presented by angie schirck-matthews broward college davie
TRANSCRIPT
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Leonardo of Pisa
• Leonardo of Pisa AKA Fibonacci was a 12th century mathematician most famously know for introducing the Fibonacci Sequence.
• The sequence was discovered as an answer to the following riddle:• “How many pairs of rabbits would there be in a year if it
was assumed that every month each pair produces a new pair, which begins to bear young two months after its own birth?”
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Rabbits
• Fibonacci's question led to the following sequence known as the Fibonacci Sequence:• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
• Each term of the sequence is the sum of the previous two terms.
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The most interesting sequence in the world!
The Fibonacci Sequence
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What’s so special about this sequence?
• I’m glad you asked!
• Eerily, many things in nature adhere to this sequence of numbers.
• Things that are visually pleasing follow this sequence.
• Things that are acoustically pleasing conform to this sequence.
• The human body yields to this sequence.
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Fibonacci in Nature
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Spirals
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Fibonacci Spiral
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Flower petals
LUCKY CLOVER!
1
2
1 2
34
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Trees and Plants
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Honey Bee Genealogy
8
5
3
2
1
1
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Ocean Life
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Ocean waves
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Fibonacci in Art
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The Golden Ratio
• The further you go out in the Fibonacci sequence, the ratio of consecutive numbers approaches the golden ratio (called )
The Golden Ratio 1.618
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A Golden Rectangle
• A rectangle with length and width consecutive Fibonacci numbers is called a golden rectangle.
• Artists and Architects knew this was an aesthetically pleasing shape and it was incorporated in ancient art and architecture.
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Art
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Art
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Architecture
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Donald in Mathemagicland 1959
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Fibonacci in Music
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Keys on a Piano
• 1 octave
• 2, 3 (groups of black)
• 5 black keys
• 8 white keys
• 13 keys total
• 5th note in a major scale is dominate, it is the 8th note of the 13 in the octave
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Consonance and Dissonance
• “Consonance is associated with sweetness, pleasantness, and acceptability; dissonance is associated with harshness, unpleasantness, or unacceptability”(Wikipedia).
• Fifth is most consonant sound in music.
• Circle of fifths used to compose, harmonize, and build chords
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Intervals in MusicFibonacci Ratio Note in Major Scale Musical Relationship
1/1 A Root
2/1 A Octave
2/3 D Fourth
2/5 F Aug. Fifth
3/2 E Fifth
3/5 C Minor Third
3/8 E Fifth
5/2 C# Third
5/3 F# Sixth
5/8 C# Third
8/3 D Fourth
8/5 F Aug. Fifth
It is interesting to note that the fourth is sometimes considered consonant and sometimes dissonant and the sixth is called an “imperfect consonance.”
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Types of Scales• Pentatonic – 5 notes (folk tunes and rock music)
• Diatonic Scale – 8 notes (many classic melodies)
• Chromatic Scale – 13 notes (symphonies and contemporary music)
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Composers
• Bach Little Fugue in G minor
• Why is this piece so pleasing to the ear?
• Some of the best composers know and use intervals of Fibonacci Numbers• Theme 34 measures
• Slow, soft 55 measures (55/34 )
• Fast, loud 21 measures
• Repeat of theme 34 measures (34/21 )
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Fibonacci and the Human Body
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Human Beauty
Brad Pitt Danny Devito
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Human Beauty
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Human Beauty
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Me
I may not be Brad Pitt, but I’m not Danny Devito either!
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Fingers and Arms
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Body and Face Proportions
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Heartbeat and Ears
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Other Places Fibonacci is Found• The face of a clock
• Squares of Fibonacci numbers• 12 = 11• 12 + 12 = 1 2 • 12 + 12 + 22 = 2 3• 12 + 12 + 22 +32 = 3 5 • 12 + 12 + 22 +32 + 52 = 5 8
• Pascal’s Triangle
• Space
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Other Places Fibonacci is Found
• Weather • Fractals
• The list goes on, but my time does not!
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