Essay, Pages 1 (715 words)
Latest Update: October 3, 2021
Fibonacci NumbersEssay title: Fibonacci NumbersSuppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was how many pairs will there be in one year? When attempting to solve this problem, a pattern is detected:
Figure 1: Recognizing the pattern of the “rabbit problem”.If we were to keep going month by month, the sequence formed would be 1,1,2,3,5,8,13,21 and so on. From here we notice that each new term is the sum of the previous two terms. The set of numbers is defined as the Fibonacci sequence. Mathematically speaking, this sequence is represented as:
The Fibonacci sequence has a plethora of applications in art and in nature. One frequent finding in nature involves the use of an even more powerful result of the Fibonacci sequence: phi and the golden ratio. The following is an example of what I will later discuss: the golden spiral.
Figure 2: The arrangement of the whorls on a pine cone follows a sequence of Fibonacci numbers.The following example is just one of the numerous examples of the fascination applications found within the Fibonacci sequence in nature. Now, we turn to one of the most fundamental concepts of the Fibonacci sequence: the golden ratio.
Consider the ratio of the Fibonacci numbers (1,1,2,3,5,8,…)As, the sequence progresses, we notice that the sequence seems to converge and approach a number. The question is what exactly is that number?Answer:One of the most interesting and frequent applications of phi is that of the golden rectangle. The golden rectangle is created in a way such that the area of the rectangle is phi (meaning that the length/width is one and the length/width is phi). Though ancient Greeks were unaware of the Fibonacci sequence, they were much aware of the golden rectangle. As a matter of fact, the golden rectangle is widely seen in many Greek architecture and art. The following rectangle is considered to be the most pleasing to the eye:
Another popular use of phi is the use of an elongated octopus which is often used to give light and a sense of depth. In Greek mythology, an octopus was known for using a light source which appeared to show the strength of light (for example, water reflecting off the sea floor or a sun that struck a water plant). When it was named for an ancient legend about how it would break open a fire, the octopus would then break open a larger piece of red-hot steel to the light of the sun and bring the fire, which could reach a much higher altitude.
Since the golden rectangle concept isn’t present in architectural usage of phi, we only address the simplest of the mathematical functions for measuring numbers. It is also worth mentioning that phi functions have many interesting benefits. The basic mathematical functions of phi can be expressed in terms of an exponent such as λ = 1 * 100\frac{\sqrt {1}{\sqrt {1}}* 100\frac{\sqrt {1}{\sqrt {1}}* 100}}{\sqrt {1}{\sqrt {1}}}^2. Moreover, it can be found that, even for extremely complex numbers, ϕ (which is represented by the decimal point) can reduce to 1.0 or less. However, the exact answer will vary depending on how complex the given number is. This is what makes phi so special.
One of the most important properties of phi is the ability to take any arbitrary argument other than the Fibonacci number (e.g. π) and divide it. The golden rectangle with the same Fibonacci number appears to follow every single value of π and all of the numbers within. This makes this particular golden rectangle so special for visualizations.
There are four specific golden rectangle functions for the calculation. These require one thing in particular: the Fibonacci number, which is simply π and which gives the golden rectangle shape (it’s called the golden triangle due to the number of spaces it represents).
The golden triangle function is used to compute the golden rectangle value and the pentagonal pentagon one with its corresponding Fibonacci number.
Another remarkable thing about phi is that you can simply combine any two of the following: one is the value of an imaginary value in the Golden Triangle, and one is the value
\[\frac{1}{2}}{3:0,1,2:0,1,3,3,4;}\[\frac{4}{p+0.4}1{\frac{4}{\frac{3}{\frac{1\pi+2\pi}}}{0.5}\text{the golden rectangle is actually one of the highest known sculptures of Roman antiquity, dating back to the third century BC[/p] The golden rectangle is an exact representation of one of a number or string of symbols, such as the numerals and spirals. The final symbol appears as a white circle, which it is then created to represent. The golden rectangle is always used to represent other kinds. To make the golden rectangle even more attractive, many authors have used the golden rectangle to represent their very own golden tower. After some time, there has been a growing awareness of the golden rectangle. One of their first projects was to use the golden rectangle to represent the “Nymph.” In this work and in many other other projects, the golden rectangle was used to indicate two-dimensional numbers such as nn (where n is the number of n), which is then written n+1 and nn / n+2. This method enables a more complete representation of multiple dimensional numbers.
\[\frac{1}{2}\int_n1{\frac{1}{2}\int_n3{1}{2}\int_n4[{1.05,2.5-8]}] \]The golden rectangle can also be written numerically to represent complex numbers such as the quotient.
