fibonacci sequence & the golden ratio click this button to continue

FIBONACCI SEQUENCE & THE GOLDEN

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MENU SLIDEFibonacci SequenceThe Golden RatioApplicationQuiz

FIBONACCI SEQUENCE

WHAT IS A SEQUENCE?•A sequence is a set of numbers that are in a specific order•An infinite sequence goes on forever, and is described by a certain rule (or rules) so that we can find any term of that sequence

FIBONACCI SEQUENCE The Fibonacci Sequence is the set of these numbers:

0,1,1,2,3,5,8,13,21,34…

FIBONACCI SEQUENCE This sequence can be described as

Where

FIBONACCI SEQUENCE Simply put, this means that the nth term of the sequence is the sum of the two terms before it.

CHECKPOINT!!!!

WHAT IS THE 6TH TERM OF THE FIBONACCI SEQUENCE?

a.6b.2c.5d.10

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ANSWER!!Continue to the next section!

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THE GOLDEN RATIO

THE GOLDEN RATIO This is the Greek letter phi that represents the golden ratio

THE GOLDEN RATIO The golden ratio can be expressed as:

𝜑=1+ 1

1+ 1

1+ 11+…

THIS IS REALLY COMPLICATED!!!!!

THE GOLDEN RATIO So mathematicians have simplified it down to

This is considered the perfect ratio in nature. Thus the name, the “golden” ratio.

THE GOLDEN RATIO The Golden Ratio is also used in art, and artist Leonardo Da Vinci used the Golden Ratio (as well as Fibonacci Numbers) in his paintings.

THE GOLDEN RATIO

Graphic obtained from http://leonardodavincihorseandrider.com/authenticity/leonardo-and-the-golden-spiral/

Paintings by Leonardo Da Vinci

THE GOLDEN RATIOActually, the golden ratio and the Fibonacci sequence are very closely related as

Where n is greater than or equal to 4

THE GOLDEN RATIO Simply stated, this means that, starting with the 4th term of the Fibonacci Sequence, the ratio between two consecutive Fibonacci numbers (with the larger number in the numerator) is approximately the same as the golden ratio!

THE GOLDEN RATIO

And 3/2=1.5

5/3=1.6666666… 8/5=1.6

13/8=1.625 ….

610/377=1.618037…

These numbers are Fibonacci numbers.

As the Fibonacci numbers get bigger, the ratio between them gets closer to the golden ratio.

CHECKPOINT!!!!

WHAT PAINTER USED THE GOLDEN RATIO & FIBONACCI NUMBERS IN HIS PAINTINGS?

a. Leonardo Da Vinci b. Leonardo DiCaprio c. Claude Monet d. Pierre-Auguste Renoir

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APPLICATION

Now this is all fine and dandy, but what does it

mean?

APPLICATION Amazingly, these numbers actually present themselves to us in nature!

APPLICATION

CHECKPOINT!!!

WHICH OF THE FOLLOWING WERE SHOWN IN THE VIDEO AS BEING OBJECTS FOUND IN NATURE THAT DISPLAY THE GOLDEN RATIO/FIBONACCI SEQUENCE?

A. sunflower B. pinecone C. both A&B D. none of the above

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FINAL QUIZ!!!!

WHICH OF THE FOLLOWING IS NOT A FIBONACCI NUMBER? A. 2 B. 21 C. 13 D. 4

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WHICH GREEK LETTER IS USED TO REPRESENT THE GOLDEN RATIO? A. (phi) B. (delta) C. ∑ (sigma) D. π (pi)

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FILL IN THE BLANKS: A SEQUENCE IS A SET OF _____THAT ARE IN A _____ORDER

A. objects, random B. objects, specific C. numbers, random D. numbers, specific

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IN THE GOLDEN RATIO, THE LETTER PHI CAN BE DEFINED AS A. B. C. 1.728303 D.

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THE RELATIONSHIP BETWEEN THE GOLDEN RATIO AND THE FIBONACCI SEQUENCE IS THAT A. Starting with the 2nd term of the Fibonacci Sequence, the ratio between two consecutive Fibonacci numbers (with the larger number in the numerator) is approximately the same as the golden ratio.

B. There is no relationship between the two C. Starting with the 4th term of the Fibonacci Sequence, the ratio between two consecutive Fibonacci numbers (with the larger number in the numerator) is approximately the same as the golden ratio.

D. The sum of two consecutive Fibonacci numbers divided by their average value equals the golden ratio

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POWERPOINT!

REFERENCES Sequences. (n.d.). Sequences. Retrieved October 23, 2013, from http://www.mathsisfun.com/algebra/sequences-series.html

Fibonacci Number. (n.d.). Wolfram Alpha. Retrieved October 25, 2013, from http://mathworld.wolfram.com/FibonacciNumber.html

Nature, The Golden Ratio, and Fibonacci too .... (n.d.). Nature, The Golden Ratio and Fibonacci Numbers. Retrieved November 9, 2013, from http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html

etareaestudios. (2010, March 11). Nature by Numbers. YouTube. Retrieved November 13, 2013, from http://www.youtube.com/watch?v=kkGeOWYOFoA

Da Vinci Horse and Rider. (n.d.). Da Vinci Horse and Rider. Retrieved November 14, 2013, from http://leonardodavincihorseandrider.com/authenticity/leonardo-and-the-golden-spiral/