Problem 28: Configurations.

Index

Recall that for a projective plane of order k,

b = k² + k + 1
m = k + 1
n = k + 1
v = k² + k + 1

where b is the number of lines, m is the number of points on any given line, n is the number of lines incident with any given point, and v is the number of points. Similarly for an affine plane of order k we have

b = k² + k
m = k
n = k + 1
v = k².

In both cases it is easy to see that the relation bm = nv holds true. A special class of geometries that satisfy this equation are called configurations:

CONFIGURATIONS 1. There are p points and l lines with n lines through every point and m points on every line. 2. Two distinct points are contained in at most one line. 3. Two distinct lines intersect in at most one point. 4. The geometry is connected (see problems on graphs for a defintion of connected).

A very special case would occur if v = b and m = n, in which case we will call the configuration a b_n configuration. We've seen such cases examples when we looked at the Fano plane (a 7₃ configuration),

Pappus and Desargues configurations (9₃ and 10₃ configurations respectively):

Please enable Java for an interactive construction (with Cinderella).

Created with Cinderella

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Please enable Java for an interactive construction (with Cinderella).

Created with Cinderella

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Can you find all possible b₂ configurations?

There are many possible b₃ and b₄ configurations, and we will only look at the simplest of cases. If we start with b₃ configurations, then we can convince ourselves that b > 7. We can rexamine the Fano plane (the unique 7₃ configuration), a homogeneous geometry, by drawing it in such a way as to showcase as many of its automorphisms as possible. We do this below where we start with the typical representation of the Fano plane (each line has its own color), and then we place its 7 points equally spaced along a circle. We then represent the lines as triangles, and it is easy to see that the automorphism group of the Fano plane contains the isometry group of the regular 7-gon (remember that the automorphism group of the Fano plane contains 168 elements, while that of the dihedral group D₇ contains 14 elements). In the last picture (bottom left corner) we show only one of the lines and we can see how to easily generate any other line via a simple "rotation" automorphsism: we will call such a diagram a generator-only diagram as the given line (or lines) generate all others via automorphisms.

This same kind of trick can be used on the affine plane of order 3 with one point and all lines incident with that one point removed (the point being removed is the center point):

As it turns out, this is the only 8₃ configuration.

What then about all possible 9₃ configurations? We begin then with the vertices of a regular 9-gon, and imagine different generators. In the first column below we show 3 different 9₃ configurations. In the second column we show the corresponding generator-only diagrams (the diagrams in each row pertain to one particular geometry - the second row corresponds to the Pappus configuration). And in the last column we connect two points eactly when they are not connected by a line.

These three 9₃ configurations are all the 9₃ configurations possible.