(Oscar Fernandez Ph.D) How mathematics helps us explain beauty.
by Oscar Fernandez Ph.D., June 24th, 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.
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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, 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:
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):
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.
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The Numbers in Beauty
If numbers can generate beautiful patterns, do beautiful objects have some underlying beautiful mathematical pattern? 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:
(You can explore this “mathematical face mask” tool yourself here.)
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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:
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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 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:
Additional techniques, including using symmetry and finding triangles in your field of vision, can also imbue your pictures with 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 (a 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 in it.
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 the answer to your question will involve some beautiful mathematics.
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References
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
Stillness in the Storm Editor: Why did we post this?
Truth, beauty, and goodness, are regarded in some schools of thought as three supreme values. If one appreciates the fullness of truth, they’ll develop the capacity to see beauty in all things, thereby capable of acting for the good in one’s life. Psychologically, we need to appreciate beauty to ensure a positive life outlook. The mind is designed to reflect reality within, which is then evaluated as good or bad. The animal mind assumes things are “bad” so that it can ensure your survival. The human, spiritualizing mind, has the capacity to see the truth, recognize beauty and appraise all of this as good, thereby creating a node of positivity in your memory. The overall disposition you feel toward life, your mood, and worldview, are informed by the tapestry of evaluations in all life’s situations. Thus, to appreciate beauty improves your world view, it makes you feel better about life in general. It helps you overcome depression, maintain inspiration, and stave off boredom. Combined with the desire to grow and evolve, to become wise, honorable, and an agent for good, the hardships of life become opportunities to bring goodness into the world, though you’re creative deeds. In the act of heeding this supreme calling, you become an agent for good, living a fulfilling life for yourself, while making the world a better place through your increasingly joy-infused acts of compassion, kindness, righteous defense, and truthfully guided ways. The following information reveals some of this power of beauty, discussing the effects of appreciating it or showing you beauty in nature or created by others.
– Justin
Not sure how to make sense of this? Want to learn how to discern like a pro? Read this essential guide to discernment, analysis of claims, and understanding the truth in a world of deception: 4 Key Steps of Discernment – Advanced Truth-Seeking Tools.
Stillness in the Storm Editor’s note: Did you find a spelling error or grammar mistake? Send an email to [email protected], with the error and suggested correction, along with the headline and url. Do you think this article needs an update? Or do you just have some feedback? Send us an email at [email protected]. Thank you for reading.
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