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 The Harmony of the Spheres.  Vitruvian Geometry. Spirals Homepage. Labyrinths Homepage.  Geodesy Homepage. Petrosperes and th Platonic Solids. 

        Sacred Geometry:

 

   What is 'Sacred' Geometry' :

The synchronicity of the universe is determined by certain mathematical constants which express themselves in the form of 'patterns' and 'cycles' in nature.

The outcome of this process can be seen throughout the natural world as the following examples demonstrate:

The Passion-flower Spiral galaxy M74 Giant's causeway, Ireland. Spiral snail shells.

These displays of mathematical and geometric constants are confirmation that certain proportions are woven into the very fabric of nature. Recognising the significance of this simple fact offers us the means to understand how and why such matters were considered sacred. They and everything around us, are the product of the delicate balance between chaos and order.

 The word 'geometry' can be traced through its component parts:

The word 'Geo-metry' comes from the Greek words Geos meaning 'Earth' and Metron meaning 'To measure', which together literally translate as the 'Measuring of the earth' or 'Earthly measurements', an art which was traditionally restricted to the priest-hood.

Sacred geometry has existed in many forms across the ages

It is often mistakenly said that geometry began with the Greeks, but before them were the Minoans, the Egyptians, Sumerians, Indus valley, Chinese, Phoenicians and of course, the builders of the western European megaliths all of whom left clear geometric fingerprints in their greatest constructions. The Greeks may well have been the first to have offered geometry to the public at large, but they were by no means the first to realise it.

 

Sacred-Geometry: The First Step.

One of the most common shapes in nature is the circle, it is therefore extremely significant to understand that all other geometric shapes can be determined from a circle...with the use of only a compass (or string) and a ruler (straight edge) as the following procedure illustrates...

Starting with the Vesica-pisces...from which one is able to produce...

 

An Equilateral triangle, Hexagon, Pentagon, Square and so on and so on..

The Vesica Pisces is one of the key starting blocks from which sacred geometry was applied to life.

A continuation of the geometric Vesica Pisces results in the geometric matrix named Ad-triangulum...as used for the design of many of Europe's greatest Cathedrals...

 

Featured Articles:

   
  • The Harmony of the Spheres.

The first confirmed record of a knowledge of the relationship between astronomy, music and geometry comes (almost predictably) from the Greeks: in particular, Pythagoras who wrote of the 'Harmony of the Spheres', and of whom it was said: '...of all men, he alone [Pythagoras] could hear the music of the spheres...'

(The Harmony of the Spheres)

 
 
Link to Labyrinths page.    
  • Labyrinths:

The symbol for the Labyrinth has been found associated with 'sacred' places for thousands of years from all around the ancient world. More recently, more especially from the middle ages onward, it has been used as a tool for pilgrimage - representing our metaphorical path through life.

(Labyrinths Homepage)

 
 
   
  • The Flower of Life:

The flower of life symbol has found its way into the human consciousness, it is no more than an elaborate extension of the Vesica pisces, but one within people have found themselves lost in a state of perpetual imaginative bloom.  It is said to contain values that depict the fundamental geometry of time and space.

(The Flower of Life)

 
 
Spirals Homepage.    
  • Spirals: Form and Function.

Spirals are one of the most a common forms of natural geometry, being a product of the sacred mean. They are associated with omphalos and earth-navels and are regularly found engraved on megaliths. Their original meaning is lost today but spiral designs at such noticeable sites as Newgrange and Chaco Canyon, has led many to believe they were primarily astronomical.

(Spirals Homepage)

 
    Geodesy Homepage
  • Geodesy and the World Grid:

This section examines the theory that geometry was applied in the placement of certain prominent (sacred) prehistoric sites, based on an understanding of longitude and latitude. The 'linear-mentality' of our ancestors seems to have had no frontiers, but what if any, was the origin and purpose of networking sacred sites, and how

(Geodesy Homepage)

 
 
The Scottish Petrospheres.    
  • Petrospheres and the Platonic Solids.

Of the hundreds of small carved stone balls fond in Scotland, over 75% have been found to conform to the five Platonic solids. This remarkable discovery suggests an association as yet undetermined. Although it is generally believed that the Greeks discovered this mathematical principle, these come from over a thousand years earlier.

(Scottish Petrospheres)

 
 
   
  • The Myth of Precession.

It has been proposed that the understanding of sacred geometry extended into time and space, as realised through the 'Platonic Year'. Santillana (6) and others have shown that certain precessionary numbers were encoded into ancient sacred buildings, texts and mythologies. these same numbers appear to be reflected in  the natural geometry of the universe.

