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Clinical ImplicationsThe results of the study do not support the use of the rule of golden proportion in an unmodified form. Despite the applicability of the rule to central incisors, marked deviation of canine width ratios identifies the need for the consideration of individual properties when planning implant therapy or conventional prosthetic treatment in the esthetic zone.

Statement of problem. In the last 2 decades, several studies have questioned the application of the rule of golden proportion in dentistry.

Purpose. The purpose of this study was to test the rule of golden proportion for the maxillary anterior region in a population of young, healthy people with excellent oral hygiene at the Faculty of Dentistry of the University of Szeged.

Material and methods. One hundred and nine participants (36 male, 73 female) were examined, 41 of whom had previously worn orthodontic appliances. A Canon 450D camera, Canon 100 mm Macro lens, and a Sigma 140 DG ring flash were used for digital documentation. The accreditation photography protocol of the American Academy of Cosmetic Dentistry was applied to standardize the images. The width of the teeth from the frontal view was measured digitally. Tooth dimension data were collected and the Mann-Whitney U test was performed (�=.05).

Results. The relative width of the central incisors, lateral incisors, and canines was 1.6:1:0.85 on both sides. The ra-tios were not influenced by either sex or previous orthodontic treatment.

Conclusions. The results demonstrated that, because of the difference in canines in this population, the golden pro-portion in its original form was not valid for the maxillary anterior region. Orthodontic treatment does not change the dentition toward the golden proportion. The results suggest that the rule of golden proportion should not be used for treatment planning without observing individual modifying factors. (J Prosthet Dent 2013;110:211-215)

Width ratios in the anterior maxillary region in a Hungarian population: Addition to the golden proportion debate

András Forster, DMD,a Róbert Velez, DMD,b Márk Antal, DMD,c and Katalin Nagy, DDS, PhDd

University of Szeged, Faculty of Dentistry, Szeged, Hungary

aAssistant Professor, Department of Operative and Esthetic Dentistry.bGraduate student, Department of Operative and Esthetic Dentistry.cAssistant Professor, Department of Operative and Esthetic Dentistry.dDean, Professor, Department of Oral Surgery.

The first written definition of the golden ratio was provided by the an-cient Greek mathematician Euclid of Alexandria.1 In mathematics and the arts, 2 quantities are in the golden ra-tio if the ratio of the sum of the quan-tities to the larger quantity is equal to the ratio of the larger quantity to the smaller. The golden ratio is an irratio-

nal mathematical constant, approxi-mately equal to 1.618.

The rule of golden proportions in dentistry is defined as follows: after measuring the width of the maxillary anterior teeth from the frontal view, and considering the width of the lat-eral incisor as 1, the width of the ca-nine compared to that of the lateral

incisor is 0.62, while the width of the central incisor compared to that of the lateral is 1.62.2 In the 1960s and 1970s, several studies were conduct-ed to find possible applications of the rule in dentistry. Ricketts3,4 was the first to define the application of the rule for the anterior maxillary teeth, stating that application of the for-

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mula was indispensable in restorative and prosthetic dentistry. Magne et al5 also found a width/length ratio of 78% for unworn central incisors, that is, length was related to width accord-ing to the rule of golden proportion.

The rule of golden proportion in dentistry essentially means that if the width of a maxillary anterior tooth can be measured from the frontal view, the width of the other maxil-lary teeth can be calculated. It is also true that according to the formula the width of each maxillary anterior tooth can be calculated on the basis of the width of the maxillary anterior region from the frontal view. Therefore, if the width of the lateral incisor is 1 unit, then the width of the central incisor is 1.6 units, and the width of the ca-nine is 0.6 units. This equals 3.2 units (1.6+1+0.6) for the half anterior arch and 6.4 units from canine to canine.

According to Lombardi6 the rule, as it is, can be applied only within nar-row limits. Measurements by Preston2 yielded results corroborating Lombardi6: the use of the rule for the maxillary ante-rior teeth would lead to narrower teeth than desired. In spite of their findings on central incisors, Magne et al7 con-cluded that by using the rule of golden proportion as it is, narrower canines than desired would result. Murthy and Ramani8 concluded that the golden percentage principle can be applied if percentages are adjusted, considering the ethnicity of the population.

