In
finite geometry
A finite geometry is any geometry, geometric system that has only a finite set, finite number of point (geometry), points.
The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based ...
, PG(3, 2) is the smallest three-dimensional
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
. It can be thought of as an extension of the
Fano plane
In finite geometry, the Fano plane (named after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and ...
, ''PG(2, 2)''.
Elements
It has 15 points, 35 lines, and 15 planes. Each point is contained in 7 lines and 7 planes. Each line is contained in 3 planes and contains 3 points. Each plane contains 7 points and 7 lines.
These can be summarized in a rank 3
configuration matrix counting points, lines, and planes on the diagonal. The incidences are expressed off diagonal. The structure is self dual, swapping points and planes, expressed by rotating the configuration matrix 180 degrees.
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