A Discrete Geometrical Gem
3. Some ramifications of the Sylvester-Gallai Theorem
This plane with 7 points and 7 lines has exactly three points on each line. However, one line looks "peculiar." It is the line L6 in the diagram and contains the three points P2, P6, and P7 which is shown as a "circle" above. It follows from the Sylvester-Gallai Theorem that no matter how we might try to position 7 points in the Euclidean plane so as to lie on 7 lines as in the Fano configuration, we can not succeed!. The Fano plane is named for Gino Fano (1871-1952), the Italian geometer who pioneered the study of finite geometries and point configuration and whose two sons (one, Ugo, a physicist and the other, Robert, an engineer) had distinguished careers in the United States.
If a finite set of points in the plane are not all on one line then there is a line through exactly two of the points(*),
can be thought of as holding in the Euclidean plane or in the real projective plane. (We saw above that the theorem need not hold in finite projective planes.) Because the usual axioms for a projective plane have the property that if one interchanges the words "point" and "line" in the axioms one gets the same axiom set back, it follows that there is a duality principle for projective planes. Duality refers to the fact that when, in the statement of a theorem, the words "point" and "line" are interchanged, then one gets another valid theorem. What is the dual of (*)? This is the statement that if one has a configuration of lines and points, not all of the lines going through a single point (i.e. not a pencil of lines), then there is some point that lies on exactly two lines. Such a point is known as an ordinary point. (To state the associated "dual" theorem in the Euclidean plane we have to deal with lines which do not pass through a single point and no two of which are parallel.)
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