folded GCF folds each pair of fractions from the unfolded GCF into one fraction, thus doubling the convergence pace. The Manny Sardina reference further explains "folded" continued fractions. Note: Using the continued fraction for cited above with the best-known Machin-like formula provides an even more rapidly, although still linearly, converging expression: with and . Roots of positive numbers The th root of any positive number can be expressed by restating , resulting in which can be simplified, by folding each pair of fractions into one fraction, to The square root of is a special case with and : which can be simplified by noting that : The square root can also be expressed by a periodic continued fraction, but the above form converges more quickly with the proper and . Example 1 The cube root of two (21/3 or ≈ 1.259921...) can be calculated in two ways: Firstly, "standard notation" of , , and : Secondly, a rapid convergence with , and : Example 2 Pogson's ratio (1001/5 or ≈ 2.511886...), with , and : Example 3 The twelfth root of two (21/12 or ≈ 1.059463...), using "standard notation": Example 4 Equal temperament's perfect fifth (27/12 or ≈ 1.498307...), with : With "standard notation": A rapid convergence with , , and : More details on this technique can be found in General Method for Extracting Roots using (Folded) Continued Fractions. 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 , and the way lattice points in two dimensions lie to either side of the line . 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. See also Gauss's continued fraction Padé table Solving quadratic equations with continued fractions Convergence problem Infinite compositions of analytic functions Notes References (Covers both analytic theory and history.) (Covers primarily analytic