Oscar Fernandez Ph.D.

The Calculus of Happiness

The Beauty in Numbers, and the Numbers in Beauty

How mathematics helps us explain beauty.

Posted Jun 24, 2019

You’ve no doubt heard the phrase "beauty is in the eye of the beholder." What if the beholder is a mathematician? How do these creatures see beauty differently than others? Are there patterns to beauty? And if so, are they mathematical patterns?

Let me whisk you away on a short adventure that answers these questions and leaves you, at its conclusion, seeing beauty as some mathematicians do.

The Beauty in Numbers

Numbers can do more than just help us count; they can help us discover—discover beautiful patterns, that is. A wonderful illustration of this is the Fibonacci sequence. Named after the mathematician Leonardo of Pisa (c. 1170 – c. 1250), this sequence begins with:

1, 1, 2, 3, 5, 8, 13, 21, …

Except for the first two numbers, each number in the sequence is the sum of the previous two. (For example, 3=1+2 and 5=3+2.) Fibonacci stumbled on this sequence when thinking about how to count the offspring produced by a pair of rabbits. Not very exciting (or beautiful), I admit. But a hidden pattern emerges when you visualize the numbers differently. The image on the left creates squares whose side lengths are the successive Fibonacci numbers; the image on the right draws circular arcs connecting opposite edges of those squares:

 Fibonacci number.
The golden rectangle (left) and golden spiral (right).
Source: Wikipedia: Fibonacci number.

The rectangle on the left is called the golden rectangle. The beautiful blue spiral on the right is called the golden spiral. I can almost guarantee that you’ve seen these in your everyday life (albeit sometimes hidden in plain view):

 Draw Paint Academy.
Left: golden rectangles superimposed on the Mona Lisa. Right: the golden spiral in Nature.
Source: Left: Pinterest. Right: Draw Paint Academy.

This connection between the Fibonacci numbers and the many beautiful objects containing patterns describable by the numbers is why I, along with most other mathematicians, think that the Fibonacci numbers are beautiful.

The Numbers in Beauty

If numbers can generate beautiful patterns, do beautiful objects have some underlying beautiful mathematical patterns? Often the answer is yes. Take human faces, for example. A 2009 study found that "individual attractiveness is optimized when the face’s vertical distance between the eyes and the mouth is approximately 36% of its [the face’s] length, and the horizontal distance between the eyes is approximately 46% of the face’s width." 

Other ratios also turn out to encode beauty. Additionally, computer-generated overlays constructed from the golden ratio—a number that one can use to generate Fibonacci numbers—have been shown to detect beauty in faces. For example, here’s a picture of Jessica Simpson (left) and the same picture with a superimposed mask constructed from the Fibonacci numbers:

Jessica Simpson (left) with an overlaid computer-generated face mask (right) based on the golden ratio.
Source: Intmath

(You can explore this "mathematical face mask" tool yourself here.)

The face masks described above have also been used to generate more attractive pictures of faces from a given input. Here's an example, courtesy of the Marquardt Beauty Mask:

Source: Goldennumber

More generally, photography provides another excellent example of mathematics hidden behind beautiful objects. Indeed, amateur photographs can be made instantly more attractive to the eye by utilizing the Rule of Thirds, which advises that we align objects in our pictures along the points and lines created by dividing the picture into thirds, both horizontally and vertically, as illustrated here:

The Rule of Thirds.
Source: Digicameras

Additional techniques, including using symmetry and finding triangles in your field of vision, can also imbue your pictures with a beauty that others will recognize but have difficulty pinpointing the origin of.

Parting Thoughts

While I've focused this article mostly on the Fibonacci numbers and the various other patterns that emerge from it, there are many other numbers that give rise to beautiful patterns and objects. A prime example (pardon the classic math pun) is the irrational number pi (approximately equal to 3.14), the ratio of a circle’s circumference to its diameter. Embedded in every circle—a beautiful shape in and of itself—and appearing so frequently in mathematical formulas describing natural phenomena, pi is a truly magical number. I urge you to invest four minutes of your time watching Rebecka Taule's beautiful video illustrating pi's majesty.

While there are many exceptions to the math-beauty connection I've highlighted, as I hope you now better appreciate, mathematics and beauty are often interlinked, and I hope this article has given you a new lens with which to view your surroundings and find the hidden beauty.

Indeed, the next time you see something beautiful I encourage you to ask, "Why do I find this beautiful?" You'll be starting to think like a mathematician—whose fundamental drive is to find and explain patterns—and I'm willing to bet that the answer to your question will involve some beautiful mathematics.


1. Wikipedia Fibonacci numbers: https://en.wikipedia.org/wiki/Fibonacci_number

2. Mona Lisa: https://www.pinterest.com/pin/468655904947901455/?lp=true

3. Draw Paint Academy: https://drawpaintacademy.com/golden-ratio-in-art/

4. Jessica Simpson: https://www.intmath.com/numbers/math-of-beauty.php

5. Marquardt Beauty Mask: https://www.goldennumber.net/beauty/

6. Digicameras: http://digicameras.weebly.com/the-rule-of-thirds-a-photography-tip-for-designers1.html