Posted September 2006.
This will involve an excursion over a wide swath of the mathematical landscape and feature a variety of distinguished contributors to mathematics from many cultures...
The word "geometry," of Greek origin, means earth or land measure. As a branch of mathematics, geometry's standard definition concerns obtaining insights into shapes and the nature of space. Geometry has evolved into a rapidly growing field which is not merely concerned with shapes and space but more broadly with visual phenomena.
Steiner Triple Systems
Mathematicians are trained to abstract and generalize. The physical origin of many mathematical problems tended to result in continuous models for problems arising outside of mathematics. Mathematical questions about counting, the collection of ideas that now belongs to the branch of mathematics called combinatorics, were developed surprisingly late in mathematics. Combinatorics became one part of the very rapidly growing part of mathematics known as discrete mathematics, which has especially blossomed under the influence of ideas from computer science. Many of these early combinatorial problems arose from attempts to solve problems that were raised as recreational mathematics questions.
(The Reverend Thomas Penyngton Kirkman)
Here are two examples of such Steiner triple systems:
(The Norwegian mathematician Thoralf Skolem)
The techniques involved in these constructions range from ideas involving graph theory to the use of Latin squares (square arrays in which the symbols in the array appear in each row and each column once), and to methods involving abstract algebra. Fans of Sudoku will be familiar with the concept of Latin squares.
Given that numbers and number systems are at the very heart of mathematics, it is surprising that the development of finite arithmetics systems by the mathematics community came so late. It was Evariste Galois (1811-1832) who in conjunction with his efforts to show that polynomial equations of the fifth degree (quintics) could not be solved by providing formulas for their roots (as is true for polynomial equations of degree 1 to 4) first discovered the idea of finite arithmetics.
Finite arithmetics which obey all the algebraic rules (e.g. associativity, commutativity of addition and multiplication, etc.) of the real numbers are now known as finite fields, and are denoted GF( pn ) where p denotes a prime integer. The GF in this notation stands for Galois Field, which is another name given to these finite arithmetics. Amazingly, it has been shown that these are the only finite arithmetics which algebraically share the properties of the real or rational numbers. They are known to exist when the number of elements is a power of a prime, and for no other values. The breadth of applications of these finite fields is staggering, including such areas as error-correction and data compression systems that are essential for HDTV (high definition television) and cell phone technology.
Steps towards finite geometry
Geometry made another big leap forward during the Renaissance when various artist-mathematicians laid down the foundations of projective geometry. One challenge these artists were trying to address was making more realistic looking images of buildings in their paintings. We are all familiar with the phenomenon that railroad tracks seem to meet in the distance when we know perfectly well they do not meet. The artists wanted to incorporate this perceptual reality into the way they represented scenes. Projective geometry, when axiomatized, a development that did not come immediately in the Renaissance, adopts two important axioms. The first of these axioms we have met before as axiom Affine 1:
The earliest work on Finite Geometries has not been well charted by historians of mathematics. Part of this may be that one of the earliest contributors was Gino Fano (1871–1952), an Italian mathematician, who wrote almost exclusively in Italian. Fano constructed examples of finite projective planes and also finite spaces.
Fano's work in the area of finite geometry included the discussion of a 3-dimensional finite geometry which consisted of 15 points, 35 lines, and 15 planes where each line had 3 points on it and each plane had 7 points. Fano had a long career as a teacher at the University of Turin, where he had also been a student. Due to his Jewish background he was forced out of his position in Turin and went to Switzerland for the war years. By the time the war was over Fano was quite elderly, but this did not prevent him from traveling to the United States and lecturing in Italy. Fano's sons Ugo and Robert had distinguished careers. Ugo earned a doctorate in mathematics but pursued a career in physics, and Robert taught engineering at MIT, where he did research on the mathematical problem of efficient data compression. The finite geometry shown in the diagram below is now called the Fano Plane in honor of Gino Fano.
(Figure 1: One way of drawing a diagram representing the Fano Plane.)
A few comments are required to clarify the meaning of the diagram above. The Fano Plane has 7 points, and in the diagram above they are represented by the dark dots P1, ..., P7, which is a finite set of points. What are the lines of the Fano Plane? A typical line, say L7, consists of a set of three points. L7, as can be read from the diagram, consists of the points P1, P7, and P3. Looking at the way the diagram is drawn it may be tempting to think that the point P7 is "between" the points P1 and P3. However, this is not the case. Line L7 is a set of points but for this geometry we will not be able to give meaning to a concept of "betweenness." Also, the way that the line L7 has been drawn may make it appear as if there are lots of points between P7 and P1 but, in fact, this line has exactly three points on it. Many geometers like to have a diagram such as this figure to illustrate the Fano Plane but for those who find the diagram "confusing," one can revert to thinking of the Fano Plane as a set of 7 points and seven lines, where the exact points that make up each line have been specified.
