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The Golden Ratio | Phi and the Esthetic Smile | Cosmetic Considerations | Putting it all together

Dental Esthetics and the Golden Ratio

By Todd R. Schoenbaum, DDS

Dental esthetics is a rapidly changing field, but like all fields that involve esthetic concerns there are some principles considered to be timeless. One such concept is that of the Golden Ratio as described by Fibonacci in the early 13th century. This may be a bit esoteric at first, but bear with me as it does become quite relevant. Long before Fibonacci, the Greeks used it to make the Parthenon pleasing to the eye, Leonardo Da Vinci used it in almost all of his later work, and even modern car designers use it.

Most people remember the Fibonacci Sequence from the days in grade school where we saw that most things in nature fall within this set of numbers whether it be numbers of petals on a flower or the number baby rabbits born in a year (this is the problem that Fibonacci himself studied in 1202).

Background on the Golden Ratio and the Golden Spiral

Fibonacci’s Sequence is a series of numbers calculated by starting with 0 and 1 and adding the last two numbers to get the next number.

Like this: 0+1=1, 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, 8+13=21, 13+21=34, and so on...
Resulting in the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34 ...

By dividing a number in the sequence by the number that precedes it you get a number very close to 1.618 (as the sequence progresses this number becomes more accurate). 1.618 is known as Phi, and its inverse (0.618) is known as phi. This is the golden ratio.

(In some of the images below there is a gauge that corresponds to the Golden Ratio of 1:1.618)

An interesting fact is that for all series that are formed from adding the latest two numbers to get the next starting from any two values (bigger than zero), the ratio of successive terms will always tend to Phi!

The Fibonacci sequence also gives rise to the Golden Rectangle (a rectangle in which the ratio of the shorter side to the longer side is 1:1.618), and the Golden Spiral. The Golden Rectangle can most easily be seen in a credit card. The measurements of a credit card are 53mm and 86mm, 53 multiplied by Phi (1.618) results in 86.

The Golden Spiral is best seen in the sagittal section of the Nautilus shell. The Golden Spiral is created by making squares out of the number of the Fibonacci Sequence, starting with a square that is 1x1.

How all this relates to Dentistry

The most useful way in which this relates to dentistry is in the use of the Golden Ratio (1:1.618) in the determination of tooth size.

For example, the two center teeth in the mouth are called central incisors.

The average width of a central incisor is about 8mm. If we look at the two central incisors together their combined width is 16mm. When attempting to determine the ideal esthetic length of the central incisors we can take this number and divide it by Phi (1.618), which results in a length of 9.89mm. This is a great asset when restoring teeth that have been ground down well short of their natural and esthetic length.

When restoring an entire smile with veneers or composite bonding, the dentist is able to change both the length and the width of the teeth being restored. The use of the Golden Ratio is useful here as well.

Another example: The width of the central incisors is a good starting place and for this case we have determined that 8.5mm is the ideal width for a particular patient. The width of both central incisors is therefore 17 and the ideal esthetic length is 10.5mm (17/1.618=10.5). If we divide the width of this central incisor by Phi, it will return the ideal esthetic visual width of the tooth behind it (the lateral incisor). Therefore 8.5mm / 1.618 = 5.3mm, and this is the ideal visual width of the lateral incisor in this case.

I stress visual width because this is not the actual width of the lateral incisor when viewed perpendicular to its surface, but rather the width when viewed from the front. We can do this again to determine the ideal esthetic width of the canine tooth: 5.3mm / 1.618 = 3.3mm, and again for the first premolar behind it: 3.3mm / 1.618 = 2.0mm, and again for the second premolar: 2.0mm / 1.618 = 1.2mm, etc.

So we now have a blueprint for this esthetic reconstruction.

Other Considerations for Creating the Esthetic Smile

The use of the Golden Ratio is merely a tool in the creation of the ideal esthetic smile. It can provide the dentist with a framework of information from which to start when designing a patient's new smile. Adherence to the Golden Ratio is not the final judge of success as there are many other factors that must be considered.

Masculine vs. Feminine Smiles

There are many subtle differences between a smile that is perceived as masculine and one that is perceived as feminine. Masculine smiles tend to have flat edges on the teeth, prominent central incisors, smaller lateral incisors, a flat smile line, flat canines, etc. Feminine smiles tend to follow the curve of the lower lip, have rounded edges on the teeth, laterals that are more equal in size to the centrals, sharper canines, etc.

Tooth Size and Shape

This may seem a bit peculiar, but generally speaking the size and shape of a person's teeth is directly related to the size and shape of that person's head. If one looks at the outline of the central incisor it is usually the same shape as their head turned up-side-down. The width and length of the central incisors also correlates to the width and length of the head. Another place to look for the width of the central incisors is the intra-alar distance (the width of the nose at its largest point). People with wider noses tend to have proportionally wider teeth.

Age

As we age our teeth no longer look like they did when we were younger. Older teeth tend to have more color to them, and they tend to be worn down a bit. Creating the ideal cosmetic smile requires taking their apparent age into consideration. A smile best suited for a 16 year old girl is going to be out of place and esthetically displeasing on a 68 year old woman.

Shade (tooth color)

There are four components that make up the final appearance of a tooth's color.
Hue: is the wavelength of the shade and this determines if a tooth looks more yellow or gray or brown. Like a person's hair color, the color of their teeth needs to compliment the color of their complexion.

Chroma: is the saturation of the hue; i.e. how yellow is it. Chroma is usually determined by age and by the specific tooth in question. As we age the teeth tend to have more color to them and certain teeth in the mouth, like the canine, tend to be more saturated as well.

Value: is the brightness of the tooth from black to white. The value used in veneers or composite restorations is determined by place in the mouth and by the desired brightness. Again, age and complexion play an important role here.

Translucence: this is the amount of light that passes through different layers of the tooth. Teeth tend to be more translucent towards the edge. Also the enamel that covers the outer surface of the teeth is glass-like and tends to be quite translucent as well. It is this characteristic of the enamel that gives teeth the natural look of depth. When this translucence is lacking the teeth no longer look natural.

Subtleties

There are many other subtle factors to consider in the creation of an esthetic smile. Teeth have characteristics like faces that make them unique which must be incorporated for a successful result. There are small irregularities in the shade, there are developmental bumps and ridges even on what generally appears to be a flat surface. There is symmetry that needs to be created contra-laterally (both sides of the mouth). There is the lip line that needs to be considered. And two other big topics are the gingival (gum) esthetics and the functional mechanics (articles on these two topics coming soon).

Putting it all Together

The final outcome of a beautiful, natural looking, healthy smile is based upon many factors which must be applied and weighed against each other.

The Final Esthetic Treatment must meet the following requirements:
1. It must compliment the functional requirements of the teeth, musculature and the TMJ (jaw joint).
2. It must be in esthetic harmony with the entirety of the face.
3. It must not adversely affect the biological health of the gingiva and supporting bone.
4. It must create symmetry within the mouth and be centered with the face.
5. It must be as conservative as possible.
6. It must bring joy to the patient!