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Original Articles

Some geometrical generalizations of Euler's formula

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Pages 461-468 | Received 22 Aug 1988, Published online: 09 Jul 2006
 

Abstract

Geometrically, Euler's formula draws a circle in the complex plane. Replacing the square root of minus one in exp () with the 2‐by‐2 matrix

draws an analogous circle in the x,y plane, exp (Liθi ), where Li are certain 3‐by‐3 matrices, gives all rotations of a 3‐dimensional object. The set of such rotations modulo ‘the set of 2‐dimensional rotations’ gives a sphere. The hypersurface of a 3‐sphere in 4‐space is obtained in a similar fashion. Finally, the notation developed for generating the above surfaces is related to Lie groups and Lie algebras. This branch of mathematics has applications in particle physics, mechanics, and control theory. The article introduces the reader to simplified ideas from Lie groups and Lie algebras via geometrical generalizations of exp ().

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