Wednesday 29 March 2006

The mathematics of looking beautiful

Looking good is important to a lot of people. Gyms, cosmetic companies, clothes stores and fashion magazines all exist because of our preoccupation with looking good. So what makes someone attractive?

Greek philosopher Plato believed he found the answer in the fourth century BC, and discovered it candidate in the most unlikely place of all: mathematics.

Plato was interested in designing beautiful shapes. As strange a hobby as this may seem, his results were even stranger. When measuring the dimensions of rectangles that he considered beautiful, he found their ratio was always the same ‑ 1.618.

Although this number is nothing special to look at, it has intrigued Plato and generations of mathematicians for centuries. The ratio 1.618 can be found in many examples of objects that we consider to be beautiful ‑ from the human body to music, architecture, nature and art. This number is so special that it has been called ‘the golden ratio’.

What exactly is the golden ratio?
The golden ratio can’t be written exactly as a decimal number, because it is irrational. It is a decimal that continues on and on, without any apparent repetition.

The best definition of the golden ratio is it is the only number that when squared is equal to the sum of itself and one. In 1996, a mathematician called Greg Fee programmed his computer to compute the golden ratio to ten million places. It took Greg’s computer approximately 30 minutes to complete.

There is an easier way to estimate the golden ratio using the Fibonacci sequence. This is a sequence of numbers where the next term is the sum of the two previous:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 … and so on.

If you divide each number in the sequence by the one before, you may notice that the quotient approaches the golden ratio:

1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666…, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615…

Fibonacci in nature
The Fibonacci sequence is often found in nature. The number of petals on a flower is often a Fibonacci number. Buttercups have five petals, lilies and irises have three petals and daisies often have 34, 55 or 89 petals. You might find some Fibonacci numbers in other places in your garden as well. Seeds on flower heads, such as on sunflowers, are often Fibonacci numbers. Patterns on pinecones are also based on Fibonacci numbers.

Sometimes these numbers are not exactly Fibonacci. In the case of pinecones, this can be due to deformities produced by disease or pests. In this case, the absence of the Fibonacci sequence indicates that the pine tree is not well.

Animals also display the Fibonacci sequence. The most spectacular of these are seashells. You can recreate a spiral sea shell pattern using a compass and a ruler.

Start by drawing two unit squares side by side. Together these squares form a rectangle. On the bottom of this rectangle, add a 2x2 square to make a new bigger rectangle. Beneath this new rectangle draw a 3x3 square. This creates a larger rectangle. Each rectangle is known as a Fibonacci rectangle, because the dimensions of each new square follow the Fibonacci sequence.

To complete the spiral, use the compass and to draw a quarter circle in each square to form a spiral. This shape is called a Fibonacci spiral. Similar spirals are found on snail shells and seashells.

The golden ratio in the human body
It may be surprising to know that the golden ratio can be found in the human body. Your hands contain three bones in each finger (you may notice this more by bending them). The ratio of the longest bone compared to the middle bone and the ratio of the middle bone compared to the smaller bone are both close to 1.618.

Another place you will find the golden ratio is your height. Measure how tall you are and compare this to the distance between your feet and navel. Their ratio is close to 1.618. Your face may also contain the golden ratio. What is the ratio of the width of your mouth compared to the width of the bottom of your nose?

Some people think that the closer certain ratios in your body are to the golden ratio, the more attractive you appear.

Can you find the golden ratio anywhere else in your body?

Art imitates life
The golden ratio can be found in art, architecture and music. The famous Greek sculptor, Phidias, was known to use the golden ratio in his sculptures of the human body. The most famous Greek building, the Parthenon, is 1.6 times as wide as it is tall. Leonardo da Vinci also used the golden ratio. He would frequently divide his canvas in this proportion.

The golden ratio has also been used by musicians. Mike Kay, an American mathematician, has examined Mozart’s sonatas and found that most of them divide into two parts exactly in the ratio of 1.618:1. Whether this was intentional or intuitive is not known.

Another researcher, Derek Haylock, found that the famous opening motto in Beethoven’s fifth symphony is repeated at the golden ratio point from the beginning of the song (32 percent of the way). Intriguingly, it is also heard at the golden ratio point from the end of the song (68 percent of the way). Other composers that appear to have used the golden ratio in their music were Bartok, Debussy, Schubert, Bach and Satie.

Is the golden ratio the key to looking good?
No. Everybody has his or her own opinion as to what is good looking. However, the golden ratio has been used in the design of various works of art and architecture to increase their appeal.

In living organisms, it seems the presence of special numbers and proportions has provided some sort of evolutionary advantage. In nature, plants with Fibonacci numbers may be healthier and pest-free. Animals with these ratios may also appear healthier and more desirable to mates. However, the golden ratio is not the whole story. Beauty still escapes a complete explanation.

The mp3 for this show can be found here


  1. Just for your information girls, I can personally verify that Mr Science not only knows about the golden ratio, he IS the golden ratio. I hope to marry him some day when he comes to Africa.

  2. I think he will marry me!

  3. The key to looking beautiful is expensive haircare products and moisturises. Duh.

  4. Cool site Marco... I think I can guess which comment is my sister's... she's right, you know. Expensive haircare products and moisturisers worked for her. Keep being entertaining... ;) x pinoy gal

  5. I hate to say this, but most of this information is incorrect. For an article on a similar subject that debunks the myths, read

  6. That's not actually correct, anonymous.

    Much of this article is correct. I am happy to concede that the Parthenon may or may not have that ratio. Most of the examples in this article are not, however, in the one to which you refer. The only comment I make regarding Da Vinci is that he often used that canvas ratio - this is certainly not debunked in that article.

    In the comments section of that page, a writer actually makes a number of comments regarding the Fibonacci sequence and how natural selection brings about the golden ratio in plants in his comments section - if you read my article, a lot of it is about the Fibonacci sequence and is supported by that comment in the article. These comments also note the reason why the golden ratio may be seen in structures, music etc. This is quite interesting. It doesn't debunk anything, its simply further information.

    Happy to debate this and happy to concede points - I'm not concerned either way - would be more happy to be correct than dogmatic. However the article you point to does not debunk this one.