The largest study9 conducted on this topic involved 376 dental stu-dents in Jordan. The authors found that the average ratio of the maxillary central incisor, the lateral incisor, and the canine was 1.53:1:0.8. Dimen-sions that corresponded to the rule of golden proportion were found in only 31.3% of women and 27.1% of men.9

Ong et al10 conducted a survey with a selected sample of individuals who had to assess 60 different smiles on a 5-point Likert-type scale. Their results demonstrated that overall dental attractiveness did not depend on any particular feature of the den-tition. In a survey involving 301 den-

tists,11 it was found that dentists pre-ferred other esthetic design principles to that of the golden proportion.

A study conducted at the Ankara University analyzed the smile of 100 dental students.12 No single recurrent or typical proportion for all anterior teeth could be determined. A signifi-cant sex-related variance of the central maxillary incisors and canines was also found, with the canines showing the greatest gender variation. When re-storing or replacing the maxillary ante-rior teeth, a complex approach should be followed, which observes the den-tofacial specificities of the individual, the anatomic and esthetic axis of the face, a wide variety of natural tooth proportions according to the facial measurements present in the scientific literature and the anatomic properties typical of the treated ethnic group.13-15

The purpose of the study was to assess the rule of golden proportions in a population of Hungarian dental students and professionals. The in-vestigation also considered whether orthodontic treatment (even uninten-tionally) changed the patient’s denti-tion toward the golden proportion.

The null hypothesis was that the rule of golden proportions is valid for the maxillary anterior region, so the width ratio of the central incisors, lat-eral incisors, and canines would cor-relate with a 1.6:1:0.6 ratio.

MATERIAL AND METHODS One hundred and nine volunteers

(36 male, 73 female) participated in this study. The volunteers were either dentists or dental students of the Fac-ulty of Dentistry, University of Szeged, Hungary. The objective was to assess young people with an outstanding level of dental health awareness to obtain results that are closest to the local anthropological optimum. Post hoc power analysis was conducted as described in Erdfelder et al16 (1-�=.97 [d=0.5; �=.05]). The mean age of the participants was 24 years (male: 23, female: 24.2; ±6 years). Forty-one participants had previously worn

orthodontic appliances. Sixty-eight participants had not received orth-odontic treatment. Those presenting with a dental prosthesis in the maxil-lary anterior region or with orthodon-tic brackets or bands were excluded from the study. Participants with crowding, spacing, or tooth rotation were also excluded. Data collection was approved by the Regional and Institutional Human Medical Biologi-cal Research Ethics Committee of the University of Szeged.

The images were made with a digital single lens reflex camera (Can-on 450D; Canon Inc, Tokyo, Japan) equipped with a 100 mm f/2.8 macro lens (Canon EF 100mm f/2.8 Macro USM; Canon Inc) and a macro ring flash (Sigma EM-140 DG; Sigma Corp, Kanagawa, Japan). Oral retractors (PTJ Intl Co, Houdemont, France) were used to displace the soft tissues.

To standardize the views, the pro-tocol for photographic presentation of the American Academy of Cos-metic Dentistry (AACD) was used as a guide.17 Since the rule of golden pro-portions requires a frontal view and a view of the entire maxillary anterior region is necessary, the 1:2 ratio re-tracted frontal view picture was select-ed (Fig. 1). All digital images were re-corded by the same person. To achieve standardization, the angles described in the AACD photo protocol were eval-uated by an additional observer.

The following camera settings were used: 1/200 shutter speed, f/22 aper-ture, ISO 200 sensitivity. As a digital camera with an APS-C size sensor (22.2 mm × 14.8 mm) was used, a 1:3.2 mag-nification ratio was applied to achieve the 1:2 magnification defined in the AACD accreditation protocol.

Analysis of the photos was per-formed digitally with software devel-oped for this purpose. After manually marking the most incisal point of the proximal contacts of the maxillary an-terior teeth in photo editing software (Adobe Photoshop CS5; Adobe Sys-tems Inc, San Jose, Calif ), the width of each tooth was computed in pix-els (Fig. 2). To mark contact points,

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1-pixel grayscale dots were used. On all examined digital images, 7 such points were determined; after this, the layer containing the actual picture was deleted, yielding a layer contain-ing only the distances necessary for the measurements. The mesiodis-tal width of the teeth was recorded, shown in Figure 2. This way, neither the corono-apical position of the contact point nor the inclination of the incisal plane compared to the midline of the face could have altered the results. Data were collected in a spreadsheet (Excel; Microsoft Corp, Redmond, Wash) and analyzed with statistical software (SPSS 18; IBM Corp, Armonk, NY).

