If you’ve ever wondered how cosmetic dentists design an attractive smile when we do a smile makeover, we can tell you that part of the answer is math. Specifically, one of the tools we use for determining the proper width and height of teeth is the golden proportion, which helps give cosmetic dentistry results a pleasing appearance.

## What Is the Golden Proportion?

The golden proportion is a mathematically defined relation between two numbers such that the ratio between them is equal to the ratio between the larger number and the sum:

a : b = (a + b) : a

This relation was defined by Euclid in his Elements, although he didn’t link it to beauty. Numerically, in the above expression a = 1 and b = 0.618 (approximately, the golden proportion creates an “irrational” number, which means it never ends and never repeats). One interesting appearance of the golden proportion in math is in the Fibonacci series. The Fibonacci series starts with 0 and 1, and each successive term is defined by adding the two previous terms together. As you get higher and higher in the series, the ratio between two terms gets closer and closer to the golden proportion. For example, the first terms of the Fibonacci series are 0, 1, 1, 2, 3, 5, 8, 13, 21. Consider a few of these ratios: 3:2 = 1.5, 5:3 = 1.667, 8:5 = 1.6, 13:8 = 1.625, 21:13 = 1.615. You can see how it gets closer and closer to 1.618.

The golden proportion is found in many places in nature. How common it is in nature is disputed, as is its role in our perception of beauty. The golden proportion was of interest to neoclassical painters and Renaissance scientists, who were looking for evidence of God’s design in the world. They imagined him as a kind of mechanical engineer who had a very mathematical approach to the world so they identified the golden proportion in many places. Some of these are true, others are imagined or mistaken.

It wasn’t until the 19th century that people really began to see the the golden proportion as an ideal of beauty. Today, we can see that the golden ratio meets many of the criteria for beauty in a smile: proportion, symmetry, and regularity, so it’s one of several tools we can use to help design your smile.

## The Golden Proportion in Smile Design

How do we apply the golden proportion in your smile design? The most common approach is to use the golden proportion to define the widths of “The Cosmetic Dentistry 6,” the 6 teeth most important to the appearance of your smile. The maxillary central incisors should have an apparent width about 1.6 times that of your lateral incisors, which should have an apparent width of about 1.6 times that of your canines.

This defines a ratio that most people will find pleasing. It is part of the reason why peg laterals look unattractive, and why even some peg lateral replacements don’t meet cosmetic expectations.

Of course, this is just one approach for determining the proper proportions for your smile. Not everyone’s smile should be identical. We have several other tools (which we may discuss in the future) that allow us to choose a smile that is ideal for you.

If you want to discuss the design of your ideal smile, please contact us by email, or call us at