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What is Geometry?

The nature and inter-relationship of various geometries

Having evolved from antiquity from often-used methods for measurement of figures drawn mainly on plain surfaces, the methods and principles became distilled, for ready reference and use, as mathematical propositions, particularly in Greece in the peace and prosperity of the few centuries following the rule of Pericles. Then Euclid, in around 300 B.C., gathered, improved, and systematically wrote down all that was known in Geometry to his day. The work - called The Elements - attempted to develop Geometry from the firm foundation of axioms and succeeded in great measure to provide rigorous demonstrations - proof - of the mathematical results loosely proved by his predecesors.

While most of the proofs in Euclid's work were correct, blemishes were discovered in some proofs one of which being the very first proposition in the Elements. However, these blemishes were not due to erroneous deduction but tacit assumptions or "intuitively obvious" facts that were not justified by the axioms.

Considerations of these blemishes culminated in 1899 A.D. with David Hilbert's proposal of a revised axiom system that would not only preserve the validity of the proofs in the Elements but which was in conformity with the modern notion of the axiomatic method as proceeding from a set of undefined terms, definitions, and the axiom statements on the undefined terms. The Euclidean Geometry of today is the Geometry based on this revised axiom system or other equivalent systems since proposed.

The beauty of Euclid's version of Geometry
Although Euclid's axioms needed to be revised to make all his proofs correct, Euclid's Geometry remained for two millenia as the example and model of the deductive method of mathematics. But the beauty of Euclid's Geometry is that it does not make use of numbers to measure lengths or angles or areas or volumes. Instead, it deals with points, lines, triangles, circles, and the relationships among these.

Neutral Geometry
Euclid's axioms without the fifth - which is the parallel axiom often referred to simply as the Fifth - or Hilbert's axioms without Euclid's Fifth is today called Neutral Geometry. In this geometry, the most important theorem is the Exterior Angle Theorem that the measure of the exterior angle of a triangle is greater than either of the non-adjacent interior angles. Compare this to the case in Euclidean Geometry which is Neutral Geometry + Euclid's Fifth axiom. In Euclidean Geometry, the measure of the exterior angle is equal to the sum of the measures of the nonadjacent interior angles. The weak-looking Exterior Angle Theorem of Neutral Geometry implies the Alternate Interior Angle Theorem. That is, if two lines are intersected by a transversal such that a pair of alternate interior angles formed are congruent, then the lines are parallel. This theorem has the immediate corollary that two lines perpendicular to the same line are parallel. This gives the following theorem which can be said to be the culminating result of Neutral Geometry:
Theorem
There is at least one line parallel to a given line through a point not on that line.
PROOF.
Let g be a straight line and P a point not on g. From P, draw line m perpendicular to g, and line n perpendicular to m. This construction is possible in Neutral Geometry. By the previous corollary to the Alternate Interior Angle Theorem, the line n is parallel to the line g. Q.E.D.

An important result of Neutral Geometry is also that the angle sum in a triangle is atmost two right angles. The corresponding situation in Euclidean Gemoetry is that the angle sum in a triangle is two right angles.

Hyperbolic Geometry
In the two millenia since Euclid the greatest ongoing mathematical discussion was not on the blemishes in some of Euclid's proofs, but on the independence of the Fifth Axiom of Euclid which meant that given a line and a point not on it, exactly one line can be drawn through the point and parallel to the line. Gauss, Bolyai,and Lobachevsky separately and independently discovered the new geometry - Hyperbolic Geometry - that assumed the negation of the Fifth Axiom. This established the independence of the Fifth axiom of Euclidean Geometry and laid to rest the longest discussion in the history of mathematics to that date.

The Geometry of René Descartés
Even as the discussion on the Parallel Postulate of Euclid was raging on, René Descartés, in the year 1637, discovered that the ruler and compass constructions of Euclidean Geometry correspond to the solution of linear and quadratic equations in algebra on a Real Cartesian Plane constructed as follows. A point is an ordered pair of real numbers. The set of all such pairs is the Cartesian Plane. The set of points (a,0) is called the x-axis, and the set of points (0,b) is called the y-axis. The intersection of these axes is called the origin. A line in this plane is the subset defined by an equation of the form ax + by + c = 0, with a and b not both zero. Descartés gave the followng four constructions using compass and unmarked ruler - starting with line segments a and b to contruct a+b, ab, a/b and √a. Since these suffice to construct the solution of linear and quadratic equations, all construction problems of Euclidean Geometry can be done in his Cartesian Plane. Since this geometry in the Real Cartesian Plane satisfies all the axioms of Hilbert, it follows that the Cartesian Plane is Euclidean Geometry using numbers.

The method of solution used in finding points from given points in the Real Cartesian Plane is called Analytic Geometry. But Analytic Geometry is not a Geometry in the sense of a theory like Euclidean Geometry that is derived from an axiom set. Rather, Analytic Geometry is a method of doing geometry problems using algebra. In this sense, Analytic Geometry is like Trigonometry which is a method of doing geometric problems involving angles.

