Eye-Catching Math: Fibonacci Spirals

Eye-Catching Math is a series about beautiful mathematical images. Read more about it here.

Some patterns ought to never show up in the real world. The sequence of Fibonacci numbers is one such example. Fibonacci numbers periodically appear in popular culture: for example, they are featured in the bestselling novel and film The Da Vinci Code. But it would seem to have nothing to do with the natural world.

The Fibonacci series goes like this:

The first Fibonacci number is 1.

The second Fibonacci number is also 1.

The third number is equal to the first number plus the second number. So the third Fibonacci number is 1+1. In other words, the third Fibonacci number is 2.

It’s not hard. The fourth number is the second plus the third, or 1+2. So the fourth number is 3.

And so on. Each number is the sum of the previous two numbers. So if the eighth number is 13 and the ninth number is 21, the tenth number will be 13+21, or 34.

The beginning of the list of Fibonacci numbers is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…

You might think this in an interesting sequence to study. Or you might not. Certainly, though, it seems these numbers would never show up in the natural world.

After all, this sequence is just a list of numbers that follow a pattern. The pattern isn’t “number of volts of electricity” or “number of oranges on a tree.” It is generated from pure arithmetic. So it seems it should have no application outside of number theory.


Fibonacci spiral 34


Nope. Fibonacci numbers are everywhere.

For example, the Fibonacci numbers tell you how to draw a particular type of spiral, called the Golden Spiral. (Each successive box above, starting from the center and working outwards, has sides of length of a Fibonacci number. The box lengths are 1, 1, 2, 3, 5, 8… One can “smooth out” this spiral with a number called the Golden Ratio. This is the limit of the ratio between successive Fibonacci numbers. If you want to learn more about this, look up the Golden Spiral.)

Mysteriously, natural spirals often resemble the Golden Spiral.

"Fibonacci spirals," by helen.2006, is licensed under CC BY-NC-SA 2.0

Spirally plants often contain a Fibonacci number of spirals, like the 5 spirals in this aloe plant:

Each of these pineapples almost certainly has a Fibonacci number of diagonal spirals.

This pinecone, too, has a Fibonacci number of spirals.

It’s almost impossible to count, but this flower has a Fibonacci pattern too.

For a mind-blowing series on Fibonacci numbers, check out Vi Hart‘s videos on YouTube. It’ll also explain most of the objects in this post.

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6 thoughts on “Eye-Catching Math: Fibonacci Spirals

    • Unfortunately I can’t get the font size on my posts to change consistently 🙁 I’ll ask the site admins if there is anything to do about it.

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