TOPICS
Search

Reciprocation


Reciprocation is an incidence-preserving transformation in which points are transformed into their polars. A projective geometry-like duality principle holds for reciprocation which states that theorems for the original figure can be immediately applied to the reciprocal figure after suitable modification (Lachlan 1893, pp. 174-182). Reciprocation (or "polar reciprocation") is the strictly proper term for duality. Brückner (1900) gave one the first exact definitions of polar reciprocation for constructing dual polyhedra, although the plane geometric version (inversion pole, polar, and circle power) was considered by none less than Euclid (Wenninger 1983, pp. 1-2).

Lachlan (1893, pp. 257-265) discusses another type of reciprocation he terms "circular reciprocation." However, the circular reciprocal figure is, in general, more complicated than the original, so the method is not as powerful as the usual polar reciprocation.


See also

Canonical Polyhedron, Dual Polyhedron, Duality Principle, Inversion Pole, Midsphere, Polar, Reciprocal

Explore with Wolfram|Alpha

References

Brückner, M. Vielecke under Vielflache. Leipzig, Germany: Teubner, 1900.Casey, J. "Theory of Poles and Polars, and Reciprocation." §6.7 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 141-148, 1888.Coxeter, H. S. M. and Greitzer, S. L. "Reciprocation." §6.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 132-136, 1967.Lachlan, R. "Reciprocation" and "Circular Reciprocation." Ch. 11 and §405-414 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 174-182 and 257-265, 1893.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, pp. 1-6, 1983.

Referenced on Wolfram|Alpha

Reciprocation

Cite this as:

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

Subject classifications