by Sofie Neudecker -

05. Facial Beauty

The Golden Ratio: Mathematics and Human Beauty

The golden ratio is also known as : 1. 618, divine proportion, extreme and mean ratio, golden mean or golden section.

In mathematics, the golden ratio, approximately equal to 1.618, is often denoted by the Greek letter ϕ or τ (Carlson, 2023). It represents the ratio of a line segment cut into two parts such that the ratio of the whole segment to the longer part is the same as the ratio of the longer part to the shorter one. This concept dates back to ancient Greece, where Euclid referred to it as the „extreme and mean ratio“ in Elements.

In modern algebra, the golden ratio is represented by the equation (x + 1)/x = x/1, where the shorter segment is one unit and the longer segment is x units. Rearranging this equation yields the quadratic equation x² – x – 1 = 0, with the positive solution x = (1 + √5)/2, which is the golden ratio.

Phi in the Human Face

The golden ratio appears frequently in the human face, providing insight into the design of facial features. According to Meisner (2012), the head forms a golden rectangle, with the eyes positioned near its midpoint. Furthermore, the mouth and nose are located at golden sections of the distance between the eyes and the bottom of the chin.

This principle can also be applied to a side profile. Similar golden proportions can be found when examining the face from different perspectives, highlighting how this mathematical ratio plays a role in facial structure.

The similar principles and proportions of the golden ratio can be discovered if we change our perspective and look at a profile of a face from the side view.

Human Beauty and the Divine Proportion

Numerous studies have linked the golden ratio to perceptions of beauty. A 2009 university study on attractiveness demonstrated that faces deemed particularly beautiful had more numerous, even proportions closely aligned with the golden ratio. In the following example, a „golden ruler“ is applied to an attractive face, showcasing the key ratios.

  • The blue line forms a perfect square encompassing the pupils and the outer corners of the mouth.
  • The golden section of the blue line defines several features, including the nose, nostrils, and upper lip, as well as key points on the ear.
  • The yellow line, a golden section of the blue line, delineates the nose’s width, the distance between the eyes and eyebrows, and the space from the pupils to the nose tip.
  • The green line, a golden section of the yellow line, marks the width of the eye and the distance from the lashes to the eyebrow.
  • The magenta line, a golden section of the green line, defines the distance from the upper lip to the nose and other dimensions of the eyes.

Phi and the Teeth

Phi can also be found in the proportions of teeth. For example, the ratio of the width of the first tooth to the second tooth is equal to phi. Similarly, the width of the smile relative to the third tooth also corresponds to phi proportions.

Variations and Other Factors in Beauty

It’s crucial to note that these observations do not imply that all beautiful faces conform to a strict set of golden ratio proportions. There are „endless variations in beauty that are as unique as each individual.“ Moreover, a positive attitude can enhance beauty, as more elements of phi appear in our faces when we smile. Interestingly, symmetry, often considered a hallmark of beauty, does not always equate to attractiveness. Perfectly symmetrical faces can appear unnatural, while small imperfections and subtle variations in the arrangement of golden proportions often enhance a person’s appeal.

For instance, many of the proportions in Angelina Jolie’s face adhere to the golden ratio. However, her unique beauty lies in the small imperfections and individual differences that set her apart. A perfectly proportional face might appear more ordinary, lacking the complexity that makes her face captivating.

For instance, many of the proportions in Angelina Jolie’s face adhere to the golden ratio. However, her unique beauty lies in the small imperfections and individual differences that set her apart. A perfectly proportional face might appear more ordinary, lacking the complexity that makes her face captivating.

The Benefits of Being Attractive

Numerous studies suggest that attractive individuals receive „unfair, beneficial treatment“ in various aspects of life, such as business, school, and social interactions (Holmes Place, 2023). The advantages of beauty are observed across cultures, as certain facial features are universally deemed attractive. A University of New Mexico study found that „beauty and symmetry are related to intelligence,“ which can lead to better career prospects, higher incomes, and greater persuasive power.

However, some research indicates that highly attractive people may also face specific challenges. For instance, a study found that attractive individuals may be at a disadvantage in job interviews when the decision-makers are of the same sex (Dean, July 2022). Additionally, attractive people can experience social rejection from their same-sex peers and may be overlooked as romantic partners due to fears of rejection. Despite these drawbacks, the positive benefits of attractiveness typically outweigh the negatives, contributing to the continuous growth of the beauty and plastic surgery industries.

For more on the universal appeal of beauty and the benefits of being attractive, watch this video here.

What is Beauty?

Ultimately, we must ask ourselves whether the golden ratio is the sole criterion for defining beauty. The Cambridge Dictionary (2023) provides multiple definitions of beauty, including „the quality of being pleasing and attractive, especially to look at“ and „the business of making people look attractive, using makeup, treatments, etc.“ However, beauty can also be viewed beyond physical appearance. For instance, beauty can be „something that is an excellent example of its type,“ which can include a person’s character, skills, or personality traits. This broader understanding reminds us that beauty is subjective and personal—everyone can feel beautiful in their own unique way.

