Complete Guide to Understanding Euclidean Geometry

The undefined terms in our axioms are point and lines.

Axiom 1: Every point has at least two lines

Axiom 2: Every line has at least two points

Axiom 3: There exist at least eight points

Prove theorem 1: How many lines are there in the axiomatic system? There exists at least one line as axiom three guarantees the existence of a line but no axiom specifically points to existence of a line. To prove the existence of a line, we consider axiom three which states that there exists at least eight points. Since each point must have at least two lines by axiom one, then there must be a line that connects the points.

Prove theorem 2: What is the minimum number of lines? The minimum number of lines is eight as there are at least eight points according to axiom three. Axiom one states that there exists at least two lines for every point. This, together with axiom three, proves that there exists at least eight lines for every model designed.

To prove Axiom 1 which states that every point has at least two lines, there must be a line that joins any two points in a given model. To join the two points again with another line there must exist another line between the points. Therefore for every point there must exist at least two lines. Every line has at least two points by axiom two. To be able to have a line joining with another line, there must be a point that connects the two on one end. On the other end also there must exist a point to complete transform the line into a shape, therefore, there must exist two points every line to successfully form a shape.

Our geometry will contain at least eight points to show that an eight point geometry does exist.

Model 1.

Model 2.

In the models above, let the darkened areas represent the points, while the conjoining areas represent the lines. The two models prove that the eight-point axiomatic system can be created independently. The models represent real-world shapes which include squares, rectangles, and triangles. The models are concrete systems demonstrating that the axiomatic system above is consistent.

The axiomatic system is categorical since the two models in our system are isomorphic. That is, there is a one-to-one correspondence between the elements which preserves all relations. Every statement from the axiom containing the undefined and defined terms for this system can be proven which means that the system is complete. The two models also prove that all the three axioms are not independent of each other, for axiom three to hold axiom two and one must be incorporate, meaning that for every point model there must be at least eight lines for the system to hold.

The system supports Euclidean system as at no point in time do the parallel lines meet in my system. It is composed of real geometric shapes which mainly include squares, rectangles, and triangles that are made up of only straight lines.  Also, the axioms are not independent which basically supports Euclidean’s fifth postulate.

Principle of duality states that an axiom system in which the dual of any axiom or theorem is also an axiom or theorem has been supported by the system.

Young’s geometry is a type of a finite geometry that satisfies the following axioms.

There exists at least one line.

Every line is linked exactly to three points in the geometry.

All points of the geometry are not on the same line.

There exists exactly one line for every two distinct points

If a point does not exist on a given line, there exists exactly one line on that point that does not intersect the given line.

When comparing and contrasting my first model to young’s axiomatic system, the first point would highlight the fact that both are finite systems. There exist at least one line for both systems, and all points in the geometry are not on the same line. Also, there exists exactly one line on both of every two distinct points. If a point does not exist on given line, there exists a line on that point that does not intersect the given line in both systems. The difference is that in my system every line is linked to exactly two points and there exist exactly two lines for every two distinct points.

Work cited

Veblen, Oswald. “A system of axioms for geometry.” Transactions of the American Mathematical Society 5.3 (1904): 343-384.

Penrose, Roger. Shadows of the Mind. Vol. 4. Oxford: Oxford University Press, 1994.

Tarski, Alfred. ”What is elementary geometry?.” Studies in Logic and the Foundations of Mathematics. Vol. 27. Elsevier, 1959. 16-29.