(The Myth of Precession)

 

Other Articles:

 

Sacred Geometry and the Great Pyramid of Giza:

The exterior angle of the 'Great pyramid of Giza' can be reproduced with the Vesica-pisces. It has been long suspected that the Great pyramid was a subsidiary of geometric knowledge. There are several other indications that sacred geometry was an important factor in the design of the pyramid.

The Sacred mean (Phi), was also recorded into the dimensions of the pyramid itself.

 

Pi and the Pyramid.

Egyptian mathematicians arrived at a figure of 3.16 (as shown on the Rhind Papyrus), written 600 to 800 years later and far cruder than the precise ratio the great Pyramid seems to express. The figure of Pi is recorded into the dimensions of the Great Pyramid several  times; As well as the Height/Perimeter ratio of the pyramid itself

(2∏x H = Perimeter)

The perimeter of its main compartment, the so called “King’s Chamber,” is also exactly 3.14 times its length, and the large granite coffer or sarcophagus in this room shows the same proportion. In the earlier, rhomboidal or “Bent Pyramid” at Dashur, a few miles to the south, the sides begin to ascend at the same 2-Pi angle as the Great Pyramid, then change half way up to 3-Pi (43 ½º ) ratio.

(More on the Geometry of the Great pyramid)

 

Pyramid Geometry and Latitude.

The exterior angle of the Great pyramid is the exact same latitude that Silbury Hill was built, and at the same time. It is perhaps no coincidence to find that exterior angle of the Silbury monument has an exterior angle of 30°, the same as the latitude of Giza.  This exact angle is also found at nearby Stonehenge in the orientation of The Avenue, which points towards the sunrise on the summer solstice. Stonehenge sits exactly 1/4 of a degree of longitude south of Avebury.

This fact is just one of many in the increasingly convincing argument in favour of the existence of applied sacred geometry in prehistory.

(British Geodesy)      (Egyptian Geodesy)

(Geodesy Homepage)

 

 

   The Sacred Mean:  'Phi'.

The Sacred mean - (5:8 or 1:1.618 or Φ) (The Divine Proportion, Golden Section, Golden ratio, Phi, )

One of the Key-stones of sacred geometry is the 'sacred mean' or 'golden section'.

 

The Mathematics of the Golden Ratio (Phi).

The golden section exists between measurable quantities of any kind where the ratio between the smallest and the next size up is equal to the ratio of the sum of the first two to the third.

Mathematically, the same proportion is generated with the following formula:

√5+1 / 2

In numerical terms, the 'Golden ratio' was first popularised by Leonardo Bigollo Fibonacci, the founder of the  'Fibonacci sequence', a numerical series which simply follows the rule that the next number is the sum of the previous two numbers.. as follows:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 etc...

 

Vitruvian Man: Leonardo Da Vinci.

Vitruvian man: Sacred Geometry.One of the fundamental products of this underlying mathematical structure is the 'sacred mean', a mathematical constant which is visible across the spectrum of the natural world. The sacred mean is one of the defining geometric qualities of life itself, as it plays an integral part of the complex process of division and variation. Leonardo da Vinci illustrated both the mathematical proportions of the human body, (which are based on ratios of 1.618), and the concept of 'squaring the circle' with his famous drawing (right).

Da Vinci was engrossed by Vitruvius, who had written that human proportions should have a relationship in architecture. Vitruvius believed that if human proportions could be incorporated into buildings they would become perfect in their geometry.

One of Da Vinci's greatest discoveries was the division of the body into proportions of whole numbers which he called 'cubits'. For example, while the body is 4 cubits high, it can be seen on the same body that 1 cubit is both the length from shoulder to shoulder and from elbow to fingertips. 

 

  

According to Vitruvius, the distance from fingertip to fingertip should be the same as that from head to toe. The sacred mean can be seen in the ratios of body parts. In the arm of the Vitruvian man for example, we can see that the ratio of A is to B is the same as that of B to C. The same rules apply throughout the human body.

 

One of the mathematical products of the sacred mean is the spiral, commonly found in nature.

(Spirals Homepage)

 

The sacred mean is also found in the geometry of the pentagram and its associated pentagon, where the ratio between the sides of the pentagon and its extension into the pentagram also demonstrate a ratio of 1:1.618. In the above diagram, Phi is found in the ratios of a:b, b:c, c:d, d:e and e:f.

 

 

 

   The Flower of Life:

The Flower of Life symbol is considered to be sacred among many cultures around the world, it is considered by some to be an 'Akashic Record'.

 

The Flower of Life is the modern name given to a geometrical figure composed of multiple evenly-spaced, overlapping circles. They are arranged to form a flower-like pattern with a six-fold symmetry, similar to a hexagon. The centre of each circle is on the circumference of six surrounding circles of the same diameter.

