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The real (or complex) projective plane and the projective plane of order 3 given above are examples of Desarguesian projective planes. The projective planes that can not be constructed in this manner are called non-Desarguesian planes , and the Moulton plane given above is an example of one.
If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2. The Fano plane is called the projective plane of order 2 because it is unique (up to isomorphism). In general, the projective plane of order n has n 2 + n + 1 points and the same number of lines; each line ...
A similar construction, starting from the projective plane of order 3, produces the affine plane of order 3 sometimes called the Hesse configuration. An affine plane of order n exists if and only if a projective plane of order n exists (however, the definition of order in these two cases is not the same). Thus, there is no affine plane of order ...
The order of a finite projective plane is n = k – 1, that is, one less than the number of points on a line. All known projective planes have orders that are prime powers. A projective plane of order n is an ((n 2 + n + 1) n + 1) configuration. The smallest projective plane has order two and is known as the Fano plane.
Steiner's Roman surface is a more degenerate map of the projective plane into 3-space, containing a cross-cap. The tetrahemihexahedron is a polyhedral representation of the real projective plane. A polyhedral representation is the tetrahemihexahedron, [4] which has the same general form as Steiner's Roman surface, shown here.
The Fano plane is the projective plane with the fewest points and lines. The smallest 2-dimensional projective geometry (that with the fewest points) is the Fano plane, which has 3 points on every line, with 7 points and 7 lines in all, having the following collinearities:
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.
L 2 (7) ≅ L 3 (2) which acts on the 1 + 2 + 4 = 7 points of the Fano plane (projective plane over F 2); this can also be seen as the action on order 2 biplane, which is the complementary Fano plane. L 2 (11) is subtler, and elaborated below; it acts on the order 3 biplane. [8]