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Duality Principle


All the propositions in projective geometry occur in dual pairs which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "point" and "line." The principle was enunciated by Gergonne (1825-1826; Cremona 1960, p. x). A similar duality exists for reciprocation as first enunciated by Poncelet (1817-1818; Casey 1893; Lachlan 1893; Cremona 1960, p. x).

Examples of dual geometric objects include Brianchon's theorem and Pascal's theorem, the 15 Plücker lines and 15 Salmon points, the 20 Cayley lines and 20 Steiner points, the 60 Pascal lines and 60 Kirkman points, dual polyhedra, and dual tessellations.

Propositions which are equivalent to their duals are said to be self-dual.


See also

Brianchon's Theorem, Conservation of Number Principle, Continuity Principle, Desargues' Theorem, Dual Polyhedron, Duality Law, Pappus's Hexagon Theorem, Pascal's Theorem, Projective Geometry, Reciprocal, Reciprocation, Self-Dual

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References

Casey, J. "Theory of Duality and Reciprocal Polars." Ch. 13 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 382-392, 1893.Cremona, L. Elements of Projective Geometry, 3rd ed. New York: Dover, 1960.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 78, 1928.Gergonne, J. D. "Philosophie mathématique. Considérations philosophiques sur les élémens de la science de l'étendue." Ann. Math. 16, 209-231, 1825-1826.Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 26-27 and 41-43, 1930.Lachlan, R. "The Principle of Duality." §7 and 284-299 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 3-4 and 174-182, 1893.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 107-110, 1990.Poncelet, J.-V. "Questions résolues. Solution du dernier des deux problémes de géométrie proposés à la page 36 de ce volume; suivie d'une théorie des pôlaires réciproques, et de réflexions sur l'élimination." Ann. Math. 8, 201-232, 1817-1818.

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Duality Principle

Cite this as:

Weisstein, Eric W. "Duality Principle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DualityPrinciple.html

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