The Fibonacci Sequence and The Goldens Ryan Breaud and Malerie Bulot

Fibonacci Sequence: Discovery Leonardo Pisano Fibonacci (1170 – 1250) was an Italian mathematician who is known for discovering the self-named Fibonacci Sequence when presented with a problem and spreading the Hindu-Arabic number system through his book Liber Abaci.

Fibonacci Sequence: Discovery Problem: how many pairs of rabbits would you have in a year given certain conditions? The enclosure is empty. A pair of newly born rabbits (Pair A) are placed inside the enclosure. 0, 1 Pair A takes one month to mature. 0, 1, 1 Pair A then gives birth to Pair B 0, 1, 1, 2 Pair A gives birth to Pair C while Pair B matures. 0, 1, 1, 2, 3 Pair A gives birth to Pair D while Pair B give birth to Pair E. Pair C matures. 0, 1, 1, 2, 3, 5 Pair A gives birth to Pair F, Pair B give birth to Pair G, Pair C give birth to Pair H. Pair D and Pair E mature. 0, 1, 1, 2, 3, 5, 8 If the process continued, there would be 144 pairs of rabbits at the end of month 12.

Fibonacci Sequence: Discovery A red rectangle designates a newborn pair, which doesn't produce offspring until the second month.

Fibonacci Sequence: Discovery Fibonacci noticed the numbers produced when doing this was a pattern. Fn = Fn-1 + Fn-2 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181,…

Golden Ratio: Discovery The Golden Ratio, commonly represented by the Greek letter φ, is approximately equal to 1.618… φ is the result of line segments when line segment AB is divided into two segments AC and BC such that the ratio of AB to the length of larger line segment AC is the same as the ratio for the length of the larger line segment AC to the length of the smaller segment BC. A C B

Golden Ratio: Discovery The Golden Number, φ , can be seen throughout history in tandem with the use of the Fibonacci Sequence. The Ancient Egyptians have used these to construct the Great Pyramids, the Greeks with the Parthenon, and are even evident in Da Vinci’s “The Last Supper”.

Golden Ratio and Fibonacci Sequence: Historical Occurrences

Golden Rectangle The golden rectangle is often referred to as the most perfectly shaped rectangle. The golden rectangle is one that has the following ratio of its length and width: w / l = l / (w + l) The closer the ratio of the rectangles length and width is to the golden number, the more appealing the rectangle is to the human eye.

Golden Rectangle For example, a rectangle with the ratio of 34:21 is a close to perfect golden rectangle, seeing as 34/21 ≈ 1.61705882352941 ≈ φ.

Golden Rectangle

Golden Ratio and Fibonacci Sequence: Relation The relationship between the Fibonacci numbers and the golden ratio can be seen by taking the ratio of every two consecutive Fibonacci numbers. As the number of terms in the sequence approach infinity, this ratio ultimately approaches the golden number.

Golden Ratio and Fibonacci Sequence: Relation 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, … EXAMPLE: 987/610 = 1.6180327868852.

Golden Ratio and Fibonacci Sequence: Relation Furthermore, the successive powers of φ obey the Fibonacci recurrence: This identity allows any polynomial in φ to be reduced to a linear expression. For example: Not a special property of φ however. For given coefficients a, b such that x satisfies the equation.

Golden Spiral First, to define the shape of our image, we’re going to draw a golden spiral. First, divide the width of the canvas by the golden ratio, to find the ‘golden section’. Draw a vertical line from this point and draw a quartercircle in the larger of the two portion. Next, divide the height by the golden ratio to find the golden section. Draw a horizontal line from this point to the vertical line, and draw a quarter-circle in the larger of the two portions. Last, take the smaller of these portions and find its golden section (as before). Draw a quarter-circle in this. Repeat three or four more times.

Golden Ratio and Fibonacci Sequence: Historical Occurrences http://britton.disted.camosun.bc.ca/goldslide/goldanim1.gif

Golden Ratio and Fibonacci Sequence: Biological Occurrences

Works Cited “Digital Art”. <http://www.digitalartsonline.co.uk/tutorials/index.cfm?FeatureID=1718> <http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm> “Golden Ratio”. <http://en.wikipedia.org/wiki/Golden_ratio#Timeline> “Phi”. <http://goldennumber.net/>