s first four postulates. 4

Basically, the discovery of Non-Euclidean geometry created conflicts on what the mathematicians are used to believe into and to what is seemed to be also true. Questions such as “Which of the two geometries is true?” arose. 5

II. DISCUSSION

The answer to the question above will now be based on what type of space is being studied. It is not always the case that a type of geometry is applicable for all types of spaces. For example, if we are to talk about linear spaces then the first four postulates of Euclid may be applied but it may not be suitable for other spaces such as spheres.

Since the first four postulates of Euclid failed to support the argument that we are existing in a surface of a sphere, a new type of geometry was introduced, we call it the Elliptic geometry. This discussion will be focused on what is Elliptic geometry and on the axioms it involves with their corresponding models.

Elliptic geometry is a geometry with no parallel lines that exist. It is interpreted using different models. The most common and comprehensive model for elliptic geometry is the surface of a sphere wherein a “point” is defined as a pair of antipodal points and a “line” is defined as a great circle of the sphere. 3

Moreover, to understand this type of geometry better, different adaptations of the postulates of Euclid and David Hilbert, a proponent of neutral geometry, will also be made. To do this, we need to restate Euclid’s postulate to definitions that are suitable for elliptic geometry. On the other hand, we must identify some characteristics of neutral geometry and then establish how elliptic geometry differs from it. 1

According to May (2012) listed below are some of the models and adaptions on Euclid and Hilbert’s postulates that are used to discuss elliptic geometry:

1. Axioms of Incidence. The axioms of incidence are used to clarify and remove any ambiguity from Euclid’s first postulate.

Figure 1. A model representing the incidence axioms.

Incidence Axiom 1. For every point P and for every point Q, where P and Q are not equal, there exists a line l incident with P and Q and that line is unique.

In Elliptic Geometry: For every pair of antipodal points P and P’ and for every pair of antipodal points Q and Q’ such that P and Q are not equal (consequently, P’ and Q’ are also not equal) there exists a unique great circle incident with both pairs of antipodal points.

Even though a circle requires three points to be defined, the relationship of Q’ to Q indicates that the circle defined only by P, P’, and Q would also definitely travel through Q’. Similarly, though two points do not define a unique circle in the plane and as we subject lines to mean great circles on the sphere then we can conclude that two distinct points define a unique circle.

Incidence Axiom 2: For every line l, there exist at least two points incident with it. The two points must be distinct.

In Elliptic Geometry: For every great circle c, there exist at least two pairs of antipodal points incident with c. The pairs of antipodal points must be distinct.

Incidence Axiom 3: There exist three points with the property that no line is incident with all three of them. The three points must be distinct.

In Elliptic Geometry: There exist three pairs of antipodal points with the property that no great circle is incident with all three of them. The pairs of antipodal points must be distinct.

Going back to Figure 1, c is incident with P, P’, and Q’, and d is incident with P, P’, R and R’. Clearly there is no great circle exists that is incident with all three pairs of antipodal points. Hence, the model satisfies the incidence axioms.

2. Axioms of Betweenness. These are used to clarify and remove any ambiguity from Euclid’s second postulate. Defining the notation for betweenness in the following way:

A*B*C? point B is between points A and point C.

Figure 2. A model representing the axiom of betweenness in neutral and elliptic geometry.

Betweenness Axiom 1: If A*B*C, then A, B, and C are three points all lying on the same line and C*B*A. A, B, and C are distinct.

Betweenness Axiom 2: Given any two points B and D that are distinct, there exist points A, C, and E lying on (BD) ? such that A*B*D, B*C*D, and B*D*E.

Betweenness Axiom 3: If A, B, and C are three points that are distinct and lying on the same line, then one and only one of the points is between the other.

Betweennes Axiom 4: (Plane Separation) For every line l and for any three points A, B, and C not lying on l:

(i) If A and B are on the same side of l and B and C are on the same side of l, then A and C are on the same side of l.

(ii) If A and B are on opposite sides of l and if B and C are on opposite sides of l, then A and C are on the same side of l.

Notice that based on how the axioms are stated, betweenness are not valid when lines are defined in circles. In fact, the concept of betweenness is irrelevant to a circle because the betweenness when travelling around a circle clockwise or counter-clockwise will just be the same. Hence, these axioms are for neutral geometry only.

Figure 2. A model representing the distances of points in neutral and elliptic geometry.

The figure above shows that the equidistance of points is well-defined only on neutral geometry, and in fact, it is arbitrary in elliptic geometry. Since the axiom of betweenness only holds for neutral geometry, then it should be redefined in order to be well-defined in elliptic geometry. Thus, the axioms will be changed to axioms of separation.

3. Axioms of Separation. In here, we are to apply the concept of separation on elliptic geometry. The separational relations is defined as:

(A,B?C,D)? points A and B separate points C and D.

Figure 4. A model representing separation axioms 1-3, 5 and 6.

Separation Axiom 1: If (A,B?C,D), then then points A, B, C, and D are distinct and are on the same line.

This means that the points not on the same line cannot separate one another. Hence, only distinct collinear points are separable.

Separation Axiom 2: If (A,B?C,D), then we have (C,D?A,B) and (B,A?C,D).

Separation axiom 2 states that separation is both reflexive and symmetric. If you cannot get from C to D without passing A or B, then to get from A to B, C or D must be crossed. Also, if you cannot get from C to D without crossing A or B, then you cannot get from C to D without crossing B or A.

Separation Axiom 3: If (A,B?C,D) then not (A,C?B,D) .

This axiom notes that separation is well-defined for a particular set of points.

Separation Axiom 4: If points A, B, C, and D are distinct and on the same line, then (A,B?C,D) or (A,C?B,D) or (A,D?B,C).

This means that only one of the graphics in figure 5 is valid. Any two or all of them cannot be true at the same time.

Figure 5. A model for separation axiom 4.

Separation Axiom 5: If points A, B, and C are distinct and on the same line, then there exists a point D such that (A,B?C,D).

This means that if given three points, then there will always be a fourth point such that the first two points separate the third point and the new point.