The earliest confirmed example of the pattern can be seen in the Assyrian rooms of the Louvre museum in Paris. The design forms part of a gypsum or alabaster threshold step measuring 2.07 x 1.26 meters (6.8 x 4.1 feet) that originally existed in one of the palaces of King Ashurbanipal, and has been dated to c. 645 BC. (7)

 

The Osireion, Abydoss:

There are five possible 'Flower of Life' patterns on one of the granite columns and a further five on a column opposite of the Osireion. Some are very faint and hard to distinguish. They have not been carved into the granite but been drawn in red ochre with careful precision.

Recent research suggests that these symbols can be no earlier than 535 B.C., and most probably date to the 2nd and 4th century AD, based on photographic evidence of Greek text, still to be fully deciphered, seen alongside the Flower of Life circles and the position of the circles close to the top of columns, which are over 4 metres in height. This suggests the Osireion was half filled with sand prior to the circles being drawn which was therefore likely to have been well after the end of the Ptolemaic dynasty.

(More about Abydoss and the Osireion)

 

 

 

The Hebrew 'Tree of Life'.

The Tree of Life is most widely recognized as a concept within the Kabbalah, which is used to understand the nature of God and the manner in which he created the world. The Kabbalists developed this concept into a full model of reality, using the tree to depict a "map" of creation. The tree of life has been called the "cosmology" of the Kabbalah.

 

 

 

The Flower of life and the Platonic Solids.

The 'Platonic Solids' are the names given to the five “perfect shapes” formed when dividing a sphere into three-dimensional forms, with each division having the exact same shape and angle.

The Greeks taught that these five solids were the core patterns of physical creation. Four of the solids were seen as the archetypal patterns behind the four elements (earth, air, fire, and water), while the fifth was held to be the pattern behind the life force itself, the Greeks' ether. These shapes predominated in the hundreds of carved prehistoric petrospheres found in Scotland with over 75% representing one of the Platonic Solids. They came from a time over a thousand years earlier than the Greeks. These same shapes are now realised to be intimately related to the arrangements of protons and neutrons in the elements of the periodic table. (5)

Scottish Petroshperes and Platonic Solids.

(More About the Scottish Petrospheres)

 

Stonehenge and The Flower of Life.

Click here for larger image.

The geometry of the 'Flower of Life' was recognised in the dimensions of Stonehenge.

Image Credits: Andrew Monkman. (The World Grid: The invisible Made Visible)

(The Geometry of Stonehenge)

 

 

 

   Celestial Geometry - Sacred Astronomy:

The mathematical harmony of the universe is visible in the proportions of the planets of our own solar system as the following examples illustrate:

 

The diameter of the sun (864,000 miles) is the same as the perimeter of the square of the moon (4).

 

Sacred Venus :  'The Synodic Cycle'

The ratio of the 'Sacred mean' can also be seen in the rotations of Venus and the Earth around the sun so that for each five years that Earth rotates around the Sun, Venus manages to rotate it eight times).

 

Venus orbits the Sun in 224.701 Earth days ( ~.615 Earth years ), moving slightly faster than Earth. Because of the two different orbital rates of Venus and Earth, Venus must orbit the sun 2.6 times while Earth orbits 1.6 times before the two planets align. This period (583.92 Earth days) is called the Earth-Venus synodic cycle (synod means "place of meeting"). (3)

The result of this motion is that Venus 'draws' a pentagon around the sun every eight years.

 

Bode’s law:  Shows the simple mathematical relationship for the distances of the planets from the our sun.

Planet                           

     Formula              

Distance from Sun (Millions of Km’s)     

 

 

Bodes prediction

Actual

Mercury
    0 + 4 / 10 = 0.4
   60
     58    
Venus
    3 + 4 / 10 = 0.7
  105
108
Earth
    6 + 4 / 10 = Au
  150
150
Mars
  12 + 4 / 10 = 1.6
  240
228
(Asteroid belt)
  24 + 4 / 10 = 2.8
  420
550 wide
Jupiter
  48 + 4 / 10 = 5.2
  780
 779
Saturn
  96 + 4 / 10 = 10
 1500
1427
Uranus
192 + 4 / 10 = 19.6
 2940
2869
Neptune
384 + 4 / 10 = 38.8
5820
4496
Pluto
768 + 4 / 10 = 77.2
11580
5899
 
 
 
 

In addition to which...

