**Geometric Structures: Session IX
Axiomatics: Games and Their Rules
**Prepared by:

Joseph Malkevitch

Department of Mathematics

York College (CUNY)

Jamaica, New York 11451

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We have looked at a variety of geometrical objects so far in an informal way. We have studied points, lines, nets, polyominoes, polygons, polyhedra, graphs, etc. We have seen that different people can use the same words in different ways and that it often is convenient to invent new words to help make distinctions between different kinds of objects. Thus, we can distinguish between different types of quadrilaterals by talking about convex and non-convex polygons. We invent words like rhombus and rectangle to distinguish between different kinds of quadrilaterals.

After a certain body of mathematical knowledge is "collected," it is often possible to organize this knowledge in a more "coherent" way. Sometimes this is done by developing an axiomatic system for the body of knowledge. An axiomatic system begins with a collection of undefined terms and a collection of axioms (rules) that involve these terms. Typically, one also defines additional concepts involving the undefined terms. Now, using the laws of logic one can prove facts (theorems) about the concepts involved. For example, point and line may be undefined but one can define the idea of a triangle, and more generally, an n-gon. One can use axiomatic systems in algebra as well as geometry. Thus, one studies algebraic systems like the real numbers, rational numbers, symmetries of a square, and the integers modulo a prime. One can abstract from these settings to the concept of a group. A group is a collection of objects on which an operation has been defined subject to a collection of axiomatic rules. The power of the axiomatic approach is that any fact that one proves about groups applies to the group of real numbers (without 0) under multiplication, the symmetries of a square, and the integers mod 7 under addition. One can prove "facts" (theorems) about groups, rather than proving these facts for the individual systems, that obey the axioms for a group. (Examples of "things" that obey axioms are referred to as models for the axioms.)

We will develop some axiom systems for geometry, but I think it is helpful to first think about the analogy between axiom systems for geometry and the rule systems that govern the sports world that many people enjoy following. There is a strong similarity between axiom systems and the rules by which baseball, American football, and soccer are played. Let me remind you briefly about one aspect of American football. When a team is in possession of the ball, they get 4 downs to move the ball 10 or more yards (or get a touchdown), otherwise they lose possession of the ball. We can think of a what a typical football game looks like in terms of these two "parameters." Suppose we let d represent downs and y represent the number of yards the ball must be moved in d downs to avoid losing possession of the ball. Ordinary football has d = 4 and y = 10. Suppose on some "alien" planet they also play American football, except that d =4 and y = 11. All other football rules apply. How will this change the game? Can you make a prediction about the typical scores of games when d = 4 and y = 11? Similarly, can you make predictions about what happens if d = 3 and y = 10? These two games are similar to football but not the usual football. Some people might like the new version better or worse than the usual version. The changes in the game from changing the rules is like the changes in geometry that will occur when we change the axioms. The new geometries are not necessarily better or worse than the geometry we are familiar with, only different!

The axiomatic development by Euclid, in the book called the Elements, that laid down the foundation of what today we call Euclidean geometry was an amazing accomplishment, considering how long ago it was done and how much less mathematics was known then. However, as impressive as Euclid's axiomatization of Euclidean geometry is, it is not quite right. It took until the later 19th century to get it right. This impressive step was taken by many mathematicians from many countries. For simplicity, let me say that one "rigorous" axiomatization of Euclidean geometry was developed by the German mathematician David Hilbert. (Pieri and Tarski were others who found axiomatizations of Euclidean geometry.) His work was summarized in his book, Foundations of Geometry (1905). To achieve all the rich detail of Euclidean geometry one has to not only have materials about "incidence" properties of points and lines (including parallelism) but such issues as betweenness congruence and continuity. Although there are aspects of axiomatics that are still being studied to this day, recently geometers are much more interested in understanding new geometric phenomena rather than the axiomatic framework in which all of the new developments are made. (In a sense, results like the art gallery theorems are just theorems of Euclidean geometry which live within the old axiomatic framework. They might well have been done by Euclid.)

To get a flavor of axiomatics and the ideas that amazed the world of mathematics, physics, and philosophy at the end of the 19th century, I am going to develop the finite geometry analogues of the Euclidean plane, the infinite projective plane, and the Bolyai-Lobachevsky (hyperbolic) geometries. The jumping off point for the infinite non-Euclidean geometries as well as their finite analogues will be Euclid's 5th Postulate:

The 5th postulate of Euclid in a recent formulation (due to Playfair) is stated:

Given a line l, not containing point P, there is a unique line m through P which is parallel to l.

Two distinct lines in a plane are defined to be parallel if they do not have a point in common. (For technical reasons it is usually stated that a line is considered to be parallel to itself. The concern is that the two lines 2x + 4y = 6 and x + 2y = 3 have different equations but are in fact the same line. They have the same slope and will be considered "parallel."

For nearly 2000 years the relatively complex statement of Euclid's 5th postulate was:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

There was thought to be a flaw in Euclid because many scholars felt it could be deduced (proved) from Euclid's other assumptions. However, today we know this is not true. If we deny Playfair's version of Euclid's 5th postulate, we have the possibilities:

Given a line l not containing point P, there is a NO line m through P which is parallel to l. (In this type of geometry there are no parallel lines. Every pair of lines meet. This type of geometry is known as projective geometry.)

Given a line l not containing point P, there are at least two lines through P which are parallel to l. (In this type of geometry there are multiple parallels. This type of geometry is known as hyperbolic or Bolyai-Lobachevsky geometry.)

Note that we do not talk about "straight lines" as is often done. The reason is that we have no way to "distinguish" between straight lines and lines. Eventually, with enough axioms one can distinguish between "curves" and "straight lines" which turn out to be the "lines" of Euclidean geometry.

Let us take a step back and see what Euclid's axioms about points and lines were.

We will view lines as being made up of sets of points, though one can also proceed by thinking of there being an incidence relation which tells one which points are on which lines.

Euclidean (affine) incidence axioms:

1. Given two points P and Q, there is a unique line which contains P and Q.

2. Given two distinct lines l and m, there is at most one point which l and m have in common.

3. Given a line l not containing point P, there is a unique line m through P which is parallel to l.

4. There exist 4 points, no three (collinear) on a line.

Projective Geometry's incidence axioms:

1. Given two points P and Q, there is a unique line which contains P and Q.

2. Given two distinct lines l and m, there is at most one point which l and m have in common.

3'. Given a line l not containing point P, there is no line m through P which is parallel to l. (All pairs of lines meet.)

4. There exist 4 points, no three (collinear) on a line.

Hyperbolic (Bolyai-Lobachevsky geometry) incidence axioms:

1. Given two points P and Q, there is a unique line which contains P and Q.

2. Given two distinct lines l and m, there is at most one point which l and m have in common.

3''. Given a line l not containing point P, there are at least two lines through P which are parallel to l.

4. There exist 4 points, no three (collinear) on a line.

Finally, we will assume a very REVOLUTIONARY axiom:

5. There exists a line with exactly n points on it.

The most important figure in developing the theory of finite geometries was Gino Fano, and the projective plane of order 2 shown below is often called the Fano Plane.

For finite affine planes, the number n is called the order the plane. For finite projective planes, the plane is said to be of the order n if it has n + 1 points on some line.

Using diagrams, we can show that such finite planes exist. The simplest examples are shown below:

Finite affine plane of order 2:

Affine plane of order 3:

(This plane above has three points on every line and a total of 9 points.)

Finite projective plane of order 2 (Fano Plane)

Finite hyperbolic plane: