While I was cleaning my room I saw my abandoned snail toy decoration at the corner of my desk. Then the shell spiral draws my attention because it seems like the golden spiral. Through some research, I noticed most people commonly have agreed on a type of seashell – Nautilus shell has the golden spiral. Afterward, I have a question: “Is the Nautilus shell spiral actually a golden spiral?”

#### Golden spiral(Fibonacci spiral)

The golden spiral is a spiral that has the proportion of the golden ratio. Golden ratio comes from the Greek letter Phi – a number approximately equals to 1.61803399. The actual equation of Phi is (1+ √5)/2. Phi is an irrational number – a decimal but its decimal place never self repeats. A method to get Phi is by cutting the line into two pieces in a particular way. So when the longer piece over the shorter piece proportion equals to the original line(short + long) over the long piece proportion, the result of these proportions will exactly be Phi.

#### Disprove

One of the ways to build a golden spiral is to draw golden rectangles as a guideline then create the spiral. To draw golden rectangles, rectangles that has the proportion of the golden ratio(height/width). Then rotate 90 degrees counterclockwise, split the remaining to a square and another golden rectangle. Afterward, repeat these step until a “center point” is created. Next starting from the center point, connect one of the squares’ opposing vertices by a quarter of a circle curve, in moving clockwise from the smallest to the biggest square. The result will be the clockwise golden spiral

after all of the squares are connected. From overlapping the golden spiral to the Nautilus shell, the two spirals don’t match up with each other.

However, the growth rate of the spiral shows its proportion(average to 1.587) which is very close to the golden ratio proportion. That explains why the Nautilus shell is commonly mistaken to have the golden ratio proportion.

Conclusion

The Nautilus shell isn’t a golden spiral because it doesn’t match up with the actual golden spiral model. Along the way, I discover that math is an important tool to help us understand the natural world by building models at the ideal circumstances. Since the natural world is complicated by having various undetermined factors, math uses the models or equations to stimulate all scenario. Even though these simulations might not be exactly the same, but it gives a great overview of the whole situation while being as accurate as possible. This is a very fascinating side of mathematics.

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