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Modern Math through Games: Notes for Discovery Seminar

Tien Chih

Section 2.2 Neutral Plane Axioms

Subsection 2.2.1 Introducing Axiom Systems and Models

Definition 2.2.1.

An axiom system has two components:
  1. A set of special words that we officially assume have no meaning. The words are sometimes called undefined terms, primitives or technical terms.
    A set of statements called axioms that we assume to be true concerning the undefined terms.
A model of an axiom system is a set of meanings for the undefined terms. These meanings are assigned without any regard for whether the axioms become true statements about the undefined terms.

Example 2.2.2.

Consider the following axiom system, the 4 point geometry. Let point, line and belongs be the undefined terms. The axioms are:
  1. There are 4 points.
  2. Given two distinct points, there is exactly one line that they both belong to.
  3. Each line has exactly two points belonging to it.
Then the following are models of this axiom system.
  1. Points are letters, lines are columns, a point belongs a line if it is on the column:
    \begin{equation*} \begin{array}{|c|c|c|} \hline A & A & A & B & B & C \\ B & C & D & C & D & D \\ \hline \end{array} \end{equation*}
  2. Points are dots, segments are lines, point belongs to a line if a segment connects them:

Problem 2.2.3.

(a)
Come up with your own model for the 4 point geometry.
(b)
Exaplain why the following is not a model for the 4 point geometry
(c)
Compare your model to everyone else’s models and the examples in [cross-reference to target(s) "example-4point" missing or not unique]. Are these models actually different?

Subsection 2.2.2 Neutral Plane Axioms

We consider the following axiom system. Let point, line and belongs be the undefined terms.

Problem 2.2.7.

Would any model for the 4-point geometry also be a model for this axiom system? Justify your assertion.

Problem 2.2.8.

What is the minimum number of points possible for a model of this axiom system? Justify your assertion.

Problem 2.2.11.

Prove or disprove: Each point belongs to a line.
Since such a “geometry” is fundamentally uninteresting, we add a fourth axiom.

Problem 2.2.13.

Let cards be points and SETs be lines. Does the grid you constructed in Activity 2.1.6 satisfy all four of our axioms? Justify your assertion.

Problem 2.2.14.

Find three different models of our four axioms (it’s easier if they are finite).

Problem 2.2.15.

Find the invalid line of the following “proof”.

Proof.

By Axiom 2.2.6, there is a line \(L\text{,}\) by Axiom 2.2.4, \(L\) has two points, \(A\text{,}\) \(B\text{.}\) So for each point \(A\text{,}\) there is a line containing \(A\text{.}\)

Problem 2.2.16.

Prove that every point belongs to a line.

Problem 2.2.17.

Is it possible that some point belongs to only one line? Prove your response.

Problem 2.2.18.

Find a model that satisfies our axiom system in which there are two different points \(A\) and \(B\) such that at least two different lines each contain both \(A\) and \(B\text{.}\)

Problem 2.2.19.

Find a model that satisfies our axiom system in which there are two different points \(A\) and \(B\) such that at least two different lines each contain both \(A\) and \(B\text{.}\)
Since this axiom system still seems flawed, we replace the fourth axiom with a stronger version.

Definition 2.2.21.

We let \(\overleftrightarrow{AB}\) denote the line containing both \(A\) and \(B\text{.}\)

Definition 2.2.22.

A collection of points are colinear if there are is a line which all the points in the collection belong to.

Problem 2.2.23.

What would we mean by the statement that points are non-collinear?

Problem 2.2.24.

Is it true that given a point that there exist two more points such that the three points are collinear? Prove your answer.

Problem 2.2.25.

Can we say each point is not in some line? Prove your answer.

Problem 2.2.26.

Suppose \(L\) and \(H\) are different lines. What can you say about the number of points that can belong to both \(L\) and \(H\text{?}\) Prove your answer.

Definition 2.2.27.

If \(L, M\) are lines, and a point \(A\) belongs to both lines, we say \(L\) and \(M\) intersect\(A\text{.}\)

Subsection 2.2.3 Parallelism

Problem 2.2.28.

Let points be ordered pairs \((x,y)\) such that \(x, y\) are real numbers. Let lines by sets of points which satisfy the equality \(ax+by=c\) for some \(a,b,c\text{.}\) (This should be familiar).
(a)
Show that this model satisfies our axioms.
(b)
Let \(L\) be a line and let \(A\) be a point which doesn’t belong to \(L\text{.}\) How many lines are there that contain \(A\) but don’t intersect \(L\text{.}\)

Problem 2.2.29.

Let points be ordered pairs \((x,y)\) such that \(x, y\) are real numbers and \(x^2+y^2 < 1\text{.}\) Let lines by sets of points which satisfy the equality \(ax+by=c\) for some \(a,b,c\) and \(x^2+y^2 < 1\text{.}\) (This may not be familiar).
(a)
Show that this model satisfies our axioms.
(b)
Let \(L\) be a line and let \(A\) be a point which doesn’t belong to \(L\text{.}\) How many lines are there that contain \(A\) but don’t intersect \(L\text{.}\)

Problem 2.2.30.

Find a model with 7 points for our axioms where every pair of lines intersect. (Let the lines contain 3 points each)

Problem 2.2.31.

For any model of our axioms, let \(L\) be a line and let \(A\) be a point which doesn’t belong to \(L\text{.}\) What can we say about the number of lines which contain \(A\) but don’t intersect \(L\text{.}\)

Subsection 2.2.4

Problem 2.2.32.

Define a plane. (The definition should rule in known examples, and rule out non-examples like 3d things.)

Problem 2.2.33.

State and prove a theorem about planes.
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