Maybe the most important subject of sacred geometry is the Golden Mean. The Golden Mean is a very special ratio and is expressed by the Greek letter Ф called Phi.

= ½ * v5 + ½ = 1.618

Phi like Pi is an irrational number, meaning you can never calculate its exact value, you can only approximate it.

The Phi ratio is expressed in the Golden Section. The Golden Section is a the length of let’s say a rope when it is divided such that the ratio of the longer part of the rope to the whole is exactly the same ratio as the shorter part of the rope is to the longer part. (Read it again)

When the Phi ratio is applied to a rectangle whereas B = 1 and A has length Ø, the rectangle is called a Golden Rectangle.

The Golden Rectangle can be used to create a spiral, the Golden Spiral. Starting with one Golden Rectangle, a second Golden Rectangle can be attached to the first using the longest side of the rectangle, side A as the shortest side B of the next rectangle. To this end the second rectangle is constructed 90 degrees perpendicular to the first rectangle. If this process is continued, called the spiralling of the Golden Rectangle, a curved line can be drawn through the corners of the rectangles creating the Golden Mean spiral. The spiralling of the Golden Mean spiral continues indefinitely in inward and outward directions, it’s getting smaller and smaller spiralling inwards and getting bigger and bigger spiralling outwards.

Golden Mean spiral

A variant of the Golden Mean spiral is the Fibonacci spiral. The difference with the Golden Mean spiral is that is does not spiral in indefinitely but starts with a Golden Rectangle of which one side has length 1 and the other length Phi.
Gradually when the Fibonacci spiral, spirals outward, there will be no distinction noticeable any more between a true Golden Mean spiral and the Fibonacci spiral. The Fibonacci spiral is based on the progression of the Fibonacci sequence.

Crop circle, fractal spirals, Milk Hill, Wiltshire august 12th 2001

Fibonnaci sequence

Leonardo Fibonacci (1175 AD), a great mathematician of the Middle Ages discovered the Fibonacci sequence by studying nature. He studied the growth of rabbit populations and the growth of leaves and petals and discovered a well-defined mathematical sequence in all of this.

This is the Fibonacci sequence:

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

Each number in the sequence is the sum of the two preceding numbers starting with the root number 1. The Fibonacci sequence progresses towards the Golden Mean if we divide two successive numbers in the sequence.

1/1       = 1
2/1       = 2.0
3/2       = 1.5
5/3         = 1.667
8/5         = 1.60
.
144/89 = 1.618

The Fibonacci sequence propagates towards Phi (Ø) but never reaches it since it is an irrational or transcendent number.

Fibonacci spirals and Golden Mean ratios appear everywhere in the universe. The spiral is the natural flow form of water when it is going down the drain. It is also the natural flow form of air in tornados and hurricanes. Here’s another beautiful example of a Fibonacci spiral in nature, it’s the Nautilus shell and every book about sacred geometry contains one:

Nautilus shell

The Golden Mean ratio is all over the human body, in the ratios between the bones, the length of your arms and legs. The Golden Mean is also the ratio in the distance from the navel to your toe and the distance from your navel to the top of your head. Michelangelo has beautifully hidden these Golden Mean ratios in his fresco on the ceiling of the Sistine Chapel in Rome:

Michelangelo and the Phi ratios in the human hand
(Courtesy of Frank van den Bovenkamp, www.frankvandenbovenkamp.com)

A classical mathematical problem dating all the way back to Plato is called ‘squaring the circle’. In the last three thousand years mathematicians have tried to come up with a solution using only a pair of compasses and straightedge on how the construct a circle and square such that they have the same perimeter. In 1882 Lindemann proved that there is no solution to this problem. Since Lindemann’s proof is rather complex, we will show in simple terms why the circle cannot be squared. The circumference of the circle with a radius of one is 2 * π and π (Pi) is an irrational number (a transcendent number, π can never be measured only be approximated!). But when Pi is irrational and cannot be measured so is the circumference of the circle! However the square’s circumference is a real number since it equals 4 times the side of the square that is a real number that can be measured. Hence the circumference of both circle and square will never be equal in the mathematical sense; however they can become infinitely close.

