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. Referenced
on Wolfram|Alpha 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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