Fibonacci Sequence and the Golden Ratio

Robert FarquharMA 341

Who Was Fibonacci?European mathematician 1175-1250Real name Leonardo of Pisa.Author of Liber abaci or Book of the Abacus.Remembered today because of Edouard Lucas

What Is the Fibonacci Sequence and Why Is It Significant?

Generalized sequence of first two positive integers and the next number is the sum of the previous two, i.e. 1,1,2,3,5,8,13,21,…Has intrigued mathematicians for centuries. Shows up unexpectedly in architecture, science and nature (sunflowers & pineapples).

Sunflower Seedhead andthe Fibonacci Sequence

Fibonacci Sequence Cont’d

Has useful applications with computer programming, sorting of data, generation of random numbers, etc.

Fibonacci Sequence and the Golden Ratio

A remarkable property of the sequence is that the ratio between two numbers in the sequence eventually approaches the “Golden Ratio” as a limit.1/1=1 2/1=2 3/2=1.5 5/3=1.66678/5=1.6 13/8=1.625 21/13=1.6154

Fibonacci Sequence and the Golden Ratio

Golden RatioFamous Irrational number phi—1.61803398…Only positive # that becomes its own reciprocal by subtracting 1Used extensively by Ancient Greeks in architecture

Golden Ratio Continued

Ratio also shows up in many famous paintingsDa Vinci studied extensively the ratio and body proportions.

Summary

Fibonacci Sequence has been around for hundreds of years.Golden Ratio has been used for thousands of years.Both concepts are still in use today and continue to interest us.

ReferencesGardner, Martin (1979). Fibonacci and Lucas Numbers. In Mathematical Circus (pp. 152-160), New York:Alfred KnopfGardner, Martin (1961). Phi: The Golden Ratio. In Mathematical Puzzles & Diversions (pp.89-100), New York: Simon and Schuster.Web Page “The Golden Ratio” by Steve Blacker, Jeanette Ploanski and Marc Schwach, http://www.geom.umn.edu/~demo5337/s97b/fibonacci.html