 

Keplers 3rd Harmonic law – Kepler initially used the geometry of the Platonic solids to calculate the distances of the planets from the sun. In doing so, he attempted to revive the ancient tradition of Sacred Geometry with astronomy. Although this met with reasonable success, proving the ancient systems to be accurate, he eventually determined that the period of a planet or comets orbit is relate to its distance from the sun in the following simple mathematical equation:

(p² = a³)

( Where p = period of revolution and a = the distance from sun in astronomical units ‘Au’).

Kepler furthered the research of Aristotle who first realised the concept of the Harmony of the Spheres in which the planets positions and 'noise' were predicted according to musical harmonic ratios.

(Archaeoastronomy)

 

Sacred Geometry and the Harmony of the Spheres.

The theory of the 'Harmony of the Spheres' was originally proposed by Plato, in which he envisioned the five 'perfect' solids to be enclosed within imaginary spheres, each placed within the other. He proposed that the distances of the planets from the sun showed similar ratios from each other as the spheres surrounding each solids did. Modern science has indeed shown that planets have unique 'vibrations', or 'sounds' supporting Plato's conjecture.

(More about the Harmony of the Spheres)

 

 

   Sacred Geometry in Architecture:

What Happens when geometric constants are placed into the dimensions of buildings.?

Some of the best examples of the application of sacred geometry can be seen in constructions from the ancient world. It has been shown (1), that ancient sacred and ceremonial sites were invariably built with dimensions that incorporate mathematical figures such as infinite numbers, astronomical or mathematical constants (such as Pi or the 'sacred' mean), and the use of geometry (3:4:5 Pythagorean triangle etc). The application of 'sacred' geometry in our most important buildings is a reflection of the importance attached to it, but exactly how early was such information realised?

 

Following his renowned survey of over 600 English stone circles, Prof. Alexander Thom concluded that geometry had been used in their design.

Thom also proposed that common mathematical units of measurement (the megalithic yard), had been used in order to achieve these geometric results. Surprising as this might sound, one finds that as well as working with common units of measurement, the Neolithic people were also apparently aware of geometric constants as the following examples demonstrate.

 

 
The geometric design above was discovered by Prof A. Thom to have been used as a ground-plan for many European 'Type I' and 'Type II' flattened stone circles (Such as at Avebury, England).

(Note: Both type I and II circles show the application of the Vesica-Pisces and 3:4:5 triangles)

Sacred geometry, involving both astronomy and geometry appears to have been  applied to prehistoric quadrangles. It has been shown by Thom etc, that the Quadrangles at both Stonehenge and Carnac have incorporated into them geometric constants which relate to the latitude upon which they were built.

(More about Quadrangles)

 

The Golden section is a ratio which has been used in sophisticated artwork and in sacred architecture from the period of ancient Egypt (1).

 

Freemasonry and Sacred Geometry.

Following the collapse of the Roman empire, architects versed in geometry grouped together into 'guild's', thus forming the roots of 'freemasonry'. The tradition of building sacred/holy structures with applied sacred (euclidian), geometry was continued into the middle ages by the 'Templars', who envisioned their (mostly round) churches as 'microcosms of the world' (1). This idea was soon adopted by the Christian church, who began to employ 'sacred' dimensions into their religious buildings. These traditions were carried in the form of 'freemasonry' until, as Pennick aptly quotes - 'The lodges of freemasons closed down one by one. The last to go was the premiere lodge of Europe - Strasbourg, which shut shop in 1777. From then on, the arts and mysteries of freemasonry were carried on exclusively by 'Speculative masons' (1).

(Chartres Cathedral and Harmonic Ratios)

 

A continuation of the geometric Vesica Pisces results in the geometric matrix named Ad-triangulum...as used for the design of many of Europe's greatest Cathedrals...

 We may never know for sure whether such geometry was identified first from the observation of natural formations, or whether it came as a result of an intellectual quest, but whichever, it is clear that these natural mathematical building blocks began to be used in the design of many important man-made structures.

 

(Labyrinths)

(Archaeoastronomy)

(Harmony of the Spheres)

(Science and Technology Homepage)

 

 

References:

1). N. Pennick. Sacred Geometry - Symbolism and Purpose in Religious Structures. 1994 Capall Bann Publ.
2). S. Skinner. Sacred Geometry: Deciphering the Code. 2006. Octopus Publ Group Ltd.
3). http://www.lunarplanner.com/HCpages/Venus.html
4). J. Martineau. The little book of Coincidence. 2002. Wooden Books Ltd.
5). Laurence Hecht, “The Geometric Basis for the Periodicity of the Elements,” 21st Century, May-June 1988, p. 18. 
6). G. Santillana and H. Von Dechend. Hamlets mill. 1983. D. R. G. Press.
7). http://en.wikipedia.org/wiki/Flower_of_Life
 

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