ABSTRACT: In the search to idealize esthetic dental restorations, many authors have advocated using geometric or mathematical proportions to aid in establishing a mathematical model to improve dental esthetics in a predictable way. Analysis of beautiful smiles has revealed repeatable, objective principles that can be applied to evaluate and improve dental esthetics. Although controversial, the Golden Proportion has been advocated as a mathematical tool to assess proportions in the frontal view of the arrangement of the maxillary teeth.
A historical appraisal of beauty reveals that man has succumbed to the power of beauty and its effect on us. "Personal beauty is a greater recommendation than any letter of reference."--Aristotle (384-322 BC) "Beauty is power; a smile is its sword."--Charles Reade.1 Beauty has a mystique that makes us wonder when some say, "beauty is the first present nature gives to women and the first it takes away." It is often said that it is easy to be beautiful; it is difficult to appear so.
Do we know what beauty is? Beauty has been described as whatever gives you joy, or as the promise of happiness, or beauty itself is but the sensible image of the infinite.2
In analyzing facial form and beauty many have discussed the forms of facial attractiveness, and indeed the form of beauty in general, as being composed of or the result of two ideal components. These components are symmetry and harmony. Symmetry can be defined as the mirror image of parts or components about an axis. By all accounts a beautiful face is symmetric. Harmony can be defined as a recurring theme.
The Greeks said that all beauty is mathematics. If that is true then perhaps there is a mathematical code, formula, relationship or even a number that can describe facial beauty.
The Greeks were the first to perform facial measurements in the quest to find the mathematical code or formula for facial beauty.3 Only one mathematical relationship has been found to be present in beautiful things. The Golden Ratio--a mathematical ratio of 1:1.618 has been used since antiquity by mathematicians and artists. The number 1.618 is called "Phi" from the Greek letter of Phidius.8 This ratio has also been referred to as the Fibonacci ratio or Divine ratio.
The golden proportion was first recorded by the Pythagoreans and later by the Greek geometrician Euclid as the ratio between two portions of a line, in which the lesser of the two is to the greater as the greater is to the sum of both (Fig.1).
When the ratio between B and A is in the golden proportion, then B is 1.618 times larger than A.
The symbol of the golden proportion is the pentagon, which was the symbol of the Pythagorean School, which was deeply involved in the study of the golden proportion. The Golden Proportion is unique in that the ratio of the smaller length relative to the larger length is identical to the ratio of the larger length to the total length (Fig. 2). The illustration demonstrates this ratio within the segments of a pentagon. The ratio AB:BC = the ratio AC:AB. The ratio is 1.618:1.4
The Golden Proportion has intrigued experts for centuries because of its connection with esthetics. It occurs in triangles, circles and spirals but most notably in the Golden Rectangle, whose sides have a golden relationship to each other. The Golden Rectangle is said to be one of the most visually appealing of all geometric forms5 (Fig. 3). It can be seen in the construction of Notre Dame in Paris, as well as in the more modern United Nations building in New York City.
This golden proportion has been defined as the ratio that is most attractive to the human eye and mind, and the Greek letter "Phi" (ø) is used to indicate the number 1.618. The proportions of the Parthenon on the Acropolis of Athens show how the Greeks used this concept (Fig. 4).
The ratio of the golden proportion was also described by Leonardo de Pisa, Fibonacci, in his development of the mathematical sequence.
The Fibonacci mathematical progression in which each number is the sum of the two immediately preceding it: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on, which converges on the golden proportion as shown in Table 1.6
Leonardo da Vinci (1452-1519) was intrigued with the Golden Proportion and wrote about the principles in 1509 and published the "Divine Proportion".
Ricketts used a golden divider to prove the harmonious faces of beautiful women followed golden proportions. The golden divider is a sliding caliper with which any given distance can be divided in accordance with the 1:1.618 ratio (Fig. 5).
These calipers always open to a constant golden proportion between the larger and smaller parts.
The biologic implications of seeing many things in nature follow the principle of golden proportionality imply that the proportion is not only linked to growth but also that it relates to optimal function.7 Nature is abundant with examples of the Golden Proportion. From the double helical form of our DNA to flowers and insects, the golden proportion is readily evident all around us (Fig. 6).
Ricketts found a soft tissue progression that followed the golden proportion from the width of the nose to the mouth, to the eyes and to the width of the head at the level of the eyebrow to be 1.618 times (Fig. 7). For vertical facial relation in the soft tissue, another progressive series is seen as confirmed from composites of beautiful faces. Starting with the larger value, the face height is taken from Trichion (at the top of the wrinkled forehead or near the hairline in young) to the bottom of the chin (soft tissue menton). If the lateral canthus level to Trichion is taken as a unit 1.0, the height of eye to chin is in the golden proportion, if the face is beautiful.6
From the eyes downward, a golden relation is seen from the nose to the chin (1.0:1.618). If the alar rim to the upper lip is taken as 1.0, the distance to the chin is 1.618 and the same distance to the eye is 1.618. Overlapping congruent areas show that there are three equal areas of the face, which is the same in beautiful faces. These are forehead to eyes, eyes to mouth, and nose to chin (Fig. 8). Another examples of soft tissue proportions can be seen in the width of the eyes to the inter-canthal distance. The outer canthus of the eye to the medial inner canthus, or the visible white eyeball is proportionate to the distance between the two eyes which is the distance between the "eye whites" (Fig. 9).