Descriptive statistics were calcu-lated and comparisons of distribu-tions carried out according to the different grouping of variables. Since a 1-sample Kolmogorov-Smirnov test revealed that the requirement of nor-mal distribution was not fulfilled for the left central incisors and left ca-nines, the use of the Mann-Whitney U test was selected for general compari-sons. The confidence level was set at �=.05. Post hoc power analyses for the Mann-Whitney U test were conducted in a software developed specifically for such purposes (G*Power 3.1.5; Uni-versity of Kiel, Kiel, Germany).

RESULTS It was evident from the descriptive

statistics (Table I) that the canines de-viated markedly from the rule (0.85 on average on the right and 0.89 on aver-age on the left side versus 0.6 as pre-dicted by the rule). To characterize the difference of these measured averages from the predicted 0.6, the 95% confi-dence intervals were calculated (0.66-1.00 right; 0.71-1.13 left). Based on these calculations, 0.6 on both sides fell below the lower limit of the 95% confidence interval, that is, the chance of the predicted 0.6 appearing in the study population was low (≤5%). It could, therefore, be concluded that the predicted value differed signifi-cantly from the measured averages and was not characteristic of the stud-ied population. Apparently this was independent of sex, but to establish that, the Mann-Whitney U test was conducted for both sides by sex. The results showed that no significant dif-ference existed between the canines of males and females either on the right (MWU= 1185, n1=36, n2=73, P=.41, 2-tailed) or the left (MWU=1068, n1=36, n2=73, P=.11, 2-tailed) side. As expected from the descriptive sta-tistics, no significant sex-related dif-ference was found for central incisors on the right (MWU=1249, n1=36, n2=73, P=.68, 2-tailed) or the left

(MWU=1246, n1=36, n2=73, P=.67, 2-tailed) side either. Although the sta-tistical power of these comparisons turned out to be only moderate by convention, the power was so close to high (1-�=.77 [d=0.5, �=.05]) that these comparisons could be accepted as valid and the sex-independence of the observation could be established.

Therefore, to test the null hypoth-esis of side-independent deviation, a Mann-Whitney U test was conducted regarding the sides, without differenti-ating between male and female teeth. The distribution of ratios was inde-pendent of side for both sets of ca-nines (MWU=5232, n1=109, n2=109, P=.13, 2-tailed) and central incisors (MWU=1068, n1=109, n2=109, p=.55, 2-tailed). From these comparisons and the basic statistics, it was concluded that canines in the sample on both sides and in both sexes deviated mark-edly from the expected ratio of 0.6. The null hypothesis was rejected.

Central incisors did not exhibit a statistically significant difference. On average, their ratios corresponded al-most exactly (1.58 on the right side and 1.59 on the left) to the expected 1.6.

As 41 participants in the sample had been previously treated orthodon-tically, it was possible, as an ancillary measurement, to analyze data for possible changes due to orthodontic treatment. The treated and untreated

2 Marking mesiodistal width of maxillary anterior teeth with digital landmarks. Measurement of width was per-formed by connecting vertical lines (black) with perpendic-ular lines (red), length of which were calculated in pixels.

1 1:2 ratio retracted frontal view according to AACD Ac-creditation Photo Protocol.

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groups were compared by tooth type and side. On the right side neither the canines’ (MWU=1289.5, n1=41, n2=68, P=.52, 2-tailed) nor the ratios of the central incisors (MWU=1286.5, n1=41, n2=68, P=.50, 2-tailed) differed significantly by orthodontic treatment. For the left side, the results were simi-lar (MWU=1233, n1=41, n2=68, P=.32, 2-tailed; and MWU=1280.5, n1=41, n2=68, P=.48, 2-tailed; canines and central incisors). The post hoc power analysis revealed near-high power (1-�=0.79 [d=0.5, �=0.05]). It was con-cluded that orthodontic treatment had not caused significant deviation from the untreated pattern. A compar-ison of the means of the treated and untreated groups is given in Table II.

Also as an ancillary measurement, Pearson r was calculated between tooth types, and the analysis revealed a significant inverse correlation be-tween the relative width of canines and central incisors (r =-0.49, P<.01).