The geometry taught today in school is a confused mixture of Euclid's and Descartes'. But the teacher is not to blame. The geometer of antiquity drew diagrams on the Euclidean Plane that was the sand surface. The teacher in the classroom uses the blackboard to represent the Euclidean Plane. The blackboard is also used to represent the real Cartesian Plane. The theorems of Euclidean Geometry can be proved either in the Euclidean Plane or algebraically, using numbers, in the Cartesian Pane. This encourages us to think that the Cartesian Plane and the Euclidean Plane are equivalent. Not so! While all theorems in the Euclidean Plane can be proved algebraically in the Cartesian Plane, there are theorems in the latter that can not be proved in the Euclidean Plane. Examples are Desargues Theorem and Pappus Theorem.

Geometries defined over Fields
We have seeen that the Real Cartesian Plane satisfies all the axioms of Euclidean Geometry. The word "Real" is important here. The points in the Cartesian Plane we discussed were ordered pairs of real numbers. The set of Real Numbers is an example of an algebraic structure called a field. So we could ask if we can use any field in place of the Real Numbers. It turns out, depending on the field, only some of the Hilbert axioms may hold in the associated Cartesian Plane. However, all of these various Cartesian Planes provide various geometries.

A natural question to ask is this. What are the fields whose associated Cartesian Planes give exactly Euclidean Geometries? It turns out that the smallest such field is obtained from the set of rational numbers by adjoining square roots as well as the addition, multiplication and square roots of the rationals [2]. The resulting field is called a Hilbert's Field and the Cartesian Plane associcated with this field is called a Hilbert Plane.

Finite Geometries
These are systems comprising a finite number of points and satisfying some of the axioms of Euclidean Geometry. An example is Fano Geometry.

Elliptic Geometry
A non-Euclidean Geometry is a Geometry that has the negation of the Parallel Postulate as one of its axioms. An example is hyperbolic geometry. It turns out that hyperbolic geometry allows more than one parallel line through a point off a given line. Elliptic Geometry allows no parallel line to a given line.

The name "elliptic" is misleading, for it has no direct connection with the curve called ellipse. The name is used as an analogy of the following result. "A central conic is called an ellipse or hyperbola according as it has no asymptote or two asymptotes. Analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity"[1]. A model of Elliptic Geometry is the surface of a sphere with great circles as lines and antipodal points identified.

Note that, if Elliptic Geometry allows no parallel lines, it can not have Neutral Geometry inside it as is the case with hyperbolic geometry. For in Neutral Geometry, we have the Alternate Interior Angle Theorem which implies that there is at least one parallel line through a point off a given line. If we can not have Neutral Geometry as part of Elliptic Geometry, we ask what axiom of Neutral Geometry is requiring the existence of parallel lines.

Since the Alternate Interior Angle Theorem of Neutral Geometry makes parallel lines necessary, we investigate its proof. We noted above that its proof depended on the Exterior Angle Theorem. So we ask "what does the proof of the Exterior Angle Theorem depend on?" It turns out it relies on three major components: triangle congruence, angle addition, and plane separation. Since triangle congruence and angle addition are at the core of geometry, we do not seek to modify these. The remaining candiate for removal is plane separation. So we remove those axioms that imply plane separation. That would result in a geometry where we are able to pass from one side of a line to the other without crossing the line. And that is the geometry on the surface of the sphere with great circles as lines. Since the model identifies antipodal points on great circles, it is called a single elliptic geometry.

Affine Geometry
Affine Geometry takes for its axioms only Euclid's axioms I and II. This Geometry is motivated by the fact that its results hold not only in Euclidean Geometry but also in Hyperbolic Geometry and the Minkowskian Geometry of time and space. The propositions that hold in Affine Geometry are those that are preserved by parallel projection from one plane to another [1] - for example the first 28 propositions of Euclid, then # 29 and 33-45, and some others in books III(1-19,25,28-30), IV(4-9), and VI(1,2,4,9,10,24-26).

Since Affine Geometry is built on only two of the axioms of Euclidean Geometry, the latter is a specialization of Affine Geometry.

Projective Geometry
This is the parent of all infinite geometries above in that one can get all those geometries by appropriate restrictions and modifications of projective geometry. For instance, an Affine Plane can be obtained from a Projective Plane by picking out any line in the latter and requiring all other lines to meet this line at a unique point with the provision that any two parallel lines will meet the special line is a single point. The line so picked out is called the "line at infinity." Not that this does not mean that the line is situated at infinity but is forced to behave as though it is.

The projective geometry in the plane begins with the following four axioms:
1. There exists at least one line.
2. Every line contains at least three points.
3. Any two distinct points lie on a unique line.
4. Any two lines meet in at least one point.
5. There exist three noncollinear points.
The most elegant property of projective geometry is the principle of duality, which means that every definition remains relevant and every theorem remains valid when we consistently interchange the words point and line (and simultaneously interchange lie on and pass through, join and intersection, collinear and concurrent, etc.)[1]. To establish the duality it suffices to verify that the axioms imply their own duals. As a result, given a theorem and its proof, we immediately can assert the dual theorem. It is a useful exercize to verify that the dual of an axiom is itself either an axiom or a theorem derivable from the remaining axioms.

The purpose of our discussion was just to give a view of the various geometries and how they are related. We rest our discussion of geometries here.


REFERENCES
[1] Coxeter, H.S.M., Introduction to Geometry John Wiley, Inc., 1989
[2] Hartshorne, Robin, Geometry: Euclid and Beyond, Springer,New York, 2000

(G.R.T.)

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Last updated - July 4, 2005