The Mathematics of a Smile

The Mathematics of a Smile

Although the human species, Homo sapiens, “the wise man”, likes to think of itself as above all other animal species, modern science is coming to the view that we humans behave a lot more like other animals than we care to believe. Much of our behaviour is of course conscious and rational, but much is also subconscious and instinctive, derived from patterns genetically determined and hard-wired into us over which we have little real control.

The prime underlying premise of life is to reproduce and continue the species, and animals need to recognize other members of the same species and select the fittest and most disease free individuals for mating, which they do by means of the senses, some visual, some audio (dolphins), and some mainly olfactory.


Humans are mainly visual in their approach to the world, and it is believed that they have an inherent mental image of what constitutes the ideal human face. It is part of their inherited genetic information. A face that comes close to that subconscious image is perceived as attractive, and evokes a subconscious positive response. There is no denying that an attractive face is a powerful asset, not only in one’s personal life, but also in business since often an attractive person is seen as a successful person, and indeed is more likely to be successful because of a more ready acceptance by others in this day and age when appearances are so important.


There is no doubt that proportion plays a part in our appreciation of beauty, and the ratio known since ancient times as the Golden Proportion, also called Phi after a Greek sculptor, is an important part, along with symmetry, of how mathematics plays a major role.


The Golden Ratio is approximately 1 : 1.62 and as humans we easily recognize and appreciate this ratio as pleasing wherever it is found in nature, which is almost everywhere. An attractive face features this proportion in many ways. The most obvious one is that the face exhibits the Golden Proportion in respect of its major features. The line of the eyes separates the upper from the lower part of the face in the ideal ratio, for example. Also, in the relaxed position, the lip line divides the lower face in the Golden Proportion – The distance from under the nose to the lip line is in proportion to the lip line to base of chin in a ratio that corresponds to the Golden Ratio.


Research has suggested that virtually all humans have similar perceptions of facial beauty with regard to proportion although there are cultural differences in regard to decoration and enhancement of the face. The extension of the neck by means of hoops and the purposely red stained teeth seen in some peoples of the world are extreme examples of these cultural differences but the underlying appreciation of facial shape and proportion seems universal.


Dr Eddy Levin is one of a few people who have studied facial aesthetics in a scientific way and confirmed that the face shows Phi (Golden) proportions in a multiplicity of ways. He devised a Golden Mean gauge to establish this relationship between the main features of a face, or of a plant or animal, and to check its existence wherever one wishes. It is quite useful when fabricating dental restorations in order to check the relative width and height of artificial teeth constructed as crowns or bridges.


He has shown that the Golden proportion is present throughout the natural world and how also how this proportion is adopted by humans in various ways when trying to achieve an intrinsically aesthetic result. The simplest example is the credit card which inevitably is constructed with sides of length that correspond to the Golden Proportion, but are set at right angles to each other. The rectangle formed just looks right because it conforms to Golden proportions.


If as dentists we look at the two central incisors in a person’s mouth we invariably find that the height of one central incisor is in Golden Proportion to the width of the two centrals. This is an example of a Golden rectangle and is seen over and over again when looking at different parts of the mouth. As far as the teeth are concerned it certainly helps when restoring teeth to ensure that the result will conform to the Golden principles, both in respect of the size and shape of the teeth and also the contour and height of gingival margins.


The four front teeth, from central incisor round the arch on one side to the first premolar, are the most important part of the smile when seen from the front, and it has been shown that the width of the teeth in this area are in Golden Proportion to one another, and indeed a grid can be constructed to demonstrate this and is useful in checking or improving a person’s smile.


Another researcher into human facial aesthetics is Dr Marquardt who has used computer programming to construct a “Golden Mask” which is a framework representing the ideal configuration for the human face, and is another tool that’s useful in cosmetic dental reconstruction. Interestingly he has noted that the human face conforms most closely to phi proportions when we smile, exhibiting the pleasing effect of teeth and lips in their ideal Golden relationship. This helps explain the ready acceptance of a smile as a facilitator in human relationships.


Someone said that “All biology is physiology, all physiology is chemistry, all chemistry is physics, and all physics is mathematics” which elegantly sums up the fact that we are all made of atoms like the rest of the known universe, and it is the arrangement of atoms in a mathematical way that produces order out of disorder, so that mathematics is fundamental. Fortunately, we don’t have to understand the mathematics in order to appreciate the beauty in the world, of which an attractive smile is just one example!