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Stephen Marquardt Phi (Golden ratio) mask formally refuted
A formal criticism of Stephen Marquardt’s Phi (Golden ratio) mask is about to be published. The electronic version was posted online a few days ago. Here is the article in its entirety. It is written for a scientific/medical audience, but its contents have been discussed in a more layperson-friendly manner at this site before.
Stephen Marquardt has never published the validity of his mask in a peer-reviewed journal though some papers have favorably reviewed it. Strictly speaking, the criticism in the following article should not be considered definitive since time must be given to Marquardt or others to critique it, and others are more than welcome to try.
In a way, I feel sorry for Marquardt since he has worked on his Phi mask for decades, is very strongly devoted to it and seems to have had an epiphany after coming up with it. I have never experienced such a moment and would not want to find out what it feels like to have a cherished idea thoroughly refuted.
Anyway, a shortcoming of the following article is that it cites a study about the thinness of fashion models using a mid-1990s sample (average BMI 17.57), but it should have been added that fashion models have gotten thinner since then, the current preference among fashion designers being for models with a BMI in the neighborhood of 16.
Marquardt’s Phi Mask: Pitfalls of Relying on Fashion Models and the Golden Ratio to Describe a Beautiful Face
Aesthetic Plastic Surgery
DOI 10.1007/s00266-007-9080-z (pdf link)
Electronic publication, ahead of print; Jan 4, 2008
Stephen Marquardt has derived a mask from the golden ratio that he claims represents the “ideal” facial archetype. Many have found his mask convincing, including cosmetic surgeons. However, Marquardt’s mask is associated with numerous problems. The method used to examine goodness of fit with the proportions in the mask is faulty. The mask is ill-suited for non-European populations, especially sub-Saharan Africans and East Asians. The mask also appears to approximate the face shape of masculinized European women. Given that the general public strongly and overwhelmingly prefers above average facial femininity in women, white women seeking aesthetic facial surgery would be ill-advised to aim toward a better fit with Marquardt’s mask. This article aims to show the proper way of assessing goodness of fit with Marquardt’s mask, to address the shape of the mask as it pertains to masculinity-femininity, and to discuss the broader issue of an objective assessment of facial attractiveness.
Generalized Procrustes analysis is used to show how goodness of fit with Marquardt’s mask can be assessed. Thin-plate spline analysis is used to illustrate visually how sample faces, including northwestern European averages, differ from Marquardt’s mask.
Marquardt’s mask best describes the facial proportions of masculinized white women as seen in fashion models.
Marquardt’s mask does not appear to describe “ideal” face shape even for white women because its proportions are inconsistent with the optimal preferences of most people, especially with regard to femininity.
Aesthetics - Golden ratio - Phi mask - Procrustes analysis - Stephen Marquardt - Thin-plate splines
The nature of beauty is an intriguing topic. It is tempting to describe beauty in terms of a simple, recurring theme such as the golden ratio. Dividing a line segment into two parts such that the ratio of these parts equals the ratio of the larger part and the line segment gives us the golden ratio (Phi).
A number of attempts have been made to describe “ideal” facial proportions in terms of the golden ratio [14,17,30,34]. However, several studies have not found any relationship between facial attractiveness and compliance with golden ratio proportions [20,22,27,33,38] or evidence of any special aesthetic preference for the golden ratio among humans . Stephen Marquardt  has tortuously derived supposedly “ideal” facial proportions from the golden ratio. Marquardt’s Phi mask has been favorably reviewed by some authors [1,14,16]. Kim  reported that Marquardt’s mask is convenient for facial analytics, and Bashour  has claimed that Marquardt’s mask is a suitable tool for developing an objective system to assess facial beauty. However, Marquardt’s mask is associated with a number of problems.
A preliminary examination of Marquardt’s mask does not suggest that it describes “ideal” face shape. Figure 1 shows the front and side views of the mask with most internal lines in the original mask erased so as not to obscure overall face shape. The mask appears to trace best the outline of an individual with European ancestry. Marquardt claims that his mask applies equally well to attractive individuals across geographic populations. However, in W.W. Howells’ data set involving 57 linear craniofacial interlandmark distances, the proportion of global variation between populations was 24% for the first principal component of shape variation and 33% for the second principal component . Large as these values are, because a correlation structure underlying shape variation across populations also exists, the face shape differences across continental populations are striking. Therefore, how can an “ideal” face shape be conceived across populations as distinct as Norwegians, Tanzanians and the Chinese?
