Finding my dream girl with the ‘Golden Ratio’
What do you think beauty is? A beautiful flower? A beautiful building? Is it “subjective” from person to person? Whom would you perceive as a beautiful girl? A blonde girl with glistening hair? Then how come there are black female models like Naomi Campbell and Tyra Banks playing leading roles in the global modelling industry? Is beauty about being of a certain colour? Can it be measured? Can beauty be defined quantitatively as a standard scale ranging from 1 to 10?
Most of the above questions didn’t carry answers until recent times, in fact we’ve defined a distinct area from Science termed “Aesthetics”, to distinguish what is subjective and that can’t be measured based on a universal standard scale. The never-ending curious research with modern Mathematics has been able to unlock the secrets behind how our eyes and brain perceive beauty in a quantitatively re-presentable manner. This article is an attempt to introduce mathematics and its amazing involvement in Nature for defining “beauty” in the human mind.
Golden Ratio
Suppose we divide any single line into 2 non-equal parts, and the ratio of lengths between the longer and shorter part is x:1, the golden ratio (denoted by greek letter φ)is defined as the value of x in which the longer part length divided by shorter part length (a/b in Figure 1) is also equal to the total line length divided by longer part length ([a+b]/a in Figure 1).
The value of the golden ratio (φ) is a constant which is approximately equal to 1.618.
Forming Simple Shapes with Golden Ratio
Now that we know what golden ratio is we can form various shapes out of it. Of course the simplest shape we could form is a “rectangle” with width:length ratio being 1:1.618.
Imagine a door or window in your house… Is the length of your door/window is the same as its width? Is the length twice of its width? Or is the length roughly 1.618 times the width? Which configuration do you find most in doors/windows? Which configuration appears more appealing to the average eye? I think you guessed it already. It is when the length of the door/window is roughly 1.618 times its width.
In fact, if we observe several images of classical architectures it becomes amazingly evident the usage of 1:1.618 ratio based rectangles, consciously or unconsciously. You may do a self exercise to find out how many significant edges in the buildings coincide with 1:1.618 rectangles that could be found in Figures 4 and 5.
the distance from the pupils to the bottom of the chin
But wait!! We started with our quest for finding a dream girl. Does the golden ratio present itself in a human face? What do you think of Figure 6?
Forming Complex Shapes with Golden Ratio
We could form more complex shapes, other than a rectangle using the golden ratio based proportion 1:1.618. A popular example is the “golden spiral”.
Figure 7 depicts an animation of a Nautilus shell. The proportion between the blue and white areas observed when traversing the spiral is approximately aligning with 1:1.618 (which is the “golden spiral”), all the time. Isn’t φ an amazing constant to appear not only in man-created architectural shapes, but also in shapes formed by nature?
Figure 8 illustrates how the spiral shape generated based on golden ratio proportions becomes evident within every little detail for a side view of the human face.
In addition to spirals we could construct triangles seen in Figures 9 & 10, based on the golden ratio. The triangle on the left (Figure 9) is also called the “golden acute triangle” (since all 3 angles are less than 90 degrees). The triangle on the right (Figure 10) is called as the “golden obtuse triangle” (since one of the angles is greater than 90 degrees). These triangles can be used as foundational building blocks to form more complex shapes originating from the golden ratio (φ).
A regular pentagon (Figure 11), is also called the “golden pentagon”. It is a composite shape formed by 2 golden obtuse triangles and 1 golden acute triangle. Hence we could still form an argument that the pentagon shape appears beautiful (consciously or unconsciously) due to its strong relationship with the golden ratio. The US department of defence is called the “pentagon” since it is located in a building having a shape of a regular pentagon when viewed from the top.
The 5-sided star (illustrated blue in Figure 12), also called “the golden star pentagram”; is built by combining 3 golden obtuse triangles together on the base of a golden pentagon. It is a very popular shape globally for emblems, flags and overall sports.