Data suggested that canines that markedly deviate from the relative width of 0.6 as predicted by the rule of golden proportion characterized the entire study population, regardless of sex, side, or previous orthodontic treatment. Figure 3 provides a sum-mary of the results.

DISCUSSION

Within the limitations of this study, the golden ratio is not valid in the max-illary anterior region because of the marked deviation of the canines in the examined population as compared to the original 1.6:1:0.6 ratio. Thus, the null hypothesis could be rejected. The findings support the argument against the routine application of the golden proportion in the maxillary anterior re-gion. In particular, the results support the argument of Preston2 and Magne et al,5 who reported that designing implants or prostheses strictly observ-ing the 1.6:1:0.6 rule would lead to canines that are too narrow. Indeed, regardless of grouping, the ratio of 1.6:1:0.85 was found in the analysis, that is, canines turned out to be mark-

3 Summary for entire studied population. RC: right canine; RLI: right lateral incisor; RCI: right central incisor. Abbreviations for left side are identical. Dashed lines represent width ratios for canines (C), lateral incisors (LI), and central incisors (CI), as predicted by rule of golden proportion. Values are given as mean +standard error of the mean. Note that canines deviated significantly from predicted ratio, while central incisors corresponded to rule almost perfectly.

Table I. Basic statistics of width ratios. Values are given as mean (SD)

Table II. Width ratios of orthodontically treated and untreated teeth. Values are given as mean (SD)

Male left

Male right

Female left

Female right

Total (M+F) left

Total (M+F) right

Golden proportion

0.92 (0.16)

0.88 (0.29)

0.88 (0.23)

0.84 (0.16)

0.89 (0.22)

0.85 (0.21)

0.60

1

1

1

1

1

1

1

Canine IncisorLateral

1.58 (0.22)

1.61 (0.39)

1.58 (0.25)

1.59 (0.20)

1.58 (0.28)

1.59 (0.24)

1.60

IncisorCentral

36

36

73

73

109

109

n

Treated left

Treated right

Nontreated left

Nontreated right

Golden proportion

0.86 (0.15)

0.89 (0.27)

0.91 (0.23)

0.83 (0.17)

0.60

1

1

1

1

1

Canine IncisorLateral

1.58 (0.16)

1.63 (0.36)

1.58 (0.28)

1.57 (0.21)

1.60

IncisorCentral

41

41

68

68

n

1.8

Num

ber

of

No

dul

es

Tooth Type

1.4

1.6

1.2

1.0

0.8

0.6

0.4

0.2

0RC LRI RCI LCI

CI

C

LI

LLI LC

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Forster et al

edly wider than predicted by the rule in the examined groups. It must be emphasized that these ratios are sex and side independent, which allows speculation that these relative widths are anthropologically characteristic of the local adult population as a whole.

The finding that orthodontic treat-ment had not modified the ratio of max-illary teeth from the anterior view (the ratios were also treatment independent) might be interpreted as further support that the golden proportion is not a uni-versal esthetic goal - at least not in the anterior maxillary region. Although no data were collected regarding the details of the planning of these orthodontic interventions, it is assumed that ortho-dontists used individual observations for planning, not an abstract ideal of universal beauty. However, it must be noted that the number of participants in the orthodontically treated and untreat-ed groups was not the same, suggesting caution in interpreting this result.

Recent studies by Ali Fayyad et al9 and Hasanreisoglu et al12 reached the similar conclusion that the general application of the golden ratio is not advisable. These authors found ratios of approximately 1.53:1:0.8, which is close to the findings of the present investigation. However, these studies showed that the central incisors also exhibited some deviation, which, again, emphasizes the salience of using data based on actual local measurements rather than following a general rule.

These data describe only the exam-ined population in a reliable manner, and generalization of these results is not suggested. Indeed, the goal was simply to test the applicability of the rule of golden proportion in an ideal population, one which was assumed to yield data closest to the local an-thropological optimum.

It cannot be assumed from these data, however, that there are no popula-tions for which the rule of golden pro-

portion applies. Obviously, the popula-tion described by Ricketts18 was such a population. Therefore, it is suggested that these findings should by no means be interpreted as an absolute argument against the use of the golden proportion, rather, as concluded in other investiga-tions, the golden percentage theory can be applied if percentages are adjusted, taking into consideration the ethnicity of the population.8,14

Despite the limitations of the pres-ent study, it is suggested that this meth-odology, given its simplicity, could be used in multiple populations to allow for generalizations.

