Fibonacci numbers and golden ratio

Math is a subject that has been studied for centuries. It has a great influence on our daily life. From basic counting to complex calculation, there are so many circumstances that we need math and our lives would not be able to function normally without math. There is no doubt that math brings us a lot of benefits. What math brings us can be divided into three parts: Ability to calculate, applications, and inspirations. I believe that most of us are familiar with the first two parts already since we encounter these two parts very frequently. Even you are not a student taking math classes, you still do basic calculation almost every day. There is hardly a day that anyone can get away from numbers completely. As for the application of math, the most obvious one is the computer. In the time when personal computers and smartphones are common, billions of people have the access to the application of math every day. At this point, I think I have convinced you that math brings us the ability to calculate and great applications. However, you may still be frowning when I say that math gives us inspirations. You may wonder what kind of inspiration can we find in those goofy notations, crazy looking formulas, and abstract theories. Well, if you think that is what math all about, you are terribly wrong. Math can be simple, elegant, and interesting. You may have a lot of doubts in this statement at this moment. That's ok. Let me give an example to prove my point.

In mathematics, there is a sequence of numbers called "Fibonacci numbers". The Fibonacci sequence follows a simple patter. It goes as the following: 0,1,1,2,3,5,8,13,21,34,55........  With some observation, we can find out that any number in the sequence is the sum of the two numbers that precede it. An interesting thing about this sequence is that if you divide any number (b) by a number (a) that precedes it. The ratio is always closed to 1.6180339887....   For example : 8/5=1.6, 21/13=1.615384615, 55/34=1.617647059.  The ratio gets closer and closer to 1.6180339887 as the number b increases. The ratio of two adjacent Fibonacci number is also known as the Golden ratio. At this point, you may find the fact that Fibonacci sequence yields Golden ratio interesting. If I tell you that Fibonacci numbers and Golden ratio appear in nature very frequently, your mind will be totally blown away. First of all, let's take a look at the numbers of petals on flowers. It turns out that the numbers of petals on different kinds of flowers are Fibonacci numbers. Examples include Lily, which has three petals, buttercups, which have five, the chicory's 21, daisy's 34, and so on. The Golden ratio, just like Fibonacci numbers, appears frequently in nature. In fact, if you look at your fingers, you will find that the length of each section increases as you look from the tip to the wrist. Interestingly, the ratio of the length of the adjacent section is close to the Golden ratio. The same pattern can be found on insects and other animals. The shade of the Golden ratio can also be found on the spiral of shells. If we draw a rectangle that encloses a shell, the ratio of the length and width of the rectangle is surprisingly close to the Golden ratio. To further elaborate that, if we do the same thing to the spiral of the Galaxy and hurricane, we can find the same pattern

.     At this moment, I don't even have to explain why Fibonacci numbers are so amazing. The fact that Golden ratio and Fibonacci numbers can be found everywhere in nature makes us wonder that if there is a rule that governs that universe. Who would ever think that mathematic pattern can be found in nature and is all around us?