## The hidden beauty of mathematics (Part 1)

Written by Kristen Tripet

A number of years ago I visited the Louvre in Paris. For me, the Mona Lisa and the other works of Da Vinci were a must-see. When I made it to the rooms that housed the works of Da Vinci, it seemed that everyone else visiting the Louvre that day had the same idea as me. The space in front of the Mona Lisa was packed.

The Mona Lisa is only a small painting; about the size of a regular poster. However, it is hung on a wall in front of a large open space to cater for the crowds that flock to see it. On this day, there were about 200 people crowded around trying to see the portrait, many trying to snap a photo or selfie that said, “I was there, I saw the Mona Lisa”. No doubt most pictures that day would have been obscured by all the other cameras and phones being held up.

I did not witness quite the same pandemonium for any other artworks in the Louvre that day.

Why is it that this particular painting is so famous?

What makes it so special that it is the artwork everyone wants to see?

It is true that there is a mysterious history surrounding the portrait and that, for its time, Da Vinci used innovative painting techniques.

But I do wonder if there is something more.

That day in the Louvre I joined in and took several pictures of the portrait. But I also took a different photo of the Mona Lisa. I moved to the back of the room and took a photo of all the people taking photos.

As I did, I asked myself, “Who is taking a picture of the art and who is taking a picture of the mathematics?”.

My desire to see the Mona Lisa and the other works of Da Vinci that day was because I wanted to witness first-hand the mathematical brilliance of Da Vinci.

My expectation is that most, if not all people that day were blind to the mathematics in the Mona Lisa. And they were blind to the mathematics in the other works of Da Vinci in the room.

What if it was you standing there in my photo looking at the Mona Lisa – would you have been looking at the mathematics?

The typical question you are asked when you meet someone new is ‘What do you do for work?’ I always take a deep breath before answering ‘I work in mathematics’. At this point, the conversation seems to go in one of two directions. First, it goes nowhere at all. People sort of say ‘oh’, there is a short awkward silence and then they find someone else to talk to. Alternatively, it seems people feel obliged to tell me about their own relationship with mathematics, which is most commonly derived from their personal experience of maths at school. Some loved maths at school, but predominantly, the stories I hear are not great. Perhaps that is why many choose to end the conversation before it really begins. Sadly, what dominates school mathematics today and in the past is learning the tools of mathematics without experiencing real mathematics, without enjoying the beauty of mathematics.

This is the focus of Paul Lockhart’s book, ‘A Mathematician’s Lament’. Lockhart, a mathematician, compares learning mathematics to learning to paint in art class. At school, we learn to paint through the experience of painting. It is not through repetitious procedures associated with naming colours and tools. It is not through gradual introduction to painting through paint-by-number tasks. And it is not just the best, the most talented students, that progress to painting actual pictures. At school, everyone paints, and we usually start painting on the first day of Kindergarten. It is true that we will get better with practice and that we will improve as we learn to best use the tools of the trade, but we learn through experiencing the art of painting. Why is it that learning mathematics in school is so different? This question is the heart of Lockhart’s lament.

It is important that our students experience real mathematics; that they experience the beauty of mathematics. Real mathematics is accessible to students; it is not beyond them. I want to suggest that real mathematics has something to offer us to better understand the beauty of the Mona Lisa. And that mathematics has something to offer us to better understand the beauty of our world.

It was Galileo who said, “Mathematics is the language with which God has written the universe.”

I would like to take you on a short journey to uncover some of the hidden and beautiful mathematics of our world. To do this, I want to share with you some of my favourite numbers.

Many of you may have heard about the Fibonacci sequence of numbers. This famous sequence was known as early as the 6th century by Indian mathematicians but was introduced to the western world in 1202 by Leonardo of Pisa, who is better known today as Fibonacci. Fibonacci was also the one who popularised the use of Hindu Arabic numbers, that is the number system that we use today.

Fibonacci numbers are a simple series of numbers where each new number is formed by adding together the two preceding numbers. So, starting with 1, we add 0 and 1 to make 1; then we add 1 and 1 to get 2; 1 and 2 to make 3; 2 and 3 to make 5; 3 and 5 to make 8; 5 and 8 give 13 and so on.

As you can see, the Fibonacci sequence is simple. It is not complex. Yet, this simple sequence creates complex beauty.

To illustrate, let’s take a look at the world of plants.

The number of petals on a flower can consistently be found within the Fibonacci series of numbers, so too the pattern of petals of a rose.