The golden rectangle defined by the incisal edge of the maxillary central incisors, if they are of normal length, to the superior tip of the eyebrow and the pupillary width can be used as a guide in determining incisal edge position (Fig. 10). Another guide to assess the incisal edge of the maxillary central incisors is the golden proportion between subnasale to the incisal edge and that to the menton (Fig. 11). Still, one can also assess incisal edge position with the golden proportion that exists between the eyes, the incisal edge of the maxillary incisors (if ideal in length), and the chin (Fig. 12).
Lombardi8 noted that the teeth have a harmonious perspective from the dominant central incisor transitioning posteriorly, with each tooth in harmonious proportion to those adjacent, but stated that "it has proved too strong for dental use."
Levin9 advocated the use of the golden proportion for establishing tooth size and stated that "the perceived width of the maxillary central incisor is in golden proportion to the width of the lateral incisor" (1:0.618). Also he found this relationship between the lateral incisor and the canine.
It must be noted that this proportion was derived from the apparent size, as viewed directly from the anterior, for Levin also stated that, "attempts to find the relationship between the measured widths of the incisors have been futile."10 This essentially means that the width of the central incisor is in golden proportion to the lateral incisor, as is the lateral to canine and the canine to first premolar, when viewed from the front. He also stated that the anterior segment is in golden proportion to the width of the smile (Fig. 13).11 Maxillary central incisors, because of their position in the front of the arch, should appear to be the widest, whitest, and therefore, the most predominant teeth when viewed from the frontal aspect.
At times the golden proportion of the anterior maxillary teeth have a larger ratio on one side of the arch relative to the other side. This is often the case when a peg lateral is present unilaterally. As a result, different numeric ratios are obtained for the relative proportion of the central incisors, despite the fact that they are identical in width (Fig. 14).
Clearly, for the golden proportion to be most useful in esthetic dentistry, it must be adapted for easy bilateral analysis of the teeth. Snow4 has advocated the use of the "Golden Percentage" as a means of applying the golden proportion across the midline to encompass the total canine-to-canine width (Table 2).
The Golden Percentage of 10%: 15%:25%:25%:15%:10% is a more meaningful tool to analyze the esthetic properties of a smile (Fig. 15). The principle of the Golden Percentage in evaluation and treatment planning appears to be of significant benefit in esthetic smile design.
In today's health- and esthetic-conscious world, the smile is considered an important component of an individual's overall appearance and well being.
As the public desire for improved dental esthetics increases,12 and as dental materials and techniques improve, the nature and scope of dental treatment is evolving to emphasize elective esthetic treatment.
Esthetics embraces the study of beauty and emotional responses to it; esthetic dental treatment involves artistic and subjective components designed to create the illusion of beauty. Thus a greater understanding of esthetic principles is needed. Scientific analysis of beautiful smiles has shown that the principle of golden proportion can be systematically applied to evaluate and improve dental esthetics in predictable ways.
Dr. Mancuso maintains a general dental practice in Welland, ON. He is a fellow of the Academy of General Dentistry, a fellow of the Academy of Dentistry International, a fellow in the Pierre Fauchard Academy and a member of the American Academy of Cosmetic Dentistry. He is President of the Ontario Academy of General Dentistry. He conceived and developed Millenium Aesthetics in Niagara, ON.
Oral Health welcomes this original article.
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2:1 = 2.00
3:2 = 1.500
5:3 = 1.666
8:5 = 1.600
13:8 = 1.625
21:13 = 1.615
34:21 = 1.619
55:34 = 1.618
89:55 = 1.618
144:89 = 1.618
The continuation of this progression to infinity will continue to yield a ratio of 1.618:1
Converting golden proportion to golden percentage.
|Maxillary Tooth||Golden Proportion Ratio||Golden % Calc. (ratio)|
|Right canine||0.618||0.618/6.472 (10%)|
|Right lateral incisor||1.000||1.000/6.472 (15%)|
|Right central incisor||1.618||1.618/6.472 (25%)|
|Left central incisor||1.618||1.618/6.472 (25%)|
|Left lateral incisor||1.000||1.000/6.472 (15%)|
|Left canine||0.618||0.618/6.472 (10%)|
The Golden Proportion has been applied to the total canine-to-canine width to become the "Golden Percentage": 10%:15%:25%:25%:15%:10%