Phi la constante presente en la Geometría, el Arte, la naturaleza y la Guerra

1. 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)

2. 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.”

3. 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.”

4. 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.

5. 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.

6. 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

7. Some Examples - People

8. Some Examples - Nature

9. 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.”

10. 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.

11. 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.

12. What about the
Parthenon?

13. Columns, anyone?
The graceful curves of the Ionic column
are designed using the Golden Section.

14. 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

15. 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.

16. 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.”

17. 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”

18. Fibonacci Bunnies
Start with one
pair
Mate during first
month
One pair born
next month and
each month
thereafter

19. 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…

20. 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

21. Operative Masonry
How did our ancient brethren construct
such mathematically rigorous designs,
using only the tools that were available
to them?

22. 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?”