Fibonacci Sequence

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# Fibonacci Sequence - PowerPoint PPT Presentation

Fibonacci Sequence. Bethany Fleischer, Clinton Weigel , and Jonathan Beard. Who was Fibonacci ?. A Great mathematician in the middle ages Born in Pisa~1175 AD Major contributions in arithmetic, algebra, number theory, and decimal system. Bee Example. YouTube-Fibonacci. Rabbits.

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### Fibonacci Sequence

Bethany Fleischer, Clinton Weigel, and Jonathan Beard

Who was Fibonacci ?
• A Great mathematician in the middle ages
• Major contributions in arithmetic, algebra, number theory, and decimal system
Bee Example
Rabbits
• How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?
Rabbits
• A born male-female pair of rabbits start.
• Rabbits can mate when they are 1 month old.
• A female produces a pair 1 month after mating.
• And so on…
Fibonacci Sequence
• The rule for the sequence?

1, 1, 2, 3, ?, ?,...

• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...
What is the Golden Ratio?
• The Golden Ratio is a unique number, approximately 1.618033989.
• It is also known as...
• the Divine Ratio
• the Golden Mean
• the Golden Number
• the Golden Section
• Its unique mathematical properties have resulted in a very long and influential history.
Golden Ratio
• This sequence converges, that is, there is a single real number which the terms of this sequence approach more and more closely, eventually arbitrarily closely.
• Compute the ratio of Fibonacci numbers:

2 ÷ 1 = 2

3 ÷ 2 = 1.5

5 ÷ 3 = 1.6666...

8 ÷ 5 = 1.6

13 ÷ 8 = 1.625

21 ÷ 13 = 1.61538...

• The Fibonacci Sequence is an infinite sequence, which means it goes on forever, and as it develops, the ratio of the consecutive terms converges (becomes closer) to the Golden Ratio, ~1.618.
• For example, to find the ratio of any two successive numbers, take the latter number and divide by the former. So, we will have: 1/1=1, 2/1=2, 3/2=1.5
Golden Ratio
• As we can see, the ratio approaches the Golden Ratio, which is also known as the Golden Section and Golden Mean. Even though we know it approaches this one particular constant, we can see from the graph that it will never reach this value.
Fibonacci Equation
• Fibonacci Equation...
The Golden Ratio in Humans
• Dr Stephen Marquardt, a former plastic surgeon, has used the golden section and some of its relatives to make a mask that he claims that is the most beautiful shape a human face can ever have, it used decagons and pentagons as its function that embodies phi in all its dimensions.
The Golden Ratio in Nature
• The petals of the different flowers also contain the Fibonacci Numbers. The examples are that the iris has 3 petals, the buttercup has 5 petals, the delphinium has 8 petals, etc..
The Golden Ratio in Nature
• Passion Flower:
• the 3 sepals that protected the bud are outermost
• then 5 outer green petals followed by an inner layer of 5 more paler green petals
The Golden Ratio in Art
• Da Vinci, a sculpture, painter, inventor and a mathematician, was the first one who called Phi the Golden Ratio. And scientifically, her face actually appears in a golden rectangle, which also makes her face appear more beautiful to human eyes.
The Golden Ratio in Architecture
• The Golden Ratio has appeared in ancient architecture. The examples are many, such as the Great Pyramid in Giza, Egypt which is considered one of the Seven World Wonders of the ancient world, and the Greek Parthenon. The ancient Egyptians and Greeks knew about the magic of the Golden Ratio.
• Half of the base, the slant height, and the height from the vertex to the center create a right triangle. When that half of the base equal to one, the slant height would equal to the value of Phi and the height would equal to the square root of Phi.