The way that tree branches grow and split mirrors the Fibonacci sequence, as do the root systems of many trees.

The Fibonacci sequence is seen in the number and arrangement of leaves on many flower stalks.

The number of spirals seen in pinecones, pineapples, broccoli, cauliflowers and artichokes is based on Fibonacci numbers. So too are the spirals in an aloe plant or the seeds in the centre of a daisy or sunflower.

The shape of these spirals is also based on the sequence of Fibonacci numbers.

This spiral starts with a square with dimensions of 1 unit. Adjoining this, another square with sides of 1 unit. The next square has side lengths of two, and then three and so on (Figure 1). Drawing a curved line between opposite corners of each square creates a spiral.

Figure 1 - Spiral created using the Fibonacci numbers

This spiral pattern is reflected in other parts of nature. It is shape of our inner and outer ear as well as the shape of shells such as a snail shell or the nautilus shell. The Fibonacci spiral is also the shape formed by clouds in a hurricane and the shape of swirling galaxies.

Why is it that we see the Fibonacci sequence all around us in the natural world? Is it just coincidence? We can confidently say that this sequence appears in nature often enough to prove that it is not chance, it reflects a naturally occurring pattern. To understand this pattern, let’s take a closer look at the sunflower.

We mistakenly think of the sunflower as one large flower. It is actually made up of many smaller flowers. These flowers are found in the centre of the sunflower. It is these centre flowers that produce sunflower seeds. This means that the more smaller flowers that grow in the centre, the more seeds it will produce. But each individual flower also needs space to grow. To make the most efficient use of space, one small flower will grow, and then the following flower will grow further around the centre of the larger flower. This continues and forms a beautiful spiralling pattern which is based on the Fibonacci spiral as we have already seen. What is even more interesting is the number of spirals in a sunflower. Typically, a sunflower has 34 spirals in one direction and 55 in the other. These are two neighbouring Fibonacci numbers. Why is this so?

Well, it all has to do with another beautiful number – phi, written as φ.

As already mentioned, as one small flower grows, the second grows further around the centre. This means that each new flower appears at a certain angle in relation to the preceding one. If flowers were to grow at 90° angles, or at a quarter turn, the result would be lines of flowers radiating out from the centre of the sunflower (Figure 2). This is not efficient use of space. Using any angle between each small flower that can be represented as a simple fraction would always result in straight lines.

It therefore makes sense for the angles between flowers to be based on an irrational number, that is a number that cannot be represented by a simple fraction. Indeed, the sunflower uses the most irrational number that there is: φ. φ is known as the golden ratio and the corresponding angle is known as the golden angle which is 137.5°. Using this angle, the optimal filling is achieved, and beautiful spirals are formed (Figure 3).

Figure 2 - Seeds growing at a quarter turn

Figure 3 - Arrangement of seeds based on φ

So, what does φ have to do with the Fibonacci numbers? The ratio between neighbouring Fibonacci numbers is very close to φ. The larger the numbers are, the closer to φ we get. So, it makes sense that the number of spirals in the centre of the sunflower would be two neighbouring Fibonacci numbers. It also makes sense that the shape of the spirals would reflect the Fibonacci spiral.

The golden ratio brings us back to Da Vinci and the Mona Lisa.

Da Vinci was a man of extraordinary talent. He was a brilliant artist. He was also a talented scientist and mathematician.  Indeed, while working on the Mona Lisa, Da Vinci is recorded as saying, ‘Let no man who is not a mathematician read the elements of my work.’

Da Vinci’s art was infused with mathematics. The golden ratio was a tool that he used to create accurate proportions. His work the Last Supper was revolutionary for its application of proportion and perspective, and it is believed that the golden ratio was the basis for the Mona Lisa. Vitruvian Man, a pen and ink drawing, is a canon on the proportions of the human body.

Da Vinci’s passion for mathematics, particularly proportion and geometry, was born through Luca Picioli. Picioli tutored Da Vinci in mathematics and the two became close friends. In 1509, Picioli wrote the book Divina Proportione. Da Vinci illustrated the work, showing how the golden ratio is applicable to geometry, art, and architecture.

Da Vinci’s use of mathematics shows that maths is so much more than what many of us experienced at school. Da Vinci saw the beauty in mathematics. He recognised applications beyond the discipline and into all aspects of life. When we think of mathematics in this way, we understand that pattern, order and structure are the results of mathematics in action.

As Galileo said: Mathematics is the language with which God has written the universe.