page_title: Finite geometry
section_title: The smallest projective three-space
hierarchical_section_title: Finite geometry / Finite spaces of 3 or more dimensions / The smallest projective three-space
caption_attribution_description: Fano three space
context_page_description: A finite geometry is any geometric system that has only a finite number of points.
  The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries.
  Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so constructed are called Galois geometries. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field.
context_section_description: The smallest 3-dimensional projective space is over the field GF(2) and is denoted by PG(3,2).  It has 15 points, 35 lines, and 15 planes.  Each plane contains 7 points and 7 lines.  Each line contains 3 points.  As geometries, these planes are isomorphic to the Fano plane.
  Every point is contained in 7 lines.  Every pair of distinct points are contained in exactly one line and every pair of distinct planes intersects in exactly one line.
  In 1892, Gino Fano was the first to consider such a finite geometry.
caption: PG(3,2) but not all the lines are drawn
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