Sofie Neudecker, 27. 12. 2023

Sources:

Carlson, Stephan C. „Golden Ratio.“ Encyclopedia Britannica, Science and Tech, November 2023. https://www.britannica.com/science/golden-ratio.

Meisner, Gary. „The Human Face and the Golden Ratio.“ The Golden Number, May 2012.
https://www.goldennumber.net/face/.

Cambridge Dictionary. „Meaning of Beauty in Essential English Dictionary.“ Cambridge Dictionary, 2023. https://dictionary.cambridge.org/us/dictionary/english/beauty.

Holmes Place. „The Benefits of Being Beautiful: Does Looking Good Help You Get Ahead?“ Holmes Place, 2023. https://www.holmesplace.com/en/en/blog/lifestyle/looking-good-helps-you-get-ahead#:~:text=Looking%20Good%20In%20men%2C%20a%20strong%20jaw%3B%20in,healthier%2C%20wealthier%2C%20more%20socially%20dominant%20and%20more%20trustworthy.

Dean, Jeremy. „5 Disadvantages of Being Beautiful (Plus 5 Advantages).“ PsyBlog: Attractiveness, July 2022. https://www.spring.org.uk/2022/07/disadvantages-of-being-beautiful.php.

„Zwischen Botox und Body Positivity: Ist Schönheit Universell?“ Der Roter Faden, ZDF Info Dokus & Reportagen, March 2020. https://www.youtube.com/watch?v=2Q1v_Gmgu54.

by Sofie Neudecker -

01. Proportions within Faces

Generalized Distributions and Ratios in the Human Face

No two faces are identical, and each face has its own specific characteristics and unique proportions. For this reason, our face is a significant part of our identity as human beings. However, the „individual characteristics“ of all faces can be simplified and „integrated into a basic structure“ (Marina Fridman, 2023). The following five reference points are used to assist learners in the process of portrait drawing. Additionally, they provide interesting insights into the general proportions commonly found in human faces:

Even though these elements might seem obvious, it is important to point them out because they often „become lost in the complexities of the face.“ Luckily, once they have been consciously noticed, they are very easy to remember. However, one must remain aware that every face is different, and these elements should be seen as guidelines rather than fixed rules in portrait drawing. Our understanding of proportions should never make us „lazy in our observation of the model.“ Always double-check with your subject, as these guidelines are not universally applicable.

Observations by Leonardo da Vinci

Leonardo da Vinci, one of the „greatest portrait artists“ (Juhi Kulkarni, 2007-23) of the Renaissance, created „thousands of drawings about anatomy.“ In his many notes, accompanied by explanatory drawings, he provides valuable insights into using proportions in portrait drawing.

To draw a head in which the features and agree with the turn and bend of the head, pursue this method. You know that the eyes, eyebrows, nostrils, corners of the mouth, and sides of the chin, the jaws, cheeks, ears and all the parts of the face are squarely and straight set up on the face.

Leonardo da vinci

This quote shows that da Vinci was well aware of the fact that all faces have something in common and seem to follow a general scheme.

In full face there are of 11 types. These are equal, take in the middle, train in the middle, with the tip thick, and at the root narrow or narrow at the tip and wide at the root full stop with nostrils wide or narrow high or low and the opening wide or hidden by the point. And you will find an equal variety in the other details, which things you must draw from nature and fix them in your mind, or else when you have to draw a face by heart carry with you a little book in which you have noted such features and when you have cast a glance at the face of a person you wish to draw you can look, in private which nose or mouth is most alike or there make a little mark to recognise it again at home.

Despite his adherence to general anatomical schemes, Leonardo da Vinci was highly aware of the unique and individual variations among people. His exceptional eye and keen perception allowed him to distinguish individual peculiarities from broader concepts. This ability is evident in the detailed categorisations he made in his notebooks, which later enabled him to draw portraits from memory. Here are a few examples of the insights he recorded:

The distance from the attachment of one ear to the other ear is equal to that from the meeting of eyebrows, to the chin and then find the width of the amount is equal to the length of the parting of the lips to the bottom of the chin.

The cut or the depression below the lower lip of the mouth is halfway between the bottom of the nose and the bottom of the chin.

From the eyebrow to the junction of the lip with the chin and the angles off the jaw where the ear joins the temple will be a perfect square. And each side by itself is half the head.

Studying Leonardo’s notes and sketches is deeply inspiring because he not only captures the nuances he perceives in nature but also connects them to broader, universal concepts. His work reveals that, while each individual is unique, there are underlying similarities that unite us all.

Sources:

Marina Fridman. „5 Proportions of the Face to Use During Your Next Portrait Drawing.“ The Drawing Source. Last modified 2023. https://www.thedrawingsource.com/proportions-of-the-face.html.

Juhi Kulkarni. „Leonardo da Vinci on Portrait Drawing.“ Anatomy Masterclass. Last modified 2023. https://anatomymasterclass.com/news/leonardo-da-vinci-on-portrait-drawing.