The Fibonacci Golden Spiral: Drawing a golden spiral from squares

You may have seen a pattern like this: several squares arranged in order from smallest to largest, with side lengths that seem to follow a specific sequence: 1, 1, 2, 3, 5, 8, 13...

Then, when we draw quarter-circle arcs inside these squares, something magical happens: what started as simple squares connects to form an elegant spiral.

This is the familiar Fibonacci golden spiral.

It all begins with a very simple sequence

The Fibonacci sequence is perhaps one of the most famous sequences in mathematics, and its rule is very simple: starting from the third number, each number is the sum of the two preceding ones.

So: 1, 1, 2, 3, 5, 8, 13, 21, 34...

You will notice:

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8

5 + 8 = 13

...

No complex formulas. Just adding the previous two numbers together.

The numbers start turning into squares

Now, let's do something very interesting: treat these numbers as the side lengths of squares.

So, take a square with a side length of 1 and place another square with a side length of 1 next to it.

Next, place a square with a side length of 2.

Then add one with a side length of 3, followed by 5, 8, 13, and so on...

In this way, we get a series of squares that keep getting larger.

Then, a spiral emerges

Now, draw a quarter-circle inside each square, connecting the arcs end-to-end.

The first segment, the second, the third, the fourth...

Gradually, a spiral line emerges. Starting from the center, it expands outward, loop by loop. This is what we commonly call the Fibonacci spiral.

Why does it look so natural?

Because the Fibonacci sequence has a very special property.

As the numbers get larger, the ratio between two adjacent Fibonacci numbers gets closer and closer to 1.618... This number is the famous Golden Ratio.

The Golden Ratio is approximately: 1.618

13 ÷ 8 ≈ 1.625

21 ÷ 13 ≈ 1.615

34 ÷ 21 ≈ 1.619

...

As the numbers increase, the ratio gets closer and closer to 1.618... which is the Golden Ratio.

What exactly is the Golden Ratio?

You don't need to memorize a formula.

You can picture it like this: a line is divided into two segments—a longer segment and a shorter segment.

If the ratio of the whole line to the longer segment is exactly equal to the ratio of the longer segment to the shorter segment, then that ratio is the Golden Ratio.

Which is: 1.618...

Why does the Fibonacci sequence relate to the Golden Ratio?

This is one of the most beautiful aspects of the story.

At first glance, the Fibonacci sequence seems like a simple addition game: you add the first two numbers to get the next one.

However, as the numbers grow larger, the ratio between adjacent numbers gradually stabilizes, eventually drawing closer and closer to 1.618...

So, a simple rule of addition ultimately gives rise to a famous geometric ratio.

This is also why the Fibonacci sequence and the Golden Ratio so often appear together.

But wait...

There is a very important detail here.

The shape we usually see—composed of 'Fibonacci squares' and 'quarter-circle arcs'—is not, strictly speaking, a true Golden Spiral.

Its more accurate name is the Fibonacci spiral; it is a spiral constructed using the Fibonacci sequence.

In the strict sense, the Golden Spiral is a type of logarithmic spiral. The two are very similar, which is why they are often discussed together in a visual context.

Why do they look almost identical?

Because the ratios within the Fibonacci sequence get closer and closer to the Golden Ratio.

As the squares grow larger, the spiral constructed using the Fibonacci sequence increasingly resembles the spiral form associated with the Golden Ratio.

In other words: the Fibonacci spiral is a highly intuitive and easily constructed approximation of the Golden Spiral.

This is why you often see the terms Fibonacci spiral and Golden Spiral mentioned together.

You can think of it as a process of 'gradual convergence.'

A spiral constructed using only the numbers 1, 1, 2, and 3 appears somewhat rough. Adding 5 and 8 makes the shape more natural. Including 13, 21, and 34 makes the spiral smoother. Continuing with 55, 89, 144... brings it ever closer to the shape of the spiral defined by the Golden Ratio.

