Models+of+incidence+geometry+and+betweenness

We started class by proving Proposition 1.7 from incidence geometry. As a reminder, the three incidence axioms are:

I-1: For any two distinct points there exists a unique line that passes through both of them. I-2: For any line there exist at least two points incident with it. I-3: There exist three distinct points with the property that no line is incident with all three of them.


 * Proposition 1.7**: //For every point P there exist at least two distinct lines through P.//

The first proof attempt was given by Bob. By Proposition 1.1, there exists a line called //l//. By I-2, there exist two distinct points called P and Q, which are incident with //l.// By I-3, there exist three distinct points called A, B, and C with the property that no line is incident with all three of them. This proof attempt was separated into two cases.

Case 1: Points P and Q are not the same as points A, B, or C. By I-1, for any two distinct points there exists a line that passes through both of them. Through I-1, we can construct lines {A,B}, {A,C}, {A,P}, {A,Q}, etc. There exist four lines that are incident with each point A, B, C, P, and Q. Therefore, there exist at least two distinct lines through each point.

Case 2: Points P and Q are two of the three points A, B, and C. The same argument for construction of lines is presented as in Case 1. Therefore, there exist at least two distinct lines through each point.

After this argument was given, we noted that it included excess information not necessary for the proof. We discussed if this proposition could be proven using only three points. We decided that there do not need to be five points to prove this proposition. The second proof, which turned out to be successful, was written by Erin.

Spurred by Ashley's idea to use Proposition 1.6, we began the proof with the fact that for every point P there is at least one line, called //l//, not passing through it. By I-2, on line //l// there exist two distinct points, called A and B. By I-1, there exists a unique line, called //m//, that passes through A and P. By I-1 again, there exists another unique line, called //n//, that passes through B and P. Line //m// is not the same as line //n// because P would need to lie on //l//, which is not true by Proposition 1.6. This proof avoided the issue of dealing with cases for the points, and the explanation was agreed to be satisfactory.

We revisited the idea of having several terms that we have not been able to define. The five terms that remain undefined are: point, line, between, lies on, and congruent.

Next we moved on to discussing **interpretations and models**.

We found several interpretations and determined they are models.

We have often reference the first model while proving incidence propositions. Points: A, B, C Lines: {A,B}, {A,C}, {B,C} These three noted points and lines are assumed to be the only ones that exist.

The second model is familiar to us from using the Cartesian plane. Points: (x,y) with x,y being elements of the real numbers Lines: ax + by = c with a,b,c being elements of the real numbers We can check that a specific point is on a specific line by checking to see that the point satisfies the equation of the line for the given a, b, and c.

We verbally noted that the axioms of incidence geometry are correct statements in these interpretations. Hence, the interpretations satisfy the axioms. These interpretations qualify as models because they satisfy all of the axioms of the system.

We then debated if a sphere is a model of incidence geometry. We tried to determine if a sphere would satisfy the three incidence axioms with our table groups. The class was initially split, with some groups thinking a sphere would be a model of incidence geometry, while others did not. Points: (x,y,z) are points on the surface of the sphere where x^2 + y^2 + z^2 = 1. Lines: great circles, which are the intersections of planes through the origin and two points Points positioned at two random places on the sphere appeared to satisfy axiom I-1 since only one great circle could satisfy both points.Antipodal points, however, have many lines passing through them. Hence, a circle cannot satisfy incidence axiom I-1. Therefore, spherical geometry was determined not to be a model of incidence geometry.

General thoughts about proofs: To demonstrate that a statement, S, cannot be proved from a list of statements, L, it is enough to find one interpretation in which all of the statements of L are correct, but S is false. Even if there is an interpretation for which a statement, S, is correct, this is not proof of S because it may not be true in all cases. A statement, S, must be true for all models in order to be valid. One false case proves that a statement, S, is not true, but one true case cannot prove that S is not false overall.

