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. Cremona,
L. Elements
of Projective Geometry, 3rd ed. Durell,
C. V. Modern
Geometry: The Straight Line and Circle. 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. Lachlan,
R. "The Principle of Duality." §7 and 284-299 in An
Elementary Treatise on Modern Pure Geometry. Ogilvy, C. S. Excursions
in Geometry. 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 https://mathworld.wolfram.com/DualityPrinciple.html
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