The mathematical formula for the perfect face!
Sydney June 11, 2004 3:37:38 PM IST
It is difficult to define perfect beauty as the parameters for a perfect face may vary according to individual preferences.

However, scientists have narrowed down to a simple mathematical ratio of 1:1.618, otherwise known as phi, or divine proportion, to set standards of beauty.

"Only one formula has been consistently and repeatedly present in all things beautiful, be it art, architecture or nature, but most importantly in facial beauty," The Sydney Morning Herald quoted US dentist Yosh Jefferson, who operates a website dedicated to divine proportion, as saying.

"Ideal facial proportions are universal regardless of race, sex and age, and are based on divine proportions," he adds.

He defines the formula and says, if the width of the face from cheek to cheek is 10 inches (25 centimetres), then the length of the face from the top of the head to the bottom of the chin should be 16.18 inches to be in ideal proportion. If you're keen to see how you measure up, keep in mind that the ratio of phi also applies to:

+ The width of the mouth to the width of the cheek.

+ The width of the nose to the width of the cheek.

+ The width of the nose to the width of the mouth.

Dr Stephen Marquardt has gone one step further to prove the correlation between the divine proportion and facial beauty by developing a phi mask that acts as an archetype of the ideal human face. (ANI)




Phi In 1D

Visualizing the Golden Ratio in 1-Dimension

Phi is only a number - and we don't really see numbers when we look at things.

However, there is a visual manifestation of Phi and the Golden Ratio - something that we can actually look at and see. It is this manifestation of that Golden Ratio which has been reported to be present in many things that are seen as beautiful.

This is a line which has been divided into two segments, the larger of which has a (magnitude) ratio to the smaller of 1.618:1

Where a=1.618 and b=1

A line which has been segmented into two parts having this 1.618:1 ratio is called:

    A golden cut line
    A golden sectioned line
    A golden divided line
    A Phi cut line
    A Phi sectioned line
    A Fibonacci sectioned line

This division of a line in such a manner is referred to variously as:

    The golden cut
    The golden section
    The golden division
    The Phi cut
    The Phi section
    The Fibonacci section



Creating the Golden Sectioned Line:

Create a line from a point
We create a line of any length.

Section that line
There are an infinite number or places that we can divide that line into two segments, and we can section (or cut) that line at any point we desire. 

"The Golden Section"
However there is one place (and only one place - a unique place) where that line can be divided or "sectioned" so that the ratio of the smaller segment of the sectioned line to the larger segment of the sectioned line is 1:1.618

This ratio of 1:1.618 is called "The Golden Ratio".

The interesting and remarkable thing about this sectioning of the line into the golden ratio is that not only is the ratio of the smaller segment or the line to the larger segment of the line equal to 1:1.618


the ratio of the larger segment of the line to the whole line is also equal to 1:1.618.

And this particular division, or sectioning, of a line into segments with a ratio of 1:1.618 is called:
    "The Golden Section"

The "Golden Sectioned Line"
And the line which is cut into the Golden Section is called:
    "The Golden Sectioned Line"

The "Golden Section Point"
This place or point on the line where this golden sectioning occurs is called:
    "The Golden Section Point"

The Repeating Phi Ratio & Phi Ratio Duplication (Growth):

Intriguingly, if we duplicate that golden sectioned line to form a new longer line, consisting of the smaller and larger segments of the original line plus a duplicate of the original line, then the ratio of the larger segment of the original line to the duplicate of the original line is also = 1:1.618.

If we delete the smaller segment of the original line, we are left with a new line consisting of the larger segment of the original line and the duplicate of the original line.

If we duplicate this new line, then the ratio of the larger segment of this new line to the whole new duplicate is also = 1:1.618.

This self-duplication can continue on to infinity (i.e. forever) with each line segment in a ratio of 1:1.618 with its adjacent and succeeding line segment along the formed line.

This self-duplication never occurs at any other division, section or ratio of any line or line segments.  This continuous self-duplication only occurs with the golden section.

Because of its unusual properties, the number "1.618" has been given its own name - that name is "Phi".







