Phi and MathematicsNote to Netscape users: nx means n raised to the x power, but superscripts may not appear properly unless this page is viewed with Internet Explorer. Deriving Phi mathematicallyPhi can be derived by solving the equation: n2 - n1 - n0 = 0 which is the same as n2 - n - 1 = 0 This equation can be rewritten as: n2 = n + 1 and 1 / n = n - 1 The solution to the equation is the square root of 5 plus 1 divided by 2: ( 5½ + 1 ) / 2 = 1.6180339... = Ø This, of course, results in two properties unique to phi:
Phi, curiously, can also be expressed all in fives as: 5 ^ .5 * .5 + .5 = Ø This provides a great, simple way to compute phi on a calculator or spreadsheet! Determining the nth number of the Fibonacci seriesYou can use phi to compute the nth number in the Fibonacci series (fn): fn = Øn / 5½ As an example, the 40th number in the Fibonacci series is 102,334,155, which can be computed as: f40 = Ø40 / 5½ = 102,334,155 This method actually provides an estimate which always rounds to the correct Fibonacci number. You can compute any number of the Fibonacci series (fn) exactly with a little more work: fn = [ Øn - (-Ø)-n ] / (2Ø-1) Note: 2Ø-1 = 5½= The square root of 5 Determining Phi with Trigonometry and LimitsPhi can be related to Pi through trigonometric functions:
Phi can be related to e, the base of natural logs: Ø
= e ^ arcsinh(.5) It can be expressed as a limit:
Other unusual phi relationshipsThere are many unusual relationships in the Fibonacci series. For example, for any three numbers in the series Ø(n-1), Ø(n) and Ø(n+1), the following relationship exists: Ø(n-1) * Ø(n+1) = Ø(n)2 - (-1)n ( e.g., 3*8 = 52-1 or 5*13=82+1 ) Here's another: Every nth Fibonacci number is a multiple of Ø(n) Given 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765 (Every 4th number, e.g., 3, 21, 144 and 987, are all multiples of Ø(4), which is 3) (Every 5th number, e.g., 5, 55, 610, and 6765, are all multiples of Ø(5), which is 5) |