Some people find another aspect of Figure 1 confusing. The line L6 also has three points on it: P7, P6, and P2. Yet this line is drawn as if it is a circle! Again, the diagram is an aid to insight and if one finds the fact that one of the lines is drawn in a "non-straight" fashion confusing, one can go back to just the set of points and the list of lines and points that make them up without using any diagram whatsoever. You may ask, why draw the line L6 as if it is circular? Is it possible to locate 7 points in the Euclidean plane in such a way that the points are in positions where all 7 lines of the Fano Plane lie along "straight" lines in the Euclidean plane? The answer is "no!" In fact, the diagram we have chosen is at least somewhat appealing because 6 of the 7 lines are represented as being on "straight" lines in the Euclidean plane.
(Figure 2: A finite affine geometry with 4 points and 6 lines.)
The geometry in Figure 2 has four points and six lines and obeys the "Playfair" parallelism axiom. To help you understand the meaning of this diagram, note that if one chooses the point P1 which is not on line L6 (which consists of points P2 and P3), then the line L5 (consisting of points P1 and P4) contains P1 and is parallel to L6. It does not look as if L5 and L6 are parallel in Figure 2 but these lines are parallel in our finite affine geometry. The reason is that the point where these lines appear to meet is not a point of the geometry. Only the four points represented by dark dots are points of the geometry!
One of the most amazing theorems in the geometry of the real projective plane (the projective geometry derived from Euclidean geometry) was discovered by Girard Desargues (1591-1661). It states that if one has two triangles ABC and A'B'C' such that the points on the lines AA', BB' and CC' go through a single point O, then the lines AB and A'B', AC and A'C' and BC and B'C' meet in points R (not shown in Figure 3, but which would lie to the right of line OB'), P, and Q which lie on a single line! You should draw a diagram of this kind and verify that the three points P, Q, and R appear to be collinear, that is, lie on a single line. You should also formulate the dual theorem and converse theorem of Desargues' Theorem and convince yourself that these statements are valid in the real projective plane.
(Figure 3: Diagram illustrating Desargues' Theorem)
It turns out that while Desargues' "Statement" is true for a plane embedded in a 3-dimensional geometry and the real projective plane, there are infinite and finite projective planes where it does not hold. A simple example of such a plane is the Moulton Plane.
(Figure 4: Pappus' Theorem)
It turns out that Desargues' Theorem holds in a projective geometry if and only if that geometry can have coordinates introduced which obey the rules for a division ring. This is an algebraic structure which satisfies the usual algebraic properties of a field (for example, the real, rational or complex numbers are fields), except that commutativity of multiplication does not hold. It is also true that Pappus' Theorem holds in a projective geometry if and only if that geometry can be coordinatized using numbers from a field. Thus, it is possible to construct infinite projective planes where Pappus' Theorem holds but Desargues' does not. It turns out that every finite division ring is, in fact, a field. This lovely theorem is known as Wedderburn's Theorem. Thus, every finite Desarguean projective geometry is also Pappian. There is no known completely geometrical proof of this fact.
Veblen's doctoral thesis deals with axiomatic geometry. He has had nearly 5000 academic descendants. Veblen and Bussey were part of a movement in America, during the relatively early days of establishing a research mathematics community in the United States, which examined the foundations of geometry. Included was the mathematician E. H. Moore, who was Veblen's doctoral thesis advisor and who was an early president of the American Mathematical Society (AMS). Veblen also was president of the AMS. An important prize for research in geometry, administered by the AMS, is named for Veblen.
(Eliakim H. Moore)
We have discussed some of the properties of finite affine and finite projective planes. We have also shown that there is always such an affine plane for each value of n where n is a prime power because we can use coordinates from a finite field. Furthermore, when there is an affine plane with n points on a line there is also a projective plane with n + 1 points on a line. Certainly one major unsolved problem in the area of finite geometries is the question:
Albert, A., Finite planes for the high school, Mathematics Teacher 55 (1962) 165 - 169.
NOTE: Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal, which also provides bibliographic services.
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