Using the ratio of the basion-prosthion length to the basion-nasion length as an index of prognathism, Hanihara  reported that the mean of this index was 94.9 ± 3.45 for Norwegians, 104.1 ± 4.45 for Tanzanians, and 96.6 ± 4.01 for northern Chinese. Similarly, when the ratio of the simotic subtense to the simotic chord was used as an index of the prominence of the nasals, the result was 52.0 ± 10.20 for Norwegians, 30.3 ± 6.49 for Tanzanians, and 34.8 ± 12.13 for northern Chinese . Well-separated population means make it difficult to conceive the “ideal” Homo sapiens face.
Fig. 1. The outline of Marquardt’s mask. See the complete version at http://www.beautyanalysis.com/
Even among individuals of European ancestry, the mask outline does not appear to describe a typical man or woman. With reference to face shape variation resulting from masculinization and feminization [12,31], the outline of Marquardt’s mask appears to be that of a masculinized woman, as evident from the eminent supra-orbital ridges, low-set eyebrows, strong nasoglabellar curvature, cheekbone placement high on the face, retracted mandibular symphysis, and a prominent and squared chin.
By Marquardt’s own admission, his mask describes a woman’s shape. Therefore, the masculine element is curious given that a strong preference for facial femininity has been described in the general population. For instance, in a metaanalysis of the effect of facial femininity on attractiveness ratings, the effect size correlation was r = 0.64, with a standard deviation of 0.39 and a 95% confidence interval of 0.51 to 0.74 .
Marquardt’s mask curiously leans toward a Class 2 profile. A Class 1 profile is reported as the most attractive [4,7,15,18,26], and minor deviations from a Class 1 profile are assigned higher ratings among individuals of European ancestry if they lean toward the Class 3 type rather than the Class 2 type [4,5,15,21,26]. Additionally, Marquardt’s mask is inconsistent with the optimal preferences of white North Americans regarding soft tissue nasion placement  and earlobe proportions . Therefore, for a number of reasons, as a first approximation, it is difficult to see Marquardt’s mask as a descriptor of “ideal” facial proportions.
There also is a problem with the method Marquardt recommends for comparing a given face with the mask. He instructs that before the mask is superimposed on a person’s photo to assess goodness of fit, the distance between the lips and the midpupil level should be equalized for the front view, and the inferior irion-lip distance should be equalized for the side view. Bashour  has carried out the superimposition by equalizing interpupillary distance. However, when control is used for face size, neither of these distances or other conceivable measures such as face length or face breadth are constant for all people, even when applied to only one specific ethnic group. Therefore, how can goodness of fit be assessed?
To compare shapes, control must be used for three factors: location, size, and rotational effects. A standard procedure is to use Procrustes analysis [2,25], which proceeds as follows. Landmarks on the shapes to be compared are identified, and their Cartesian coordinates are measured. Then, the centroid of each shape, which is the center of mass of a physical system with unit mass at its landmarks, is computed. The centroid size of each shape, which is the sum of squared distances of a shape’s landmarks around its centroid, also is computed.
After this, each shape is scaled to unit centroid size. Subsequently, the shapes to be compared are aligned with respect to centroid position and rotated so that the sum of squared distances between corresponding landmarks is minimized. Now the shapes can be compared, as in computation of the Riemannian distance (0 ≤ ρ ≤ π/2) between the shapes. The shape differences can be visualized by using thin-plate spline analysis .
This article aims to use the geometric morphometric methods of generalized Procrustes analysis and thin-plate spline analysis to illustrate how goodness of fit can be assessed with Marquardt’s mask, where it stands with respect to masculinity-femininity, and how valid Marquardt’s claims are. It also aims to address the issue of an objective evaluation of facial beauty.
The landmarks used are shown in Fig. 2. If a point approximately homologous to a landmark on Marquardt’s mask lay along a curve, making it difficult to pinpoint, then the y coordinate was defined at the point of intersection of surface tangents corresponding to the angular outline of Marquardt’s mask. A better method exists , but because few such semi-landmarks exist, this article is well served by the aforementioned approximation. The choice of the landmarks is based on a need to focus on masculinity-femininity and the limitations of a paper by Hennessy et al. .
Fig. 2. Landmarks used to compare face shapes
Hennessy et al.  provided front and profile views of a very masculine northwestern European male average, a northwestern European male-female (population) average, and a very feminine northwestern European female average. The masculine and feminine extremes provided by the authors exaggerated the average male-average female shape difference threefold. These averages were produced via geometric morphometric methods, as opposed to the typical composite made by blending faces with interpupillary distance equalized, and hence were suitable for this article.
To facilitate a better understanding of Marquardt’s mask, a three-dimensional (3D) face roughly corresponding to it was generated using Poser (http://www.e-frontier.com/) and Victoria 4 from DAZ3D (http://www.daz3d.com/) and contrasted with a more feminine face (Figs. 3 and 4). Generalized Procrustes analysis and thin-plate spline analysis were carried out using the Shapes package  for the R software environment for statistical computing and graphics (http://www.r-project.org/).
