The Golden Ratio is 1.61803398874989484820… going on endlessly without any sort of pattern, an irrational number. As stated in The Golden Ratio and The Fibonacci Numbers, “while
the Golden Ratio may not be as important as other mathematical constants, it does have its claim to fame and does have its own unique properties” (Ross). The Golden Ratio was found utilizing a formula of Fibonacci numbers. So the higher the ratio between consecutive Fibonacci numbers, the closer it gets to the golden ratio. A good example of this is the fact that 3/5=1.666, 13/21 is 1.625 and 144/233 is 1.618. The higher the two Fibonacci numbers in the ratio, the closer it is to the golden ratio. The easiest way of solving for the Golden Ratio is (sqrt.of5 + 1)/2. Inputting this into a calculator yields a number extremely close to the Golden Ratio. The Golden Ratio is only one of the names for this figure. It can also be called the Golden mean, divine proportion and many other names. This ratio has subconsciously been used to create many of man kinds greatest works of art. It is widely believed that this is because the human mind searches everywhere for some sort of pattern, or order in which to categorize things This is in order to make our understanding of the world and our environment more efficient. Some examples, of these man-made structures include The Great Pyramids. While many of these are considered to be close, but not quite exactly the Golden Ratio. Some of the reasons that these creations use the Golden Ratio is because the creators/designers of these architectural masterpieces realized that they were more aesthetically pleasing following these concepts of the Golden Ratio. The final truly interesting fact of the Golden Ratio can be found in Mathematics of Phi, 1.618, the Golden Number. It states that “If you square Phi, you get a number exactly 1 greater than itself: 2.618…, or Φ² = Φ + 1. If you divide Phi into 1 to get its reciprocal, you get a number exactly 1 less than itself: 0.618…, or 1 / Φ = Φ – 1.”
Fibonacci Neighbors is really a simple concept, it relates to how the Fibonacci numbers and sequences work together. The Neighbors are just the numbers that are next to each other. For example, with flowers the numbers in the series they sit next to each other in the sequence, mostly in the curvature of the petals, are neighbors. The Neighbors are then added together to get the next prime number in the sequence. The formula to find both neighbors is: F n +F n-1 =F n+1 . It can be calculated then that, F n =F n+1 -F n-1 . So essentially this formula helps to figure out both of the Fibonacci Neighbors with relative ease and very little algebra. An example of this for the Fibonacci Number 55 would be, 55+ F n-1 =F n+1, knowing that the smaller neighbor of 55 is 34, 55+34= F n+1 Therefore F n+1 =89. The Fibonacci Neighbors of 55 are 34 and 89. This simple equation can be used to figure out both Fibonacci Neighbors and the next Fibonacci Number in the sequence.
You've reached the end of your free preview.
Want to read all 13 pages?
- Winter '15
- Andrew Martino
- Math, Fibonacci, Fibonacci number, Golden ratio, Fibonacci Neighbors