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The Golden Ratio ‐ a Masonic Number

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The Golden Ratio ‐ a Masonic Number Powered By Docstoc
					        The Divine Proportion
         – a Masonic Number



      “Geometry has two great treasures…one is the theorem of
 Pythagoras; the other, the division of a line into extreme and mean
ratio. The first we may compare to a measure of gold; the second we
           may name a precious jewel”. (Johannes Kepler)
    Why should Masons know about
        the Divine Proportion?
    The Divine Proportion, or Golden Section,
    represented by the Greek letter Φ (phi), is one of
    those mysterious natural numbers like π (pi) that
    seem to arise out of the basic structure of the
    universe.
    Φ appears clearly and regularly in the realm of things
    that grow and unfold in steps, especially living things
    – but also in art and architecture.

•   “To the Greeks therefore, and not to the Romans, we are indebted for all that is
    great, judicious, and distinct in architecture.”
    Why should Masons know about
        the Divine Proportion?
    The Greeks (and others, like Renaissance artists such as
    Botticelli, Lippi, Michelangelo) may have felt that when a
    building or artwork was designed to incorporate Φ that it had the
    purest possible proportions and was the most pleasing to the
    eye – we will see why in a minute.




•   …allude to a proper application of the useful rules of architecture, whence a
    structure will derive figure, strength, and beauty, and whence will result a due
    proportion and a just correspondence in all its parts.”
The Golden Section – A
        Ratio
The Golden Section is a RATIO – like 2:1. It is also called the
“Golden Mean”
If there is a piece of string, and you divide it into a 2:1 ratio, then
1 part is twice as long as the other.
Also, the short part is 1/3 the length of the whole string, and the
long part is 2/3 the length. The ratio of the shorter to the longer
is 1:2, and that of the longer to the whole is 2:3.
  So then what is the
   Golden Section?

But…the Golden Section is a special
ratio – where the ratio of the short part
to the long part is the same as the long
part to the whole.
   So then what is the
    Golden Section?
So, “a” is to “b” (a:b) as “b” is to “c” (b:c)
a:b = b:c
Some Examples - People
Some Examples - Nature
                       Hmmmm…
    It seems as though the GAOTU may
    have laid out the designs for creation on
    his trestleboard using, among other
    things, the Divine Proportion.


•   “While we are employed in the study of this science we must perceive
    unparalleled instances of wisdom and goodness, and, through the whole
    creation, trace the Glorious Author by his works.”
 So what is this ratio?
The Golden Section/Golden Mean/Divine
Proportion is an irregular number – like π,
and cannot be expressed fully in decimal
form (i.e. π = 3.14128…)
Φ = 1.618033… or (1+5)/2
Somehow it seems fitting that we cannot
represent the root of Sacred Geometry by an
ordinary number.
          So what about
          Architecture?
The Divine Proportion was used by the Greeks – and is still
being used by architects today to design buildings that are
aesthetically pleasing.
There is evidence that the Great Pyramid incorporates Φ – in
the so-called “King’s Chamber” and also in its overall
dimensions.
What about the
 Parthenon?
   Columns, anyone?
The graceful curves of the Ionic column
are designed using the Golden Section.
VOSL…
 Exodus 25:10 – “Have them make a
 chest of acacia wood = two and a
 half cubits long, a cubit and a half
 wide, and a cubit and a half high…”
 (ratio 2.5:1.5 = 5:3 = 1.666)
 Genesis 6:15 – “And this is the
 fashion that thou shalt make it of:
 The length of the ark shall be three
 hundred cubits, the breadth of it 50
 cubits, and the height of it 30
 cubits…” (50:30 = 5:3 = 1.666)
 Φ = 1.618033… or (1+5)/2
           Anything Else?
It is likely that Virgil’s “Aeneid” and other great works of classical
poetry used φ to determine metrical structure of the poem.
Mozart’s sonatas tend to divide in parts exactly at the Golden
Section of total time of the work.
In Beethoven’s 5th Symphony the opening motto is repeated at
exactly the Φ point through the Symphony (Bar 372) and also at
the start of the recapitulation 1-Φ of the way through.
Stradivarius placed the “f” holes in his violins at the Φ point of
the body structure.
     Hmmmm… (revisited)
    It now seems that the Divine Proportion
    was and still is used by men and
    women to build beautiful monuments
    and other works that are pleasing to the
    senses.

•   “…so as to compose delightful harmony by a mathematical and
    proportional arrangement of acute, grave, and mixed sounds.”
A little more math for anyone that’s still
                awake…

The Fibonacci series (1, 1, 2, 3, 5, 8,13, 21…)
which describes the growth pattern of a
population, is connected to the Golden Mean,
because the ratio of any 2 terms tends
towards Φ…
For instance.. 2:1 = 2.000, 8:5 =1.600, 13:8 =
1.625, 21:13 = 1.615…
Φ = 1.618033… or (1+5)/2
Each number in the series is called a
“Fibonacci Number”
Fibonacci Bunnies
            Start with one
            pair
            Mate during first
            month
            One pair born
            next month and
            each month
            thereafter
Building a Golden Spiral
             Draw 2 squares of 1
             unit each, side-by-side
             Next draw a 2 unit
             square, and then a 3,
             and then a 5, etc
             Draw quarter circles in
             each square, joining
             them up…
Building a Golden Spiral




The spirals increase in distance from the centre by phi every
quarter turn…sea shells, snails, ferns, and many other living
creatures are built to this specification
   Operative Masonry
How did our ancient brethren construct
such mathematically rigorous designs,
using only the tools that were available
to them?
 So what does this have
to do with Freemasonry?
 The Golden Section and its derivative constructs can be
 calculated using only 2 tools: the Square and Compasses.

 If the S&C allow the calculation of the proportion used by the
 GAOTU when laying out the design for Creation on His trestle
 board, then I feel that there is some demonstrable validity to the
 spiritual symbolism that they are supposed to display.

 I ask you now, Brethren: “Is it still true in your minds that there
 are no secrets in Freemasonry?”

				
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