The Divine Proportion

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The Divine Proportion

• Geometry has two great treasuresone 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)

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What is so interesting about the Divine
Proportion?

• The Divine Proportion, or Golden Section,
  represented by the Greek letter F (phi), is one
  of those mysterious natural numbers like p (pi)
  that seem to arise out of the basic structure of
  the universe.
• F 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.

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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 F 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.

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The Golden Section A Ratio

• The Golden Section is a RATIO like 21. It is
  also called the Golden Mean
• If there is a piece of string, and you divide it
  into a 21 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
  12, and that of the longer to the whole is 23.

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So then what is the Golden Section?

• Butthe 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.

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So then what is the Golden Section?

• So, a is to b (ab) as b is to c (bc)
• ab bc

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Some Examples - People
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Some Examples - Nature
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Leonardo di ser Piero da Vinci
An Old man by Leonardo Da Vinci
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Leonardo De Vinci
The Vetruvian Man"(The Man in Action)" by
Leonardo Da Vinci
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Leonardo De Vinci
Mona-Lisa by Leonardo Da Vinci
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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.

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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.
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So what is this ratio?

• The Golden Section/Golden Mean/Divine Proportion
  is an irregular number like p, and cannot be
  expressed fully in decimal form (i.e. p
  3.14128)
• F 1.618033 or (1?5)/2
• Somehow it seems fitting that we cannot represent
  the root of Sacred Geometry by an ordinary
  number.

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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 F in the so-called Kings
  Chamber and also in its overall dimensions.

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What about the Parthenon?
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Columns, anyone?

• The graceful curves of the Ionic column are
  designed using the Golden Section.

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Another Historical Tidbit

• Exodus 2510 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
  2.51.5 53 1.666)
• Genesis 615 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 (5030
  53 1.666)
• F 1.618033 or (1?5)/2

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Anything Else?

• It is likely that Virgils Aeneid and other
  great works of classical poetry used f to
  determine metrical structure of the poem.
• Mozarts sonatas tend to divide in parts exactly
  at the Golden Section of total time of the work.
• In Beethovens 5th Symphony the opening motto is
  repeated at exactly the F point through the
  Symphony (Bar 372) and also at the start of the
  recapitulation 1-F of the way through.
• Stradivarius placed the f holes in his violins
  at the F point of the body structure.

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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.

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A little more math for anyone thats 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 F
• For instance.. 21 2.000, 85 1.600, 138
  1.625, 2113 1.615
• F 1.618033 or (1?5)/2
• Each number in the series is called a Fibonacci
  Number

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Fibonacci Bunnies

• Start with one pair
• Mate during first month
• One pair born next month and each month thereafter

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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

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Building a Golden Spiral

• The spirals increase in distance from the centre
  by phi every quarter turnsea shells, snails,
  ferns, and many other living creatures are built
  to this specification

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