8. Sacred Geometry: the Square, Part 4

​This is the eighth article in a series of articles discussing Sacred Geometry.  This week we’ll take a look at how the succession of polygons takes shape and how the numbers 2, 3, and 5 are fundamental properties that are inherent in the creation of all form.  The square roots of these three numbers are innate within the structure of the Vesica Pisces, the formative power of Creation.  
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  • The  = the diagonal of the square
  • The 3 = the length of the Vesica Pisces (top to bottom) = diagonal of the cube (giving us 3-dimensional form)
  • The 5 = the diagonal of the double-square which is the Golden Rectangle that gives us the Golden Mean proportion (or Phi).  It is the 5 that has the regenerative quality producing growth.  In the metaphysical world, the Golden Mean represents the life-giving qualities.[1]
 
Leonardo da Vinci is the one who’s drawing of the 5 Platonic solids helped 15th century Europe bring forth the hidden mathematical treasures of the ancients.  It is Leonardo da Vinci who also helped us understand the significance of the Golden Mean.  Note the following interesting properties of the Golden Mean:
  • It is believed by many that dimensions divided in the ration of 1 to Phi (1:φ) are more aesthetically pleasing.  Dozens of psychological tests beginning in the 19th century tend to support this belief for invariably the subjects in these tests chose the rectangle whose sides are in the Golden Mean ratio.
  • Mathematically 2xφ = φ+1 = 2.618033988 and 1xφ = φ-1 = .0618033988
  • In the Fibonacci sequence of numbers where each number is the sum of the two previous numbers, the ratio between successive numbers tend toward φ.
                                                                      1, 2, 3, 5, 8, 13, 21, 34….etc.

The ratio between any number in this sequence and the one immediately preceding it will always be close to .618033988.  For example 8:13 or 8 divided by 13 = .615384615.
  • The Golden Mean is the only number whose square can be produced simply by adding 1 and whose reciprocal is gotten by subtracting 1.
  • The Golden Mean’s elusive nature was discussed in a three-volume treatise entitled The Divine Proportion authored by 15th century mathematician Luca Pacioli.  Pacioli was influenced heavily by his contemporary Leonardo da Vinci and the Greek mathematician, Euclid (300BC).  Pacioli used the Golden Mean as a metaphor for the incomprehensibility of God.
The Golden Mean gives birth to the spiral that is so prevalent in nature, be it plants, sea shells, cloud formations or the galaxy as a whole.

​Recall that the Golden Mean begins with the diagonal of a square (¸) or of a cube (¶3).  Doubling the square produces the rectangle with Golden Mean proportions (5).  
  • The Golden Mean has the reputation of being the most irrational of irrational numbers.  An irrational number is a real number that cannot be written as a simple fraction; its decimal equivalent goes on forever without repeating.
  • The Golden Mean is also the slowest of all continued fractions (irrational numbers) to converge.  It is this quality built into inherent structure of plants that allow leaves to grow in such a way as to maximize exposure to the sun’s rays.
  • Quasi-crystals--crystals that look the same when rotated by multiples of one-fifth of 360 or 72o-- are another example of the Golden Mean.  In the 1990s quasi-crystals were conceived of by scientists as being the space-filling form of matter.  If particles make up the matter that we perceive as 3D concrete reality, the quasi-crystals make up the space on the quantum level between the particles.
  • The black holes spinning in space also include the Golden Mean as a critical value in their existence.  According to scientists, the Golden Mean serves as the critical value that determines if the heat produced by the black hole is growing hotter or colder.  It’s as if the black hole can regulate its “centrifugal force” to prevent shrinkage that would lead to total annihilation.
  • The Golden Mean is witnessed in the Great Pyramid of Egypt.  The ratio of the length of a face of the Great Pyramid (from centre of the bottom of a face to the apex of the pyramid) to the distance from the same point to the exact centre of the pyramid's base square is about 1:φ. 
​In the ancient Egyptian creation myth coming from Heliopolis, Nun, the Cosmic Ocean, represents pure, undifferentiated spirit-space, without limit or form. Nun is prior to creation. It is pure potentiality. By the seed or will of the Creator, who is implicit within this Nun, the undifferentiated space is impelled to contract or coagulate itself into volume. Thus Atum, the creator, first creates himself or distinguishes himself from the undefinable Nun by volumizing (moving from 2D to 3D form), in order that physical creation might begin.  This myth is represented exquisitely in Sacred Geometry--moving from the Vesica Pisces through the square and cube to the Golden Mean.
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[1] Some would say that the Golden Mean is the Christ-force energy.