Thursday, June 6, 2013

The Golden Rectangle/Ratio


The faces of people contain set measurements that follow a set ratio; the golden ratio. These measurements tie-in with the Fibonacci sequence and the Lucas numbers. By using either set (not both at the same time) a specific rectangle is formed that when the longer side is divided by the shorter side, the answer is approximately 1.618...; the golden ratio. Within the overall rectangle, a square can be cut which in result creates another rectangle whose ratio approximately matches the larger rectangle. By using this method on the face helps to reveal some features of the face. For example, a square in the initial rectangle reveals the top of the nose, right before the bridge to the rest of the nose. Within the next rectangle (the one that was made when the square was set) a square can be made to reveal the edge of one of the nostrils. Obviously, not everyone’s face is the same but some faces do share the same number set.


Number Set: Fibonacci




Number Set: Lucas






Number Set: Fibonacci




Research:


Jeremy Stewart
Math
Research Paper
May 31, 2013



The golden ratio; a number used to approximate a curve. Since the renaissance era, people have used this number to hone their techniques in art and architecture. For example, Leonardo Da Vinci applied the golden ratio to many of his paintings. One of these paintings was The Last Supper. From the placement of each of Jesus’ disciples, to the placement and angles of the tables, and the placement of the windows. Another piece that Leonardo made was the Mona Lisa. The curvature in her head aligns with a section in the golden angle. The people in Greece had a letter that represented this number; phi (φ).

In Pisa, Italy, there was a man named Leonardo Pisano Bigollo (also known as Fibonacci). He had constructed a sequence of numbers that when the previous number is added to the following number, the next number in line would be made. Oddly enough these numbers relate heavily to the golden ratio. When the following number was divided by the previous number, the result would approach the golden ratio, but would never actually meet the ratio. For example, the first few numbers to the fibonacci sequence are 1,1,2,3,5,8,13,21, and 34. 2 divided by 1 is 2, 3 divided by 2 is 1.5. When 5 is divided by 3 it becomes 1.6 and the 6 repeats. This number gets close but isn’t there. The golden ratio is 1.618. 21 divided by 13 is 1.615. Again, the number comes close but doesn’t meet the ratio. The was another person who also had constructed a sequence of numbers that came close to the golden ratio but never actually reached the golden ratio either. His name was François Édouard Anatole Lucas. He created his set of numbers in relation to Fibonacci’s. A few of Lucas’ numbers are 2,1,2,3,4,7,11,18, and 29. Both sets of numbers infinitely continue through adding the previous number to the following to get the next number in line.

Nature follows this same method. Many plants abide by the same rulings. A flower and its petals, the branches of a pine cone, the leaves on a stem of many plants, each follow this idea. But on these plants, the sections of them that follow these numbers tend to spiral. This spiral, is created through relativity of the golden ratio. For example, two 1x1 squares are place side by side, creating a rectangle. The next square is 2x2 and is placed along the longer edge of the two squares, creating another rectangle with approximately the same rational proportions. One more square, one that is 3x3, is placed along the longer edge of the newer rectangle which, like the prior case, created another rectangle whose side ratio doesn’t exactly match the previous rectangle but is close. When the longer side is divided by the shorter side, the quotient is close to that of the golden ratio, but it never actually meets.

The golden ratio is a number that not only never ends, but it also doesn’t repeat. φ, the greek letter that represents the golden ratio, is equal 1+(1/φ). This is the same as saying that φ is equal to 1+(1/1+(1+... because when φ is placed into the equation, it starts to repeat and never end, although the resulting number(s) never repeat. Phi is a number that is greater than 1 but less than 2.




Citations:
1) "Fibonacci Numbers, the Golden Section and the Golden String." Fibonacci Numbers, the Golden Section and the Golden String. Dr. Rob Knott, 1996. Web. 31 May 2013. <http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html>.

2) Hart, Vi. "Doodling in Math: Spirals, Fibonacci, and Being a Plant." Khan Academy. Salman Khan, 2006. Web. 31 May 2013. <https://www.khanacademy.org//>.