The Circle, The Square and a new way to find Phi ... oh my

[Don Barone]

Hi all in my never ending geometrical pursuit to find the answers at Giza I never cease to be surprised that I continue to find new and exciting "proofs" of the connections between the three pyramids at Giza and the geometrical values Phi, Pi , square root of 3 etc. With this background I have made it a point that whenever anyone finds a new way to find Phi I immediately look for how it may tie into The Giza Rectangle and the three main pyramids there. I have found many unusual supposed co-incidences at Giza and each time I find something new I always think that there is no more to discover along these avenues and then as always then and again now I am astounded when I found another new eloquent proof of what I have proposed .

One of my major discoveries was that The Great Pyramid and The Pyramid of Khafre (G1 and G2) were in the relationship of the square root of 3 (The Great Pyramid) and Phi (The Pyramid of Khafre and G2). The proof was fairly obvious with G1 being 440 cubits and G2 being 411.04 cubits and dividing these together we get 440 / 411.04 = 1.070455430 and when we divide square root of 3 by Phi we get 1.732050808 / 1.6180339887 = 1.070466269 checking to:

1.070455430
-------------- = 0.9999899 or 99.9989 % accurate
1.070466269

If we use inches we get average of 4 sides of The Great Pyramid equal to 9068.8 and north side of Pyramid of Khafre or G2 as 8471.9 inches and dividing these together we get
9068.8 / 8471.9 = 1.070456450 and dividing this by 1.070466269 (or sq rt 3 / Phi) we get 0.999991 or 99.9991 % accurate of in my opinion a virtual certainty but it seemed that it would have to rest there, however I had failed to consider the total genius of the builders as they put even more blatant proof before our eyes and it is truly amazing to behold. The builders had decided to put so much obvious proof into this relationship that only a fool would deny that this is what was meant and here is what they did to prove that G2 or The Pyramid of Khafre was indeed designed to show us and represent the value Phi.  As always I find these things while searching other things and so it was that I chanced upon a PDF file A New way To Find Phi written by Janusz Kapusta in 2005 where it was shown that using a circle and a square and two important tangent lines the Phi point was marked on one of them, Here is a series of diagrams that get us to that point.

We start with a simple circle ...



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And now let us enclose it in a square ....



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Now let us slide the square up to the diameter point of the circle ...



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Now let us draw the tangent line at the top of the circle ... NOTE: The tangent line is simply the line drawn 90 degrees to any given radius and just touching the point where the radius touches the circle.. In this case the radius points directly up so the tangent is exactly horizontal to the plain or diagram)



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Now from the top left corner of our square let us draw a straight line to the origin or center of our circle ...



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Now from the midway point of our square or the far left of our pink tangent line we will draw a line to the top right corner of our square. We will then draw a parallel line to that line, tangent to the circle ... NOTE: Please note that both these tangent lines are 90 degrees to the line we drew in the last diagram.



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The point where the two tangent lines intersect is the Phi point of the horizontal pink tangent line ...



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Final diagram ...



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But strange things lurk in the midnight sun ... indeed.

Whenever I see something new like this I try to explore it from all possible angles and see if I can find the proof myself. I still have not proved that this is so, that the line is cut in the phi ratio however I did find some interesting things which some might enjoy.

Observe what can be deduced form Janusz Kapusta's diagram.

The original:



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And what can be deduced from it with a little simple math ...



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and then a few more lines ...



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Please take a few moments and study this diagram for there are some amazing Phi ratios here. I have colour coded the lines to make them easy to visualize.

Not content with this diagram I decided to do a little more exploring of what this set up could produce. here is what I found ...

0.618034 is of course 1/Phi
0.381966 is 1 - 1/Phi
0.309017 is 1/Phi divided by 2 (0.618034 / 2)
0.190983 is 1/Phi squared divided by 2 (0.381966 / 2 )
1.118034 is Phi - 0.5 (1.618034 - 0.5 )
0.690983 is 1 - 0.309017 OR 0.5 + 0.190983 ( which is 1/2 of 381966)

I think you can see the point by now. They are all showing us Phi but always remembering the 1, 2 and sq rt of 5 right angled triangle which is everywhere within The Great Pyramid.

But as I said I am always looking for a bit more and this is what I found. Not sure if this is common knowledge among the mathematics geeks out there but I have never seen it before and I have been playing with numbers for a fairly long time. so on with another side of this diagram.

The first thing I did was turn it sideways and complete the 0.5, 1 , 0.5 configuration for as some might know this is a very common design in flags of various nations to use along with the phi rectangle. But onward. Here is the first diagram.





