Incident Geometry

We start with axioms that describe the relationships between points and lines. We have three, and these are the only statements we may take for granted. Below are three incidence axioms. I1: For any two distinct points there exists a unique line that contains both of them. I2: For any line there exist at least two distinct points lying on it. I3: There exist three distinct points with the property that no line contains all three of them. Interpretation vs. Model For each interpretation below answer the following questions: Do we have a model for these three axioms? If not, explain which axioms fail to be satisfied. 1. Consider a fictitious goldfish bowl sitting on a table containing two goldfish in water. Suppose there is one goldfish on the table beside the bowl. By point we mean one of the goldfish and by line we mean the two goldfish in the bowl. 2. 3 distinct points, 3 lines each containing exactly two of the points. 3. 4 distinct points, 6 lines each containing exactly two of the points 4. 7 distinct points, with 3 points on every line and 3 lines through every point. (This is known as Fano Geometry) 5. What else can be proved? For Fano Geometry, prove that “If two distinct lines exist, then they intersect in exactly one point.” And -“you have exactly 7 points” or you can’t add an eight-point without violating the axioms.” More details and explanations are given in the attached document for further consideration and discussion. Further Exploration: Let us again consider the following set of axioms. I1: For any two distinct points there exists a unique line that contains both of them. I2: For any line there exist at least two distinct points lying on it. I3: There exist three distinct points with the property that no line contains all three of them. 1. In the following questions include explanations of your claims: (a) Find an interpretation which satisfies I1 and I2 but not I3. (b) Find an interpretation which satisfies I1 and I3 but not I2. (c) Find an interpretation which satisfies I2 and I3 but not I1. The fact that it is possible to find models in which two of the axioms are true and the third is not tells us that the third axiom is independent of the other two. 2. Find a model of the system given by I1-I3. 3. Is the sphere a model of I1-I3? You are expected to share your models and responses in your shared GoogleDrive Folder.