Fibonacci Sequences and the Golden Ratio

Carl WozniakNorthern Michigan University

Who was Fibonacci?

• Leonardo da Pisa (1170-1240)– First true mathematician

since the Greeks– Liber Abbaci (Book of

Calculation, 1202) introduced the nine numerals and the concept of zero to Europe

Who was Fibonacci?

• In the same book Fibonacci presented a word problem concerning breeding rabbits.

The rabbit problem

• “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?”

The rabbit problem

• So, in five months we have:

12358 pairs

We can continue the sequence123581321345589144…

Notice that each number is equal to the sum of the previous two numbers. This is the Fibonacci Sequence.

The really neat thing is that we find these numbers in many places in nature.

Fibonacci numbers in nature

• Flower petals– lilies and iris have 3 petals; buttercups have 5

petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.

Fibonacci numbers in nature

• Seed heads– Counting along the spirals

of seed heads normally leads to a Fibonacci number.

Fibonacci numbers in nature

• Pine cones– Pine cone scales are also

normally arranged in a Fibonacci spiral

Fibonacci numbers in nature

• We also find Fibonacci numbers in:– The scales of a pineapple– The number of leaves around the circle of the

stem– The number of leaves until another leaf is

directly above the leaf where we started counting

– About 90% of all plants exhibit some form of Fibonacci sequencing

So why are they there?

• What are your thoughts?

So why are they there?

• Reasons others have given– In the case of plants, the arrangement maximizes

the exposed area of each leaf– Provides maximal surface to continue growth

(coiled shell growth)

So why are they there?

• Reasons others have given– Maximizes space in seed heads

The Golden Ratio

• Divide each number in the Fibonacci sequence by the number immediately preceding it.

• 1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538...

• You find that you get closer and closer to the number 1.618….

• The ratio of 1.618:1 is the Golden Ratio, and it is also frequently found, not only in nature, but in human constructions.

The Golden Ratio

• The Golden Ratio is also known as the Golden Mean, Golden Section and Divine Proportion. It is a ratio or proportion defined by the number Phi ( = 1.618033988749895... )

• In the following illustration, A is to B as B is to C. This occurs only where A is 1.618 ... times B and B is 1.618 ... times C.

Aesthetics

• Which of the following rectangles do you find most appealing?

Aesthetics

• Well, how about this grouping?

The Golden Ratio in art

In daVinci’s The Last Supper In the front view of the Acropolis

In the construction of a violin

The Golden Ratio in nature

• In a nautilus shell, each subsequent chamber is approximately 1.68 times larger than the last.

The Golden Ratio in you

• You can find a number of instances in your own body that approximate phi

daVinci’s Vitruvian Man

The Golden Ratio in you

• The lengths of your finger joints• The distance from the floor to your navel

relative to your height• Front two incisors height to width• Ratio of forearm and hand length