CONCLUSION Based on the results of this study,

the following conclusions were drawn:1. The rule of golden proportions is

not valid for the examined population. The ratio of maxillary anterior teeth in this group of people is 1.6:1:0.85. The canine is markedly wider than predict-ed by the golden ratio (1.6:1:0.6).

2. No sex-dependent or side-depen-dent variation of the width ratios of the maxillary anterior teeth was found.

3. In the examined population, orthodontically treated participants did not exhibit a significant differ-ence in width ratios of the maxillary anterior teeth as compared to the corresponding teeth of untreated participants.

REFERENCES

1. Livio M. The golden ratio: The story of phi, the world’s most astonishing number.1st ed. New York: Broadway Books; 2002. p. 1-11.

2. Preston JD. The golden proportion revisited. J Esthet Dent 1993;5:247-51.

3. Ricketts RM. Divine proportion in facial esthetics. Clin Plast Surg 1982;9:401-22.

4. Ricketts RM. The biologic significance of the divine proportion and Fibonacci series. Am J Orthod 1982;81:351-70.

5. Magne P, Gallucci GO, Belser UC. Anatomic crown width/length ratios of unworn and worn maxillary teeth in white subjects. J Prosthet Dent 2003;89:453-61.

6. Lombardi RE. The principles of visual perception and their clinical application to denture esthetics. J Prosthet Dent 1973;29:358-82.

7. Magne P, Belser U. Bonded porcelain resto-rations in the anterior dentition: a biomi-metic approach. Chicago: Quintessence Publishing, Co; 2002. p. 57-99.

8. Murthy BV, Ramani N. Evaluation of natural smile: Golden proportion, RED or Golden percentage. J Conserv Dent 2008;11:16-21.

9. Ali FM, Jamani KD, Agrabawi J. Geometric and mathematical proportions and their relations to maxillary anterior teeth. J Con-temp Dent Pract 2006;7:62-70.

10.Ong E, Brown RA, Richmond S. Peer assess-ment of dental attractiveness. Am J Orthod Dentofacial Orthop 2006;130:163-9.

11.Ward DH. A study of dentists’ preferred maxillary anterior tooth width proportions: comparing the recurring esthetic dental pro-portion to other mathematical and naturally occurring proportions. J Esthet Restor Dent 2007;19:324-37.

12.Hasanreisoglu U, Berksun S, Aras K, Arslan I. An analysis of maxillary anterior teeth: facial and dental proportions. J Prosthet Dent 2005;94:530-8.

13.Mahshid M, Khoshvaghti A, Varshosaz M, Vallaei N. Evaluation of “golden proportion” in individuals with an esthetic smile. J Esthet Restor Dent 2004;16:185-92.

14.Isa ZM, Tawfiq OF, Noor NM, Shamsudheen MI, Rijal OM. Regression methods to inves-tigate the relationship between facial mea-surements and widths of the maxillary ante-rior teeth. J Prosthet Dent 2010;103:182-8.

15.Behrend DA, Harcourt JK, Adams GG. Choosing the esthetic angle of the face: ex-periments with laypersons and prosthodon-tists. J Prosthet Dent 2011;106:102-8.

16.Erdfelder E, Faul F, Buchner A. GPOWER: A general power analysis program. Behav Res Meth Instrum Comput 1996;28:1-11.

17.Boyd JG. AACD accreditation process. Tex Dent J 2005;122:976-9.

18.Ricketts RM. The golden divider. J Clin Orthod 1981;15: 752-9.

Corresponding author: Dr András ForsterUniversity of Szeged, Faculty of DentistryDepartment of Operative and Esthetic DentistryTisza Lajos krt. 64, Szeged, H-6720HUNGARYE-mail: andras.forster@me.com

AcknowledgmentsThe authors thank Péter Kovács, associate pro-fessor of the Faculty of Economics and Business Administration at the University of Szeged for the statistical analysis, and Professor Michele G. Shedlin, Dr Kinga Turzó, Dr Gábor Braunitzer, and Dániel Velez for their valuable contribution to this article.

Copyright © 2013 by the Editorial Council for The Journal of Prosthetic Dentistry.