We next discussed serval examples of determining if a statement is independent from other statements.

Is I-1 independent of I-2 and I-3? Interpretation 1: Assume there are three distinct points, called A, B, and C. Assume there are no lines. I-2 and I-3 are satisfied, but I-1 is not satisfied. Interpretation 2: Assume there are three distinct points, called A, B, and C. Assume there are three distinct lines, called {A,B}, {B,C}, and {A,C}. I-1, I-2, and I-3 are all satisfied. Based on these two interpretations, I-1 is independent of I-2 and I-3.

Is the statement "there are 4 points" independent of I-1, I-2, and I-3? Interpretation 1: Assume there are three distinct points, called A, B, and C. Assume there are three distinct lines, called {A,B}, {B,C}, and {A,C}. I-1, I-2, and I-3 are satisfied. "There are 4 points" is not satisfied. Interpretation 2: Consider the Cartesian plane. I-1, I-2, and I-3 are satisfied. "There are 4 points" is satisfied. Based on these two interpretations, "there are 4 points" is independent of I-1, I-2, and I-3.

Next we interpreted several of the betweenness axioms.


 * B-1**//: If A*B*C, then A, B, and C are three distinct points all lying on the same line, and C*B*A (B is between A and C).//

We drew a line with the three distinct points A, B, and C in order, respectively. We noted that if you look at the order of the points starting at either end of the line, B is still between the other two points. A*B*C cannot also be B*A*C, or equivalently C*A*B, because of axiom B-3.


 * B-2**//: Given any two distinct points B and D, there exist points A, C, and E lying on the line through B and D such that A*B*D and B*C*D and B*D*E.//

We drew a line and first chose two points B and D incident with that line, drawn in order from left to right. A*B*D tells us that there is a distinct point A that lies on the line to the left of both B and D. B*C*D tells us that there is a distinct point C that lies on the line between B and D. B*D*E tells us that there is a distinct point E that lies on the line to the right of both B and D. After we placed these five points on the line, we noticed that this axiom could be applied over and over again using other choices of sets of two points to start with. Repeated application of this axiom can lead to infinitely many points existing on a line. We also noted that a circle would not satisfy axiom B-2.


 * B-3**//: If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two.//

This axiom came up naturally earlier as we discussed B-1. B-3 rules out the consideration of a circle as a line because for three points on a circle, any of the points is between the other two. A circle with an infinitely small piece missing or a semicircle, not including the end points, could still be considered a line since they satisfy this axiom.


 * Definition 2.1:** //Let// l //be any line, A and B any two points that do not lie on// l. //If A=B or if segment AB contains no point lying on// l, //we say that A and B are on the same side of// l, //whereas if A is not equal to B and segment AB does intersect// l, //we say that A and B are on opposite sides of// l.

If point A is the same as point B, then the points are clearly on the same side of //l.// We drew a picture to show that a segment AB that does not intersect //l// then A and B are on the same side of //l.// We define segment AB = {C: A*C*B} U {A,B}.

COMMENT: A few questions/suggestions from the notes include Case 2 at the beginning of the notes, it is stated that "Points P and Q are //the same as// two of the points A, B, and C". Maybe this is saying the same thing, but I was wondering if we need to state it as "P and Q __are two of the three__ points A, B, and C".

Just after the transition to interpretations and models, there is a sentence that states "//We found several models for points and lines and interpreted them//". I understand that we find interpretations, then determine whether or not they are indeed models. Could we reword that sentence to say "__We found several interpretations, then determined that they were models__".

COMMENT: There is a statement before the discussion of axioms: "Next we proved several of the betweenness axioms." What exactly was proved about the axioms? I thought that we interpreted what the axioms state. I understand that axioms are statements are true and require no proof as they are used in proving various statements.

In B-2, would it be better if we stated that "repeated application of this axiom can lead to infinitely many points existing on a line"?

I have noted these comments and made the indicated changes.