Here is a question for you… what would be the resulting 2D shape formed; if 2 pentagrams rotated 180 degrees to each other and coincided them in the same 2D plane? Both pentagrams rotated 180 degrees to each other have 5 vertices each (shown in blue in Figure 13). You guessed it right! 5 + 5 = 10. Hence the 2 pentagrams in Figure 13 form a “golden decagon” if they coincide into a single 2D plane (which is a regular polygon with 10 equal sides as in Figure 14).
Surprisingly, for a decagon; the ratio between the distance to a vertex from the center versus the length of a side is φ (1.618). Isn’t mathematics amazing?
Have I forgotten to talk about beautiful women? Absolutely not. We’re very close to inventing a mathematical representation for defining beauty.
Defining Beauty
As illustrated in Figure 15, the entire subset of line segments formed after joining each of the 10 vertices (shown in red) in a regular decagon, is called the “golden decagon matrix”.
In fact the most beautiful things in Nature follow a geometry, adhering along the gridlines of the “golden decagon matrix” and curves formed with spirals derived from the golden ratio.
Can we construct a mathematical representation to assess beauty of a human face based on the golden spiral, golden triangles, pentagons and golden decagon matrix?
Fortunately, Dr. Stephen Marquardt from Southern California has already done this for us.
The generalised beauty mask for the human face derived based on the decagon matrix and complex shapes formed from golden ratio (Figure 18) is called the “Marquardt Beauty Mask” or the “Phi Beauty Mask”.
Figure 19 is a brain-refresher for the reader illustrated from sub-figures a to g on how the “Marquardt Beauty Mask” was derived based on what we discussed up to this point.
Applying the Marquardt Beauty Mask (Phi Beauty Mask)
Is Jessica Simpson beautiful? Well, yes at least for me. I know that most of you will say yes too. In fact when we compare the features of her face with the Phi Beauty Mask (Figure 20); the eyes, chin, nose, cheeks etc. are well aligned. Do you agree that this is why we see her as beautiful?
Figure 21 compares beautiful faces from different continents of the world with the Phi Beauty Mask and prove them to be beautiful. Do all of them appear beautiful for you too? Beauty is independent of race, ethnic group or skin colour. Hmm, I guess this explains why Naomi Campbell & Tyra Banks compete head to head with Heidi Klum & Paris Hilton in the modelling industry.
You can convert an average looking face to a beautiful face by slightly adjusting it using the Phi Beauty Mask. So be careful dating those beautiful women you see on dating sites, it could be a very different experience in person.
Is the Phi Beauty Mask applicable, only for the modern world? Absolutely not. The beauty of historical characters existed before thousands of years ago can be validated right, using the Phi Beauty Mask. We’re immortal. But φ is eternal.
Is beauty a part of Aesthetics only? Isn’t it possible to define a standard scale from being very beautiful to extremely unattractive.
Figure 24 shows how you could compare and contrast faces with different levels of attraction against the Phi Beauty Mask. As you may notice, the further the attraction diminishes from a face; the further the original facial feature lines (such as nose, eyes, chin etc.) don’t comply with Phi Beauty Mask.
For all those girls curious to rate their own level of beauty and attraction at this point, the apple appstore hasn’t let you down. “Golden Ratio Face” is a free app that you can try out yourself.
Oh! Sorry, I totally forgot. Last but not least, we’ve got to amend; “Snow-white and the 7 Dwarfs”. Would “Mirror, Mirror! Who’s the fairest in the land?” sound correct anymore? Absolutely not. The wicked queen would rather ask “Mirror, Mirror! Who’s the most Phi Beauty Mask compliant in the land?”. Fortunately for the prince and the 7 dwarfs, but unfortunately for the wicked queen; the ever sleeping Snow-white still has the highest “Phi Beauty Mask” compliance score among all in the land.
Meanwhile, the poor mirror is severely horrified about his future job security thinking the queen would throw away the mirror and buy a “Phi Beauty Mask” calculator; since the sleeping Snow-white is always the first, according to the Mirror’s assessment. Come on Mirror, life is not always the same. It’s time to get ready to live in the dungeons.
