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A projective plane of order N is a Steiner S(2, N + 1, N 2 + N + 1) system (see Steiner system). Conversely, one can prove that all Steiner systems of this form (λ = 2) are projective planes. The number of mutually orthogonal Latin squares of order N is at most N − 1. N − 1 exist if and only if there is a projective plane of order N.
The affine plane of order three is a (9 4, 12 3) configuration. When embedded in some ambient space it is called the Hesse configuration . It is not realizable in the Euclidean plane but is realizable in the complex projective plane as the nine inflection points of an elliptic curve with the 12 lines incident with triples of these.
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 ...
It follows that the fundamental group of the real projective plane is the cyclic group of order 2; ... degenerate map of the projective plane into 3 ... 12 (4 ): 51 ...
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 ...
Hanfried Lenz gave a classification scheme for projective planes in 1954, [6] which was refined by Adriano Barlotti in 1957. [7] This classification scheme is based on the types of point–line transitivity permitted by the collineation group of the plane and is known as the Lenz–Barlotti classification of projective planes.
A polarity π in a projective plane of square order n = s 2 has at most s 3 + 1 absolute points. Furthermore, if the number of absolute points is s 3 + 1 , then the absolute points and absolute lines form a unital (i.e., every line of the plane meets this set of absolute points in either 1 or s + 1 points).
The projective plane of order 3 has 13 points and 13 lines, each containing 4 points. The Mathieu groupoid can be visualized as a sliding block puzzle by placing 12 counters on 12 of the 13 points of the projective plane.