Fig. 3. Rough three-dimensional approximation of Marquardt’s mask in front view (left) and a more feminine face
Fig. 4. Rough three-dimensional approximation of Marquardt’s mask in side view (left) and a more feminine face
Tables 1 and 2 list the Riemannian distances between Marquardt’s mask and other faces. Table 1 shows that Marquardt’s mask is closest to its rough 3D approximation, which should not be surprising. Marquardt’s mask also appears to be closer to the masculine side of the population average for northwestern Europeans, but caution is required here because of the limited number of landmarks used and the large distances between all the northwestern European averages and Marquardt’s mask.
Table 1 Riemannian distances between Marquardt’s mask (front view) and other faces
Very masculine northwestern European average man
Northwestern European male-female average
Very feminine northwestern European average woman
Rough three dimensional approximation of Marquardt’s mask (Fig. 3, left)
The face on the right in Fig. 3
Table 2 Riemannian distances between Marquardt’s mask (side view) and other faces
Very masculine northwestern European average man
Northwestern European male-female average
Very feminine northwestern European average woman
Rough three dimensional approximation of Marquardt’s mask (Fig. 4, left)
The face on the right in Fig. 4
In Table 2, we again observe that Marquardt’s mask is closest to its rough 3D approximation. Marquardt’s mask also appears to be closer to the population average for northwestern Europeans than to the very masculine and very feminine shapes. Again, however, the large distances between all the northwestern European averages and Marquardt’s mask should be noted.
How Marquardt’s mask differs from the other faces can be visualized using thin-plate splines, whereby the deformation of a 2D grid in the background shows how Marquardt’s mask can be deformed to make it approach the other shapes. Figure 5 shows the thin-plate splines for the front view of the face, and Fig. 6 shows those for the side view of the face.
Fig. 5. Thin plate splines showing how Marquardt’s mask can be deformed to transform it to various faces addressed in Table 1
Fig. 6. Thin plate splines showing how Marquardt’s mask can be deformed to transform it to various faces addressed in Table 2
Figure 5 shows that some features of Marquardt’s mask are on the masculine side of the population average for northwestern Europeans, particularly the high placement of the cheekbones, the facial narrowing, and, to a lesser extent, the interexocanthion length and the width of the chin. The data from Hennessy et al.  show that the superior medial orbital margins are displaced inferiorly with increasing masculinization. The closest analogs in Marquardt’s mask are the low-set eyebrows (Fig. 1), clearly masculine in their placement. Figure 6 shows that the nasoglabellar curvature, the retraction of the mandibular symphysis, and the prominence of the chin in Marquardt’s mask are on the masculine side of the male-female northwestern European average. Figure 6 also shows that Marquardt’s mask leans toward a Class 2 profile compared with the average northwestern European profiles.
The thin-plate splines show, within the limits of the landmarks used, how the 3D approximation of the Phi mask can be fine-tuned to make it better match the mask’s proportions. The chin must be made more prominent in both the side and front views and slightly wider. Additionally, the prominence of the zygomatic arches must be increased. Although the eyebrows are not addressed in this article, the 3D approximation would need to have its eyebrows lowered as well. These changes are consistent with a slightly more masculine look and a more robust appearance of the cheekbones.
Marquardt’s claims regarding his mask are not consistent with the extant literature on correlates of facial beauty, particularly regarding placement along the masculine-to-feminine discriminant, mandibular profile and soft tissue nasion placement. The masculine element in a mask supposedly describing the “ideal” female face is curious given how strongly above average femininity in women is preferred by most people . But then, in coming up with his mask, Marquardt relied heavily on female fashion models as a reference standard of beauty. Therefore, a brief review examining the looks of female fashion models, henceforth “fashion models,” is pertinent.
Marquardt has shown that fashion models, on the average, have masculinized faces. Because sex hormones have a global effect, the masculinized faces of fashion models would be expected to correspond to masculinized physiques as well. Indeed, this is readily observed by a perusal of mainstream fashion magazines such as Elle and Vogue.
However, Tovee et al.  have described fashion models as “hourglass shaped” by virtue of a reported average circumferential waist-to-hip ratio of 0.7 and a circumferential bust-to-hip ratio of 0.99. An hourglass approximation applies to a front view, not a 3D view (i.e., the circumferential measurements should be viewed with caution). Given the typically small breasts of high-fashion models, the data of Tovee et al.  show that they have a larger rib cage for a given hip size than normal women and glamour models. A larger rib cage will stretch out the waist in front view, but the thinness of high-fashion models will translate to a small waist circumference, not much higher than that of shorter glamour models. A large rib cage is not consistent with an hourglass approximation because an hourglass has a narrow midsection. Therefore, Tovee et al.’s  conclusion is mistaken.
