1. THE GOLDEN RATIO NIKI DEMONEY MAT 113 - 02 PROFESSOR SOLLITTO

2. Description/Goals What is the Golden Ratio History Mathematics Nature Art & Design

3. What is the Golden Ratio  Also know as the Devine Proportion or Golden Mean  Ratio that gives an irrational number of 1.61803399… also know as Phi  Best shown by dividing a line segment into two segments where the ratio of the whole line segment to the longer segment is the same as the ratio of the longer segment to the shorter segment.

4. History of the Golden Ratio  Pythagoras, Greek Philosopher and Mathematician, 500BC.  Fascinated with beauty, harmony and symmetry  Experimented with different sizes of strings to produce the most pleasing sound  Discovered the concept of the Golden Ratio

5. Golden Ratio in Mathematics  Golden Geometrical Shapes  Golden Rectangle  Greeks believed most visually pleasing rectangle  Ratio of length to the width  Cornerstone of beauty in Greek Philosophy

6. Golden Ratio in Mathematics  Fibonacci Numbers or Sequence  Mathematician studying population in rabbits reproduction “If one male and one female reproduced another pair of rabbits (1 male & 1 female) and each pair reproduced another pair. How many would there be after a year?”

7. Golden Ratio in Mathematics  1, 1, 2, 3, 5, 8, 13, 21, 34, 55…  Rule of Fibonacci numbers The sum of a number is equal to the sum of the previous two numbers Fn + 1 = Fn + Fn1 (8 = 5 +3) (34 = 21 + 13)

8. Golden Ratio in Mathematics  Relationship between Golden Ratio and Fibonacci Numbers Ratios of the successive numbers equal Phi 1.61803… when calculated Fibonacci Rectangle: sequence is used to create a Golden Rectangle Golden Spiral: created from the Fibonacci Rectangle

9. The Golden Ratio in Nature  The Golden Spiral Nautilus Shell Sunflowers, Coneflowers & Pinecones

10. The Golden Ratio in Nature  Fibonacci Numbers Petals of plants Family Tree of Honeybees: Male bees are developed from an unfertilized queen bee egg. 1 Parent. His mother would have 2 parents giving him 2 Grand-Parents. He would have 3 Great Grand-Parents. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

11. The Golden Ratio in Art & Design  Architecture Parthenon, Athens Greece, 430BC. Dimensions of the Golden Rectangle Greeks were founders of the ratio. Speculated that they used it in most their designs The Great Egyptian Pyramid Giza Golden Pyramid: ratio of the slant height to the distance from its center is 1.61804…, different from Phi by one unit in the 5 th decimal place

12. The Golden Ratio in Art & Design  Architecture cont. The College of Engineering at California Polytechnic State University Designing a new Engineering Plaza based on Fibonacci numbers and Golden Spiral The United Nations Building in NYC Designed using the Golden Rectangle

13. The Golden Ratio in Art & Design  Art: Paintings & Sculptures Leonardo De Vinci had a passion for mathematics Golden Section: umbilicus divides us into the golden ratio Other Artists: Dali, Rembrandt, Michelangelo and Raphael Phidias, famous Greek sculptor Phi is named after for using it in his work Created the sculptures in the Parthenon

14. The Golden Ratio in Art & Design  Design Credit cards, photographs, boxes, posters and modern furniture have golden proportions  Music Mozart’s sonatas and Beethoven’s Fifth Symphony divided up into golden sections Modern composers use the golden ratio and Fibonacci numbers when writing ballots

15. Conclusion  Statistical Study Gustav Fechner, German psychologist Experimented with group of different rectangles with different ratios 35% chose the golden rectangle However nobody picked the golden rectangle as the least attractive Although the golden ratio may not be the absolute answer to beauty it is still a fascinating principal that has inspired many artists and mathematicians.

16. Sources  Bennett, Jeffrey, and William Briggs. Using and Understanding Mathematics: A Quantitative Reasoning Approach. 3rd ed. 2005. 645-651.  Posamentier, Alfred. Math Wonders to Inspire Teachers and Students.Virginia: Association for Supervision & Curriculum Development, 2003.  Pheasant, Stephen. Bodyspace: Anthropometry, Ergonomics and the Design of Work. London: CRC Press, 1996. <http://site.ebrary.com/lib/empire/Doc?id=10057184 &ppg=21>.  “Golden Ratio”. Wolfram Research Inc. 1999. CRC Pres LLC. 21 Mar. 2005.http://mathworld.wolfram.com/GoldenRatio.html  “Golden Ratio, Fibonacci Sequence”. Ask Dr. Math. 1994. The Math Forum @ Drexel University. 21 Mar. 2005 <http://mathforum.org/dr.math/faq/ faq.golden.ratio.html>.  Knott, Dr. Ron. “Fibonacci Numbers”. Department of Mathematics, Surrey University. 1996. 12 Oct. 2004.  Britton, Jill. “Golden Section in Art and Architecture”. 19 Feb 2005 Department of Mathematics, Camosun College. <http://ccins.camosun.bc.ca/~jbritton/goldslide/ jbgoldslide.htm>.