#### Halloween 2024: Mystery Coloring and Graphing Activities!

We’re getting ready for a spooktacular Halloween with some math-y at-home activities!

Jul 25, 2024 | Almaden

Math is more than just numbers and equations—it's a language that nature speaks fluently. If we look closely at the world around us, we’ll find math in unexpected places, from the spirals of seashells to the arrangement of petals on a flower.

One fascinating example of mathematical beauty is the golden ratio, a number that appears repeatedly in nature's design.

In this blog, we'll explore what the golden ratio is and how it shapes the intricate patterns of plants, animals, and even our own bodies.

The golden ratio, represented by the Greek letter phi (Φ), is a special number approximately equal to 1.618033988749895. The golden ratio is also known as the divine proportion, the golden mean, or the golden section.

We find the golden ratio when we divide a line into two parts so that the longer part (let’s label it as *a*
) divided by the shorter part *(b) *is equal to the whole length divided by the longer part: *a ÷ b = (a + b) ÷ a*
.

**Find Out More About Ratios and Proportions in Our Article**.

A Golden Rectangle is a rectangle whose side lengths are in the golden ratio.

Imagine you have a rectangle, and inside it, you place a square. The sides of this square are the same length as the shortest side of the rectangle. When you do this, you create another, smaller rectangle.

This process can continue forever, making a sequence of rectangles, each following the golden ratio.

As the sequence goes on, we compose the famous golden spiral – an image often used to represent the golden ratio:

**The Fibonacci sequence** is a series of numbers that goes on forever. It starts with 0 and 1, and then each subsequent number is the sum of the two numbers that precede it: 0,1,1,2,3,5,7,13, and so on.

But what does this have to do with the golden ratio?

If you take any two consecutive numbers in the Fibonacci sequence and divide the larger number by the smaller one, you'll notice that the ratio between them gets closer to the golden ratio as you progress along the sequence.

Let’s see this in action:

If we take the consecutive numbers 5 and 3 from the sequence and divide them, we get:

5 ÷ 3 ≈ 1.666

Now, let's take the consecutive numbers 21 and 13:

21 ÷ 13 ≈ 1.61538

We’re getting closer to the golden ratio.

Let's continue to demonstrate with the consecutive numbers 55 and 34:

55 ÷ 34 ≈ 1.617647

Pretty close.

After 55 in the Fibonacci sequence comes 89. So, let's divide 89 by 55 and see what we get.

89 ÷ 55 = 1.618181818

Remarkably close to the golden ratio.

If we go further along, this ratio approaches the golden ratio more and more closely.

The relationship between the Fibonacci sequence and golden ratio is why many tend to view the golden ratio and the Fibonacci sequence as **almost synonymous**.

From honeybee families to flower petals, let's take a closer look at some fascinating examples of the golden ratio in flora and fauna.

Believe it or not, but the honeybee’s family tree, specifically how they inherit their DNA, closely follows the Fibonacci sequence and the golden ratio.

Based on studies , female worker bees inherit half of their DNA from the mother, also known as the Queen, and half from the father.

The male drone bees receive all their DNA only from their mother.

As a result, while male bees have only one parent (the Queen), they still have two grandparents (the Queen's mother and father), three great-grandparents (the maternal grandparents and the paternal grandmother), and so on.

This pattern perfectly matches the Fibonacci sequence. Of course, as this sequence goes on, the ratio between successive numbers approaches the golden ratio, approximately 1.618.

Sunflower seeds inside the center of the sunflower are arranged in spirals that follow Fibonacci numbers.

When you count the spirals, you might notice that you can count either 21 spirals clockwise and 34 spirals counterclockwise, or 34 spirals clockwise and 55 spirals counterclockwise.

If you count spirals in both directions and divide the larger number by the smaller one, you'll get a value close to the golden ratio.

34 ÷ 21 = 1,61904762

55 ÷ 34 = 1,61764706

This special arrangement helps sunflowers pack in as many seeds as possible and makes sure they have plenty of room to grow and reproduce.

*Nature's design: Sunflower seeds spiral in Fibonacci patterns.*

Similarly to sunflowers, pinecones also have spirals that follow the Fibonacci numbers.

Each scale on a pinecone is arranged in a spiral pattern. When you count the spirals going in one direction, you might find, for example:

- 5 spirals going in one direction and 8 in the other
- or 8 spirals in one direction and 13 in the other

If you divide these numbers, you'll get values close to the golden ratio!

*Pinecone spirals follow the Fibonacci numbers.*

Many flowers have a number of petals that match Fibonacci numbers. For instance, the lily often has three petals, the buttercup five, the chicory eight, the delphinium thirteen, and the daisy twenty-one, and so on.

The golden ratio may appear in the arrangement of flower petals.

There are 360 degrees in a circle. If we divide a circle into a 137.5 piece and a 222.5 piece, the ratio between the longer arc and the shorter arc is about the same as the golden ratio.

The exact angle, measuring about 137.507764°, is called **the golden angle**.

Look how flowers' petals often align with Fibonacci numbers:

Nautilus shells are often cited as perfect examples of golden ratio in nature, but this claim is not accurate.

While it is true that some seashells might expand in proportion to the golden ratio in a pattern we mentioned earlier – the golden spiral, not all follow this pattern. Instead, their shells typically form a logarithmic spiral, rather than directly reflecting the golden ratio.

*Nautilus shell: Strikingly similar yet not exactly the golden spiral.*

The golden ratio isn't just found in plants and animals; it is woven into our own anatomy in ways we’ve never imagined. Let’s look at a few examples.

Several studies suggest that the Golden Ratio might be seen in various proportions of the human face. Here are some examples:

**Face Width to Height**: The height of the face (from the top of the head to the chin) divided by the width (from the left cheek to the right cheek) can approximate the golden ratio.**Mouth Width to Nose Width:**The width of the mouth compared to the width of the nose should approximate 1.618.**Eye Distance to Nose Width:**The width of the nose is about 1.618 times the distance between the eyes.**Nose to Lip Ratio:**The distance between the top of the nose and the center of the lips tends to be around 1.618 times the distance from the center of the lips to the chin.

Hand and arm proportions may also show a correlation with the golden ratio.

For example, when looking at the length of your forearm and the length of your hand (from the wrist to the tip of your middle finger), the ratio between these could approximate the golden ratio.

Similarly, when comparing the distance from the forearm to the top of the arm (shoulder), this ratio might also be close to the golden ratio.

If you look at your index finger, you'll notice it's made up of smaller sections, like your fingertip, the middle part, and where it connects to your hand.

Now, if you measure the length of each section, you'll find something interesting: each section is about 1.618 times longer than the one before it.

So, if you start with the length of your fingernail as 1 unit, the next section would be about 1.618 units longer, and the one after that would be 1.618 times longer than the previous one, and so on.

The golden ratio's influence extends even to the blueprint of life itself, DNA.

In the structure of the DNA molecule, each complete cycle of its double helix spans 34 angstroms in length (representing its helical length) and 21 angstroms in width (representing its diameter). These numbers follow the Fibonacci sequence and when 34 is divided by 21, it results in the divine proportion.

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