The typical thinness of fashion models also is noteworthy. Most individuals prefer women fleshier than fashion models [8,28]. In a sample of 300 fashion models, the average body mass index (BMI: weight divided by the square of the height) was 17.57 kg/m2 , whereas the optimum BMI of women rated by Western adults has been reported to be 2 to 3 kg/m2 higher [35,36]. In short, female fashion models obviously do not reflect the aesthetic preferences of the general public and should not be used to develop an aesthetic archetype.
A perusal of fashion models shows a preponderance of Nordic features. Therefore, it may have been better to compare Marquardt’s mask to Nordic facial norms instead of a population with a dominant Celtic element (northwestern Europeans). However, I am not aware of data on any Nordic population similar to the data provided by Hennessy et al. , and there is a significant Anglo-Saxon element in northwestern Europe, too. Additionally, because Marquardt claims that his mask applies to Homo sapiens, the differences among various northern European populations are not relevant to his argument.
It could be argued that Marquardt is mistaken in believing his mask describes the ideal proportions of a woman, and that instead, using objective methodology, he has come across a template for Homo sapiens modified into male and female forms as a result of sex differentiation. However, why are parts of Marquardt’s mask on the masculine side of the population average for northwestern Europeans? Additionally, why does this template most closely approximate, with some distortions, an outlier variety of Homo sapiens? For instance, a tendency for greater flattening of the nasal bones/midface to correlate with more protruding jaws has been clearly documented in humans, and northern Europeans, with the most prominent nasal bones and the most regressed jaws, are at an extreme of face shape . How can a mask with prominent nasals and a regressed jaw describe the “ideal” face of Homo sapiens?
More importantly, how objective is Marquardt’s methodology? Using the golden ratio to describe facial beauty does not make the attempt objective. The human face presents a tremendous number of proportions, and with a hard enough look, some of them are bound to approximate the golden ratio. Marquardt’s methodology, detailed in his U.S. patents (nos. 5,659,625 and 5,867,588), is similar to that of a researcher carrying out a huge number of statistical tests on the data he possesses in the hope of coming across a statistically significant find, which, if found, cannot make the researcher confident that the find is of any real world significance. For instance, even after years of working on a 3D version of his mask, Marquardt has yet to come up with it, obviously because he has not come across an intuitive and simple way of putting together geometric structures derived from the golden ratio to form an aesthetically pleasing human face in 3D that resembles his 2D masks in front and side views. If it is possible to derive the outline of a beautiful face from the golden ratio, then Marquardt certainly has not achieved this.
The preceding discussion leads to the issue of an objective assessment of attractiveness. Can this be achieved? If a reference template or face outline representing the “ideal” existed, then geometric morphometric methods could be used to compare individual faces with it and to describe the discrepancies numerically and visualize them graphically. A detailed comparison would require the use of a sufficiently large number of landmarks. This could be cumbersome, especially in 3D comparisons (e.g., photogrammetry or laser surface scanning), and could pose a problem if homologous locations need to be identified for points lying along a curve. These points could not be considered true landmarks but rather semi-landmarks or sliding landmarks. However, geometric morphometric methods that deal with semi-landmarks  and interpolate pseudo-landmarks  between landmarks have been developed. In addition, assessment of Cartesian coordinates is not more cumbersome than Bashour’s  method. Therefore, an objective comparison can be made, but how is a reference standard objectively obtained in the first place?
Development of an aesthetic reference standard should start with well-documented correlates of beauty, namely, averageness, above average femininity in women, and low fluctuating asymmetry . For example, development of an aesthetic female mask for a given ethnicity could involve assessing population norms for women, developing a mask with zero fluctuating asymmetry to reflect this average, feminizing it to varying degrees, and assessing a large and random sample of individuals from this ethnic group to determine what degree of above average feminization is regarded as the most appealing by most individuals. This modified average could be altered further on a computer by judges to obtain a more pleasing mask and the central tendency of the alteration noted. The final product would be a mask describing the face shape of a woman that most individuals of her ethnicity would find very attractive, which in reference to Kim’s  contention, would serve much better than Marquardt’s mask with respect to facial analytics. This method will not produce a reference standard based on a simple, recurring theme, but it remains to be shown that the latter is possible.
Stephen Marquardt has shown how misleading overenthusiasm for the golden ratio and a reliance on fashion models can be. His mask is unable even to describe aesthetically pleasing proportions for white women, especially because of its masculinization.
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