Vitruvian man

Now here’s an interesting drawing from Leonardo Da Vinci. What he is showing in this sketch is that the human body ‘squares the circle’. When man stretches his arms and holds them horizontally, man’s body will perfectly fit into the square. On the other hand, when he spreads his legs and raises his arms as in the sketch, man’s body can be perfectly circumscribed by a circle. The circumference of the square ‘equals’ that of the circle.

A lot has been written about this sketch alone, it contains a whole lot of hidden sacred geometry. We will not go into all the details here, but I want to show you some very amazing things. Ancient wisdom, the Hermetic tradition tells us that the human body can be regarded as a blueprint for the universe by means of all the ratios that are found within the body. This may indeed be true. Let’s have a look at the following picture:

This is the same picture as above, in the picture there are now two red circles added. The biggest red circle fits perfectly and is inscribed by the square. The smaller red circle’s is centered between the outer circle and the inner red circle and tangentially touches both.

Much to our surprise, the upper red circle represents the moon and the lower red circle represents the Earth!  In mathematical terms: The ratio of the diameter of the smaller red circle to the bigger red circle r / R equals the ratio of the diameter of the moon to the diameter of the Earth! Now let’s prove it:
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Radius of the moon : r
Radius of the Earth : R
Side of the square is : 2R
The perimeter of the square is : 8 R
Radius of the outer circle is : r + R
The circumference of the outer circle is : 2 p ( r + R )

Now ‘squaring the circle’ makes the perimeter of the square equal to the circle:

8 R = 2 p ( r + R )

8 R - 2 p R = 2 p r

R (8 – 2 p) = 2 p r

r / R = (8 – 2 p) / 2 p = (4 – p) / p

Earth radius = 6,370,973 m
Moon radius = 1,738,000 m
Moon to Earth ratio = r / R = 0.27279977
r / R = (4 – p) / p = 0.273239544 (p = 3.14159265)

Quod Erat Demonstrandum!

There is another mysterious relation to be discovered in the sketch of the Vitruvian man by Leonardo Da Vinci. The Great Pyramid at the Giza plateau in Egypt called after the pharaoh that is supposedly buried there named Khufu (Cheops in Greek), holds a perfect geometrical relation to the squaring of the circle and the Vitruvian man as depicted by Leonardo Da Vinci!

Look at the picture below:

Great Piramid at Giza in relation to Vitruvius man

The triangle in the picture is the exact geometrical proportion of the Great Pyramid at the Giza Plateau near Cairo, Egypt. The angles between the base and the apex (top) of the pyramid are exactly 51 degrees and 51 seconds. (51º 51’). 
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The Great Pyramid and in fact the complete layout of the Giza plateau with all its pyramids, sacred temples and the sphinx contains a lot of hidden sacred geometry of which we will reveal more in this book. The point I want to make here is that the Egyptians were aware of the art of sacred geometry and how it relates to the universe, as the Giza plateau and in particular the Great Pyramid proves.

Let there be light

While explaining the Genesis pattern of sacred geometry we mentioned that on the second day of creation God created the Vesica Pisces and that the Vesica Pisces is the geometry of the light particle the photon. The Holy Bible mentions the second day of creation as the creation of the light. By the way did you notice that the Vesica Pisces has the shape of the eye?

It was discovered by Buckminster Fuller, who did a lot of the groundwork in re-establishing sacred geometry, that the geometry of the photon must be two tetrahedrons joined at a common face.

Now the geometrical shape of the double tetrahedron is perfectly circumscribed by the Vesica Pisces whereas the vertices of the tetrahedrons neatly touch the face of the Vesica Pisces. This has been confirmed by Drunvalo Melchizedek another sacred geometry architect!
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The tetrahedron is also the hidden geometry in the electromagnetic wave (wave form of light) itself. The electrical and magnetic fields are perpendicular to each other and a spiral can be drawn connecting the electrical field with the magnetic field that exactly traces over a tetrahedron!

Tom Bearden, the inventor of the MEG machine has proof that James Clerk Maxwell must have known this but that Oliver Heaviside removed the knowledge of the hidden tetrahedron in the simplified version of electrodynamics.
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Notre-Dame de Chartres

The knowledge of sacred geometry has been preserved in the architecture of churches and cathedrals all over Europe. The cathedral of Chartres is famous for its sacred geometry that was used in its design. Sacred geometry can be found for example in the layout of the ground plan and the stained glass windows that contain the Phi ratio.
                    

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