A Finite Projective Space

This is a model of the smallest projective space in existence, the projective space over the 2-element field P(3,GF(2)). In general, P(n,k) denotes the n-dimensional projective space over the field k. Naturally, if the field k is finite, then any finite-dimensional projective space over that field must be finite. The space P(3,GF(2)) has 15 points, represented by white polystyrene balls and 35 lines, represented by chenile stems of different colors.

This projective space also contains 15 (Fano) planes, and these are represented by some particular subsets, each containing 7 lines and 7 points. The planes contained in each of the faces of the bounding tetrahedron are probably the easiest to see. However, there is also a unique plane determined by each edge and a unique plane determined by each vertex. Finally, the yellow and orange lines, taken with their intersection points, comprise a single plane.

The projective space possesses a self-duality, meaning that all of the incidence relations remain valid after interchanging the words "point" and "plane". For example, notice that there are 7 lines meeting at every point and there are 7 lines on every plane.

The model has the symmetry of a regular tetrahedron, but the group of the geometry is GL(4,2). This means that the group is the set of invertible 4x4 matrices with entries from the two-element field GF(2). The fact that the model has tetrahedral symmetry is a reflection of the fact that the tetrahedral group is a subgroup of GL(4,2). One should refer to the group of this geometry as the "projective" group PGL(4,2) instead of GL(4,2). However, there is only one element in the center of GL(4,2), so these groups are isomorphic. Applying a counting argument, one can check that the order of GL(4,2) is

15*14*12*8=20160.

As it turns out, GL(4,2) is a simple group, meaning that it has no non-trivial normal subgroups. As it turns out, this group is isomorphic to the group A(8) of even permuations on 8 letters.

References.

Burkard Polster. A Geometrical Picture Book. Springer-Verlag Inc., New York, 1998.


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