Zero system

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null system

An involutory correlation of an -dimensional projective space with an anti-symmetric operator. Suppose that the null system has the form

Then the scalar product , which is



[1] B.A. Rozenfel'd, "Multi-dimensional spaces" , Moscow (1966) (In Russian)


A null system is also called null polarity, a symplectic polarity or a symplectic correlation. As is clear from the above, it is a polarity such that every point lies in its own polar hyperplane.

In projective -space, a correlation is a dualizing transformation (cf. Correlation), taking points, lines and planes into planes, lines and points, while preserving incidence in accordance with the principle of duality. If every range of points on a line is transformed into a projectively related pencil of planes through the new line, the correlation is said to be projective. There is a unique projective correlation transforming five given points, no four in a plane, into five given planes, no four through a point.

A polarity is a projective correlation of period two (cf. Polarity). In other words, it transforms each point into a plane and each point of into a plane through . One kind of polarity transforms each point on a quadric surface into the tangent plane at that point. Another kind, a null polarity, transforms every point of space into a plane through that point. It may be described as the unique projective correlation that transforms five points (no four collinear) into the respective planes . The line is self-polar, since it is the line of intersection of the polar planes and of and . In fact, all the lines through in its polar plane are self-polar: there is a flat pencil of such lines in every plane, and the set of all self-polar lines is a linear complex.

In terms of projective coordinates, a null polarity takes each point to the plane , where

and and . In terms of the Plücker coordinates of a line, , where

the linear complex of self-polar lines in the null polarity has the equation


[a1] K.G.C. von Staudt, "Beiträge zur Geometrie der Lage" , Korn , Nürnberg (1847) pp. 60–69; 190–196
[a2] H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 65–70
[a3] D. Pedoe, "Geometry: a comprehensive course" , Dover, reprint (1988) pp. §85.5
How to Cite This Entry:
Zero system. D.D. Sokolov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098