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Squaring the Circle

The square represents the physical. The circle represents the spiritual. Squaring the circle is an attempt to unite/bring together the physical with the spiritual. The Great Pyramid is a physical place where one can bring one's physical body and experience the spiritual - this, of course, is the purpose of all sacred spaces. All sacred geometers have attempted the impossible: to square the circle (create a square who's perimeter is equal to the circumference of a circle.) It can not be done exactly (because we are working with irrational numbers), but we can get pretty darn close.

Here is the first of two valiant attempts:

Great Pyramid

This squaring of the circle works with a right triangle (YZE). Let's begin with a line drawn from the centre of the base of one side (Y) to the centre of the base itself (E). Then a line from the centre of the base (E) to the top of the pyramid (Z) creates the right angle (YEZ). The hypotenuse of this right triangle (line ZY) is called the "apothem." {On a pyramid, the apothem is a line that splits vertically one side or face of the pyramid (line ZY) - a line drawn from the base of the centre of the base of one side to the top.}

We will find that is is line (EZ) that will help us to square the circle, but the apothem has an interesting characteristic as well.


Pyramid seethrough

Now let's look at this pyrmid  in 2D, at an oblique angle from above.

Squaring the Circle with the Great Pyramid

(ABCD) is the base of the Great Pyramid. This is lettered similarly to the wire frame version on the previous page.

We also find the centre of this square is at (E), and we divide the square into four equal smaller squares (AiEh), iEfD), (fEgC) and (hEgB)

For the purpose of this exercise, the side (CD) of the base equals 2.

Construct square (i JKD), thus creating double square (JKfE).
Create diagonal (EK) which intersects (i D) at (l). iD = 1, therefore the diameter of the circle is also 1. Remember, (EK), the diagonal of a double square, = (√5) = .618 + 1 + .618.
Put the point of your compass at (E) and extend it along the diagonal (EK) to point (m) where the circle intersects (EK), and draw the arc downward to intersect line (K D f C) at (n).

If (EK) = (5), and lines (lm) and (lD) and (li) all = .5, the diameter of this circle is 1. This makes line (Em) = .618 + 1, or 1.618.

Line (Em) is called the apothem.

Draw line (En) which intersects line (AilD ) at (o).

Put compass point at (f) and extend it to (n). Again put your point at (E) and draw the circle which happens to have the radius (E o). (f n) is the height of the Great Pyramid.

This circle comes remarkably close to having the same circumference as the perimeter of the base (ABCD).

Let's go back to the original right triangle (EYZ)

if (EY) = 1, then (YZ) = phi

and (EZ) = (√phi)

Pyramid seethrough

EY = 1, The apothem is phi/1.618. This makes the famous 51+° degree angle.
This makes the height of the Great Pyramid the square root of phi (ÿ).

Using a2 + b2 = c2
or (EY)2 + (EZ)2 = (YZ)2
or 12 + √ø2 = ø2
Is this possibly true? - Yes!

12 + 1.272019652 = 1.618033992
or
1 + ø = ø2
or phi plus one = phi square!!!

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