The Golden Mean:

   The Golden Mean, or Golden Ratio as it is known, is an irrational number just like other important numbers such as Pi.  This means that it cannot be completely represented by our currently used number system, except as a formula (Sqr(5)-1)/2.  Just like Pi (approx. 3.1416) - Phi, or the Golden Ratio, has an endless number of digits after its decimal point and with no repetition of the digits sequences.  Therefore, like other "Transcendental" numbers, its value can only be approximated (using our number system).



What is the Golden Ratio, and why is it important?

The Golden Ratio is approximately 1 : 1.618


   Besides for possessing some remarkable and unique characteristics, the Golden Mean is found in ALL living creatures on Earth.  Along with the Fibonacci Sequence (which is a whole-number system approximating the Golden Ratio, discovered by Leonardo Pisano Fibonacci), this ratio is found in plants and animal life wherever one looks.  For example, this ratio can be found in fingers one's hand, amongst many other places, and it is prevalent in the skeletal structure of all creatures.



The Fibonacci Sequence is as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...


The sequence is calculated as follows.

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8

5 + 8 = 13



The Fibonacci sequence is an approximation of the Golden Ratio, and as one will see, the higher one goes in the Fibonacci Sequence the closer ones gets to the Golden Ratio.

8 / 5 = 1.6

13 / 8 = 1.625

21 / 13 = 1.615


233 / 144 = 1.618


As you can see, each consecutive number in the sequence is derived from the sum of the previous two. The Fibonacci series is a Fractal sequence (a fractal is a mathematically defined system that, when represented graphically, usually forms self-replicating or recurring patterns).

The Fibonacci sequence is the formula that plants use when deciding how many branches to grow next, but this series can be found everywhere in nature.


Examples of Fractals


An example of a Fractal - source: Bruce A. Rawles



Properties of the Golden Ratio:

The Golden Ratio can be expressed as 1.618 and 0.618 and is known as Phi and phi, respectively; phi being the reciprocal of Phi... This is a very unique property that only the Golden Ratio possesses:


1 / Phi = phi  (1 / 1.618 = 0.618)


1 / phi = Phi  (1 / 0.618 = 1.618)


Also, Phi Squared = Phi + 1  (1.618 ^2 = 1.618 + 1)

...and Phi multiplied by phi = 1  (1.618 * 0.618 = 1)


Phi is not a fraction: In other words, there is no way to express Phi as using two integers, e.g. (2/3)



Deriving Phi:

Phi = Square root of 5 + 1 / 2...  or  (Sqr(5)+1)/2



Phi to 31 decimal places: 1.6180339887498948482045868343656



Geometry in Nature and the natural world:

   Our reality is very structured, and indeed Life is even more structured.  This is reflected though Nature in form of geometry. Geometry is the very basis of our reality, and hence we live in a coherent world governed by unseen laws.  These are always manifested in the natural world.  The Golden Mean governs the proportion of our world and it can be found even in the most seemingly proportion-less living forms.


   Clear examples of geometry (and Golden Mean geometry) in Nature and matter:

Another, less obvious, example of this special ratio can be found in Deoxyribonucleic Acid (DNA) - the foundation and guiding mechanism of all living organisms:



The geometry of DNA - source: Dan Winter



Geometry and Phi


   The understanding of geometry as an underlying part of our existence is nothing new and in fact the Golden Mean and other forms of geometry can be seen imbedded in many of the ancient monuments that still exist today.  The Great Pyramid (the oldest of these structures) at Giza is a good example of this.  The height of this pyramid is in Phi ratio to its base.  In fact, the geometry in this particular structure are far more accurate than that found in any of today's modern buildings.



The following geometric shapes contain the Golden Ratio:


The Hexagram

The average of two ratios found in this form = Phi


The Pentagram (inscribed in a pentagon)

Relation of the sides of the outer pentagon to diagonals = Phi


Equilateral triangle (inscribed in a circle)

Extended line in relation to side of inner triangle = Phi, click here for details


Squares (inscribed in a circle)

Relation of the sides of the squares to the line extended to the border of the circle = Phi



The Golden Rectangle

A rectangle in proportion to Phi


The Golden Triangle

A triangle in proportion to Phi


Golden triangle

A rectangle in proportion to Phi


Golden triangle

A rectangle with base and height in proportion to Phi


The Logarithmic Spiral

A function of fractally recurring Golden Rectangles



The Kathara Grid

The 'Fractal' upon which matter is built



Links to Info.