Some geometrical generalizations of Euler's formula

Geometrically, Euler's formula draws a circle in the complex plane. Replacing the square root of minus one in exp (iθ) with the 2-by-2 matrix draws an analogous circle in the x,y plane, exp (L i θ i ), where L i 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 (iθ).