The Ever-Blooming Flower

Men are always full of curiosity, and nature always provides a mystery that is waiting to be solved. One of them is in the plants. Look at the flowers above! They are very different: the colors, kind of petals, habitats. Obviously, they are distinct kind of flowers — not even close. However, if you look closer you will find one thing that is similar among all of them. They have a single pattern to grow: spiral! This pattern, it does not only occur in five of the plants above, but also in many more! Isn’t it astonishing how there are many plants that have one growing pattern despite different growth factors that affect them? So then many people try to find out if the plants just do it randomly or they, actually, do it in purpose, and why such of this pattern could happen.

Now imagine that you are a flower, and you are willing to be a beautiful flower that all your petals are visible. Moreover, it is also beneficial for you if all of your petals are visible because every petal will get the same amount of sunshine. Then, if you have to follow one rule to grow your petals, where will you put your second petal after you put the first one? Let say you choose to put it as far as possible from the first one: 180 ° — since you are a flower then you have a circle shape so you would use degrees to measure distances. It might look good at the beginning, but you also have to use the same angle to put your third petal. Then, when you put your third angle, it will be under the first petal so the third petal will be covered by the first. You still want to find a way, now you put your second petal right beside the first one, and it works! At least at the beginning… After you have grown your petals on a full circle, you want to and one, but it should be under the first petal (again) so if you use this way, you will only be able to have a single layer of petals.

Flowers are resilient. They keep trying to find the way to achieve what the want until someday, they found a particular angle to make their goal achievable. They call it the golden angle. This angle is amazing! They put their second petals after the first one as far as the golden angle, then the third one, the so on. As a result, the petals will never really overlap with each other. Their wish comes true, and even better, they have a chance to keep growing forever! Well, we know that the flowers will eventually stop growing someday. However, if they want to, they are able to add their petals over and over again to infinity without breaking the rule because there will always a space for them to do so. That is why they eventually become the ever-blooming flowers! The way they follow the angle is also make them more beautiful because somehow the flowers will make spiral pattern (if you don’t believe it, I will tell you how much golden angle is, and you can draw petals by yourself following the golden angle.) There will always a room to grow, so that these flowers actually have an opportunity to be eternal. Since this angle is very useful, it also uses the golden angle to pack its seeds efficiently and to place the leaves so all the leaves can get the sunshine.

Nobody knows how the flowers found the golden angle, but humans, a long time after that, finally find this out too! It was started by somebody called Fibonacci. He made a sequence like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, …

Can you see the pattern there? It is the sum of the two numbers before it (1=1+0, 2=1+1, 3=2+1, …) It looks simple, but if you look for the ratio of one number and the number before, it will be like this:

the ratio between Fibonacci Numbers

The bigger the numbers get, the closer its ratio to a particular number: 1.61803398875… and since the number is too long let’s just call it 𝟇 (phi) or the golden ratio. Then, if we divide a full circle (360°) by 𝟇, we will get 222.5°. The golden angle used by plants to grow themselves is in the same position in the circle: 360° — 222.5°= 137.5°. Some other plants also use 180°/𝟇 and 90°/𝟇. It is okay as long as they are being consistent they still able to be the ever-blooming flowers!

Do you realize that flowers are also mathematizing to keep growing and being efficient? Indeed, they found the golden angle first before us. Doesn’t like numbers and equation that we used to learn every day, mathematics is beautiful and even becomes the base on how plants grow themselves. It is not only in flowers, but you can also see mathematics formed beautifully anywhere! In music, in the painting and sculpture, in the Rome ancient architecture, or simply in your home furniture. You just have to take a look.


Donald Duck — Mathmagic Land [Video file]. (2012, September 15). Retrieved from

Doodling in Math Class: Spirals, Fibonacci, and Being a Plant [2 of 3] [Video file]. (2012, January 9). Retrieved from

Golden Angle — John Edmark. (n.d.). Retrieved from

What Is the Fibonacci Sequence?. (2018, October 24). Retrieved from

Zeng, L., & Wang, G. (2009). Modeling golden section in plants. Progress in Natural Science, 19(2), 255–260. doi:10.1016/j.pnsc.2008.07.004




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