From numbers to squares, then to the spiral

The entire process is actually quite simple.

We can summarize it as: Fibonacci sequence → Squares → Quarter-circle arcs → Connecting → Spiral → Golden Ratio

This is the most fascinating aspect of the Fibonacci Golden Spiral: it connects numbers, geometry, proportions, and curves.

Why does it so easily evoke images of nature?

Upon seeing a spiral, many people immediately think of seashells, plants, flowers, hurricanes, or galaxies. Indeed, a wide variety of spiral structures can be observed in these natural phenomena.

However, a crucial point must be noted: spirals in nature do not necessarily strictly adhere to the Golden Spiral; the real world is often far more complex than a simple mathematical formula.

Therefore, we should not simply claim that 'all spirals in nature are Golden Spirals'—this is a very common misconception. A more accurate statement is: the Golden Ratio and spirals frequently appear in mathematics and nature, and the Fibonacci sequence often manifests in the growth patterns and arrangement of plants.

How does it relate to Fibonacci phyllotaxis?

If you have already explored Fibonacci phyllotaxis, you may have noticed that the name 'Fibonacci' appears in both contexts. While both are indeed linked to the Fibonacci sequence, they illustrate completely different concepts.

Fibonacci phyllotaxis examines how plant leaves, seeds, or other structures are arranged; here, you are looking at points, leaves, or flower heads.

The Fibonacci Golden Spiral examines how the Fibonacci sequence constructs a spiral geometric shape; here, you are looking at squares, circular arcs, and spirals.

In short, you can think of it this way: one studies 'how things are arranged,' while the other studies 'how a spiral is formed.' Both originate from the Fibonacci sequence but lead to distinct visual worlds.

Why is such a simple sequence so beautiful?

This is perhaps the most thought-provoking aspect of the Fibonacci golden spiral.

We start with just 1, 1, 2, 3, 5, 8...—simply by repeatedly adding numbers together.

Then, these numbers transform into squares, the squares into arcs, and the connected arcs form a spiral.

As the numbers grow larger, the ratio between adjacent numbers increasingly approaches the Golden Ratio.

Thus, a simple sequence ultimately links numbers → ratios → geometry → spirals—demonstrating the truly fascinating nature of mathematics.

Simplicity reveals the pattern

The most interesting thing about the Fibonacci golden spiral isn't its complexity; quite the opposite—it is remarkably simple.

You can sketch its general shape using nothing more than a pen and a piece of paper—no computer programs required.

First, write down the sequence 1, 1, 2, 3, 5, 8, 13...; then draw the squares and arcs; and finally, a beautiful spiral emerges.

Start with numbers and end with spirals

Looking back now, 1, 1, 2, 3, 5, 8, 13...the numbers themselves look very ordinary.

But after putting them into the geometric world, they began to produce squares, squares produced arcs, arcs finally produced spirals, and behind the spirals, there was a golden ratio hidden.

This is the most fascinating thing about Fibonacci's Golden Spiral: A simple sequence of numbers can actually lead you all the way into the world of geometry.

Continue to explore the world of mathematics

The Fibonacci Golden Spiral is an interesting entry point into connecting numbers and geometry.

If you like this kind of Simple rules → complex and beautiful shapes, you can also continue to explore other mathematical works. For example, Fibonacci leaf sequence and see how the Fibonacci sequence appears in natural arrangements.

You can also explore the Pythagorean tree and see how simple geometric rules branch out to form a mathematical tree.

Or check out the Dragon Curve to see how a simple line, through repeated folding, eventually transforms into a complex fractal structure.

They are all telling us a very interesting truth:Mathematics is not just about numbers. Numbers can be turned into shapes, shapes can be turned into patterns, and patterns can eventually turn into the beauty before our eyes.

You can continue to enjoy more wonderful mathematical graphics, or view more introductions about the golden spiral on Wikipedia.