Higher Dimensions
Another meaning for generalized continued fraction is a generalization to higher dimensions. For example, there is a close relationship between the simple continued fraction in canonical form for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx. Generalizing this idea, one might ask about something related to lattice points in three or more dimensions. One reason to study this area is to quantify the mathematical coincidence idea; for example, for monomials in several real numbers, take the logarithmic form and consider how small it can be. Another reason is to find a possible solution to Hermite's problem.
There have been numerous attempts to construct a generalized theory. Notable efforts in this direction were made by Felix Klein (the Klein polyhedron), Georges Poitou and George Szekeres.
Read more about this topic: Generalized Continued Fraction
Famous quotes containing the words higher and/or dimensions:
“Neither years nor books have yet availed to extirpate a prejudice then rooted in me, that a scholar is the favorite of Heaven and earth, the excellency of his country, the happiest of men. His duties lead him directly into the holy ground where other men’s aspirations only point. His successes are occasions of the purest joy to all men. Eyes is he to the blind; feet is he to the lame. His failures, if he is worthy, are inlets to higher advantages.”
—Ralph Waldo Emerson (1803–1882)
“Why is it that many contemporary male thinkers, especially men of color, repudiate the imperialist legacy of Columbus but affirm dimensions of that legacy by their refusal to repudiate patriarchy?”
—bell hooks (b. c. 1955)