Transcript The Golden Ratio – a Masonic Number

The Divine Proportion
“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)
What is so interesting 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.”
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
Leonardo di ser Piero da Vinci
An Old man by Leonardo Da Vinci
Leonardo De Vinci
The Vetruvian Man"(The Man in Action)" by Leonardo Da Vinci
Leonardo De Vinci
Mona-Lisa by Leonardo Da Vinci
Deoxyribonucleic acid (DNA)
• The DNA spiral is a Golden Section
• The DNA molecule, the program for all life,
is based on the golden section. It
measures 34 angstroms long by 21
angstroms wide for each full cycle of its
double helix spiral.
• 34 and 21, of course, are numbers in the
Fibonacci series and their ratio, 1.6190476
closely approximates phi, 1.6180339.
DNA
DNA in the cell appears as a
double-stranded helix
referred to as B-DNA.
This form of DNA has a two
groove in its spirals, with a
ratio of phi in the proportion
of the major groove to the minor
groove, or
roughly 21 angstroms to 13 angstroms.
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.
Another Historical Tidbit
• 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…
• 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