Projective Properties
As systems of points and lines in which all points belong to the same number of lines and all lines contain the same number of points, the complete quadrangle and the complete quadrilateral both form projective configurations; in the notation of projective configurations, the complete quadrangle is written as (4362) and the complete quadrilateral is written (6243), where the numbers in this notation refer to the numbers of points, lines per point, lines, and points per line of the configuration. The projective dual of a complete quadrangle is a complete quadrilateral, and vice versa. For any two complete quadrangles, or any two complete quadrilaterals, there is a unique projective transformation taking one of the two configurations into the other.
Karl von Staudt reformed mathematical foundations in 1847 with the complete quadrangle when he noted that a "harmonic property" could be based on concomitants of the quadrangle: When each pair of opposite sides of the quadrangle intersect on a line, then the diagonals intersect the line at projective harmonic conjugate positions. The four points on the line deriving from the sides and diagonals of the quadrangle are called a harmonic range. Through perspectivity and projectivity, the harmonic property is stable. Developments of modern geometry and algebra note the influence of von Staudt on Mario Pieri and Felix Klein .
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