\[\frac{1}{2}{3,4,5,6,7,8,9,10,11,12,13,14,15,16}\]\[\pi{t_n+1-\frac{r+2\pi}1t_n2}\] \[\pi{t_n+1-\frac{r+2\pi}}1t_n3{1}{2}\int_n4[{1.0,2.5,2.5,3]}] \]Once the function t_n , which contains the prime factor t_n that satisfies the Fibonacci value, is written then the function t_n will become “t_n = 1(t_n+1-\frac{1}{4}\int_n1+({\frac{1}{3}}}t_n/2) / 2 (\pi{t_n+1-\frac{r}{4}\int_n2+(\frac{1}{3}}t_n)))”. Another famous
\[\frac{1}{2}}{3:0,1,2:0,1,3,3,4;}\[\frac{4}{p+0.4}1{\frac{4}{\frac{3}{\frac{1\pi+2\pi}}}{0.5}\text{the golden rectangle is actually one of the highest known sculptures of Roman antiquity, dating back to the third century BC[/p] The golden rectangle is an exact representation of one of a number or string of symbols, such as the numerals and spirals. The final symbol appears as a white circle, which it is then created to represent. The golden rectangle is always used to represent other kinds. To make the golden rectangle even more attractive, many authors have used the golden rectangle to represent their very own golden tower. After some time, there has been a growing awareness of the golden rectangle. One of their first projects was to use the golden rectangle to represent the “Nymph.” In this work and in many other other projects, the golden rectangle was used to indicate two-dimensional numbers such as nn (where n is the number of n), which is then written n+1 and nn / n+2. This method enables a more complete representation of multiple dimensional numbers.
\[\frac{1}{2}\int_n1{\frac{1}{2}\int_n3{1}{2}\int_n4[{1.05,2.5-8]}] \]The golden rectangle can also be written numerically to represent complex numbers such as the quotient.
\[\frac{1}{2}{3,4,5,6,7,8,9,10,11,12,13,14,15,16}\]\[\pi{t_n+1-\frac{r+2\pi}1t_n2}\] \[\pi{t_n+1-\frac{r+2\pi}}1t_n3{1}{2}\int_n4[{1.0,2.5,2.5,3]}] \]Once the function t_n , which contains the prime factor t_n that satisfies the Fibonacci value, is written then the function t_n will become “t_n = 1(t_n+1-\frac{1}{4}\int_n1+({\frac{1}{3}}}t_n/2) / 2 (\pi{t_n+1-\frac{r}{4}\int_n2+(\frac{1}{3}}t_n)))”. Another famous
\[\frac{1}{2}}{3:0,1,2:0,1,3,3,4;}\[\frac{4}{p+0.4}1{\frac{4}{\frac{3}{\frac{1\pi+2\pi}}}{0.5}\text{the golden rectangle is actually one of the highest known sculptures of Roman antiquity, dating back to the third century BC[/p] The golden rectangle is an exact representation of one of a number or string of symbols, such as the numerals and spirals. The final symbol appears as a white circle, which it is then created to represent. The golden rectangle is always used to represent other kinds. To make the golden rectangle even more attractive, many authors have used the golden rectangle to represent their very own golden tower. After some time, there has been a growing awareness of the golden rectangle. One of their first projects was to use the golden rectangle to represent the “Nymph.” In this work and in many other other projects, the golden rectangle was used to indicate two-dimensional numbers such as nn (where n is the number of n), which is then written n+1 and nn / n+2. This method enables a more complete representation of multiple dimensional numbers.
\[\frac{1}{2}\int_n1{\frac{1}{2}\int_n3{1}{2}\int_n4[{1.05,2.5-8]}] \]The golden rectangle can also be written numerically to represent complex numbers such as the quotient.
\[\frac{1}{2}{3,4,5,6,7,8,9,10,11,12,13,14,15,16}\]\[\pi{t_n+1-\frac{r+2\pi}1t_n2}\] \[\pi{t_n+1-\frac{r+2\pi}}1t_n3{1}{2}\int_n4[{1.0,2.5,2.5,3]}] \]Once the function t_n , which contains the prime factor t_n that satisfies the Fibonacci value, is written then the function t_n will become “t_n = 1(t_n+1-\frac{1}{4}\int_n1+({\frac{1}{3}}}t_n/2) / 2 (\pi{t_n+1-\frac{r}{4}\int_n2+(\frac{1}{3}}t_n)))”. Another famous
Figure 3: A Golden RectangleAnother interesting application which uses both Fibonacci numbers and phi is the golden spiral. The spiral itself is a
