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One can similarly construct projective planes over any other finite field, with the Fano plane being the smallest. Using the standard construction of projective spaces via homogeneous coordinates , the seven points of the Fano plane may be labeled with the seven non-zero ordered triples of binary digits 001, 010, 011, 100, 101, 110, and 111.
A projective plane is a linear space in which: Every pair of distinct lines meet in exactly one point, and that satisfies the non-degeneracy condition: There exist four points, no three of which are collinear. There is a bijection between P and L in a projective plane. If P is a finite set, the projective plane is referred to as a finite ...
The Fano plane, discussed below, is denoted by PG(2, 2). The third example above is the projective plane PG(2, 3). The Fano plane. Points are shown as dots; lines are shown as lines or circles. The Fano plane is the projective plane arising from the field of two elements. It is the smallest projective plane, with only seven points and seven lines.
The Fano plane. This particular projective plane is sometimes called the Fano plane. 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
The Levi graph of the Fano plane is the Heawood graph. Since the Heawood graph is connected and vertex-transitive, there exists an automorphism (such as the one defined by a reflection about the vertical axis in the figure of the Heawood graph) interchanging black and white vertices. This, in turn, implies that the Fano plane is self-dual.
A commercial passenger plane bound for Reagan National Airport collided with a US Army Black Hawk helicopter over the Potomac River just outside of Washington, DC, on Wednesday. There were 64 ...
The crash raises questions about whether the helicopter crew was looking at different aircraft or whether the crew misjudged the position of the plane, mistaking other lights for the aircraft the ...
In incidence geometry, most authors [16] give a treatment that embraces the Fano plane PG(2, 2) as the smallest finite projective plane. An axiom system that achieves this is as follows: (P1) Any two distinct points lie on a line that is unique. (P2) Any two distinct lines meet at a point that is unique.