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Proportions Using Cross Products

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A Golden Smile
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Credit: eperales
Source: http://www.flickr.com/photos/96799823@N00/99860428/

Do you know where ratios can be found at the dentist's office? You might not be thinking of math as you have your teeth cleaned, but in order to determine a beautiful smile, dentists can use ratios.

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The Greeks believed that the key to all beauty could be found in mathematics. They began by taking geometric measurements in search of this code and discovered that there was indeed a particular ratio found in beautiful things. That ratio, called the golden ratio (also referred to by the Greek letter phi $\varphi$), is approximately $1.618:1$ and can be found anywhere from nature, to art and architecture, to the human body. In 1509, Luca Pacioli, an Italian mathematician, published a book exploring the golden ratio and showed how it could be applied to the human face, as depicted in the illustration below.

Credit: Luca Pacioli, edited by Laura Guerin
Source: http://commons.wikimedia.org/wiki/File:Divina_proportione.png

More recently, esthetic dentists (dentists interested in developing beautiful smiles) studied this ratio and found that it describes an ideal relationship between certain teeth. When the golden ratio is present, the teeth come together in a beautiful smile.

Credit: Laura Guerin
Source: CK-12 Foundation

How does this work? Well, an esthetic dentist can measure the combined width across both an upper central incisor and its neighboring lateral incisor, as well as the width of the central incisor alone. If the relationship between these two widths forms the golden ratio, then the teeth are in proportion and so is the smile. If not, then the esthetic dentist can use the measurements to adjust the person's teeth until the correct proportion can be found. Now that is a lot to smile about!

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1. [1]^ Credit: eperales; Source: http://www.flickr.com/photos/96799823@N00/99860428/; License: CC BY-NC 3.0
2. [2]^ Credit: Luca Pacioli, edited by Laura Guerin; Source: http://commons.wikimedia.org/wiki/File:Divina_proportione.png; License: CC BY-NC 3.0
3. [3]^ Credit: Laura Guerin; Source: CK-12 Foundation; License: CC BY-NC 3.0

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