Human Figures to the Human Eye
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For the purpose of class notes we have included information from "The mathematical formula for the perfect face". A word of caution however - this is an interesting mathematical theory, but true beauty will always be found in the eyes of its beholder.
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.
They define the formula and say, 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. 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.
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".
GEOMETRY IN THE NATURAL WORLD
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. 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.
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)
and...
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:
- All types of crystals, natural and cultured.
- The hexagonal geometry of snowflakes.
- Creatures exhibiting logarithmic spiral patterns: e.g. snails and various shell fish.
- Birds and flying insects, exhibiting clear Golden Mean proportions in bodies & wings.
- The way in which lightning forms branches.
- The way in which rivers branch.
- The geometric molecular and atomic patterns that all solid metals exhibit.
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
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.
Additional links:
- http://en.wikipedia.org/wiki/Golden_ratio
- http://www.jyi.org/volumes/volume6/issue6/features/feng.html
- http://www.gutenberg.org/etext/3751
