Fibonacci Sequence and Golden Ratio
Fibonacci Sequence and Golden Ratio
  • Lee Hyun-jun (ST Reporter)
  • 승인 2024.07.08 13:34
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Math is close to those who are students but it's also something we want to keep away from. So, why is math important? You will find out more about it through the article. ................Ed

     Every student in Korea has tried learning math. However, after calculating, memorizing, and doing tests, many may hate numbers, give up math, or are still skeptical about "Why am I learning this?" Math is the closest subject in our daily lives, but for students, it is obvious that they want to get far away from math. Then, why is math so important, and why are we learning this?

1 1 2 3 5 8 13 21 34 55 ··· Can you find the pattern?

     Guess you have found it, but let me rephrase the pattern, it is a sequence in which each number is the sum of the two preceding ones.

1+1=2, 1+2=3 ··· And so on.

     Then how about square each number? (Obviously, who hates squared numbers)

1 1 4 9 25 64 169 ···

1+1=2, 1+4=5, 4+9=13

     Again, can you see the pattern? If you can’t find it, let's arrange more.

9+25=34, 25+64=89

     Now you can see the sum of squared one is the odd number of elements in the original Fibonacci sequence. Now, sum the squared elements from the beginning.

1+1+4=6, 1+1+4+9=15, 1+1+4+9+25=40 ···

     We can also find the pattern here. Can you see it?

2x3=6, 3x5=15, 5x8=40, 8x13=104

     Yes, it is the multiplication of Fibonacci numbers following the Fibonacci patterns. This time, we will divide the following numbers.

3/2=1.5, 5/3=1.666, 8/5=1.6, 13/8=1.625, 21/13=1.615, 34/21=1.619

     These numbers are convergent to 1.618033··· which is called the 'Golden ratio.' Let’s figure out how the golden ratio attracts so many artists, musicians, designers, and marketers in everyday life.

     The Fibonacci sequence is an everyday formulation that is even seen in nature. Among them, a representative and most common example is the number of petals. Most of the number of petals consists of Fibonacci numbers. A lily has three petals, a rose moss has five petals, and a cosmos has eight petals. As such, petals usually consist of three, five, eight, and 13 petals. The number of petals is the arrangement of numbers in which the sum of the first two numbers is the last. This is related to the way petals grow. Whenever a new petal is formed, it grows by rotating at an almost golden ratio angle, forming a Fibonacci sequence. Like petals, leaves follow the Fibonacci sequence. This is called the 'phyllotaxis,' which is the arrangement of leaves attached to the leaf stem of a plant. There are two phyllotaxis: 'alternate phyllotaxis' and 'verticillatum.' Among them, alternate phyllotaxis has spirals of leaves around the stem. This is called a 'spiral phyllotaxis,' and the number of rotations until a leaf appears in the same way is when the first leaf expressed as the ratio of the number of rotations divided by the number of leaves. For example, Cherry blossoms have 2/5 phyllotaxis, and roses have 3/8 phyllotaxis.

     One of the things humans made by using the Fibonacci sequence is the piano. One octave on the piano consists of eight scales: do, re, mi, fa, sol, la, si, and do. Eight white keys and five black keys come together to make 13 negative sounds. Among them, two black keys and three white keys are a set, and three black keys and five white keys are a set. You can find a numerical array of 2, 3, 5, 8, and 13. Therefore, the octave of the piano is also a Fibonacci sequence.

     Then asking again, why math? Why are students so close to math? To sum up, math is the field of calculation, application, and inspiration: finding patterns in the disorderly world of numbers, finding logic in illogic, and establishing rules by synthesizing and organizing them. Try to remember that math is not just about finding x, but discovering wh‘y.’

 

Lee Hyun-jun (ST Reporter)

ace309314@soongsil.ac.kr

Lee Eun-seo (ST Cub-Reporter)

les0404123@soongsil.ac.kr

 


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