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What I then decided to do was to draw the tangents going in the opposite directions as so ..



I then decided to calculate the angle at the bottom of the image and since all triangles add up to 180 degrees and in a 1, 2 and sq rt 5 triangle the angles are 26.565 and 63.435 and if we add up the two 63.435 (one either side we get 126.870 and from 180 leaves us 53.130 WHICH IS THE ANGLE OF THE PYRAMID OF KHAFRE !

Here are the series of diagrams that led to the final exciting conclusion,

In this first diagram we extend the diagonal to meet the tangent line on the right



Please note this has now formed a triangle with one of the angles the angle at the bottom of the image.

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We now extend a new tangent line on the left going upward from the 0.309017 point as shown in the diagram below...



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In this diagram we have drawn a vertical line for the opposite point of a 0.309017 point



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From the bottom of this line we have just drawn we draw another tangent line ...



We then draw a tangent line from the intersection of the diagonal line and the far right tangent line heading toward the "northwest" as shown below ...



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We then complete the "pyramid or triangle" ...



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Here is the final view of the pyramid or triangle in question ...



PLEASE NOTE THAT THE BOTTOM ANGLE IS 53.13 AND PRECISELY THE ANGLE OF THE PYRAMID OF KHAFRE ... [/u][/b]

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Here is the triangle/pyramid turned 63.434 degrees to the left ...



So now I had manged to duplicate the pyramid of Khafre starting from the simplest of starting points a simple circle and a square and both of these are said to be found in The Great Pyramid. But now I decided to figure out what the size of this "pyramid" or triangle was in relation to the other phi measurements I had on my diagram and this is what revealed the sheer genius of the Master Builders once again. Behold genius at work ...

Here is the final measurements of the triangle/pyramid of Khafre ...



I was amazed to see that the diameter of the circle is equal to 1/2 the base of the pyramid or if we use the real Pyramid of Khafre which is what I have claimed all along we get 4235.95 inches or about 205.52 cubits (1/2 of north base which is 8471.9 inches as per Petrie). I then decided to see what the other distances would equal if we assigned "1" a value of 4235.95 or 1/2 north side of G2 according to Petrie.

So if 1 = 4235.95

(Line a) 0.618034 = 2617.9611223 inches
(Line B ) 0.381966 = 1617.9888777 inches
0.309017 = 1308.9805612 inches
0.190983 = 808.99443885 inches
1.118034 = 4735.9361223 inches
0.690983 = 2926.9694389 inches

PLEASE NOTE HERE !!!!!!!!!

In order to make the first relationship or (Line a) Phi squared times 1000 (or 2618.034 inches = 0.618034 in our original square and circle diagram) we simply need our "1" unit to equal 4236.068 inches which would give us a value for the base of G2 or The Pyramid of Khafre of 8472.136 and this checks to what Petrie got for the north side of G2 of 8471.9 or 7/100ths OF ONE SINGLE INCH or an accuracy of 99.997 %

This also of course makes Line B 1000 x Phi inches or 1618.034 inches.

Very interesting indeed. Who would have thought that G2 could tie into Phi so well. And interestingly I had long stated that G1 equals or represents the square root of 3 and G2 equals or represents Phi. Looks like I may have been correct all along.

So what exactly does this mean. Well what it means is that:

a) the measurement of G2 is directly related to Phi. I do not believe that this can ever be disputed again and what the builders have done is that taking a distance of 1/Phi or 0.618034 units and multiplying it by Phi squared gives us the measurement of G2. It is truly remarkable and is going to be very tough to argue for co-incidence of this one so here is the formula for the distance of The Pyramid if Khafre IN ANY UNITS !!!!!

(1 / Phi x Phi squared) x 1000 = distance of  1/2 the base of The Pyramid of Khafe in inches
 


2.618034 / 0.618034 = 4.236068 x 1000 = 4236.068

or even simpler

Phi cubed x 1000 = precisely the number of inches in 1/2 of the base of The Pyramid of Khafre in inches ...

The sheer genius of it is so humbling !!!!!!!!!!

[Don Barone]

Hi all ...

These numbers seem to be the final conclusions based on many years of research. They are as follows.

PLEASE NOTE: None of the pyramids are exactly square nor are all the sizes the same. Was it error or were they trying to tell us a multitude of things. Regardless here is what at least one of the measurements were and are for each of the three main pyramids at Giza ...

G1 = 9069.13 inches

G2 = 8472.14 inches

G3 = 4154.25 inches


Solution:

G1 = (Phi cubed x 2 x 1000 / Phi) x square root of 3 = 9069.13

G2 = Phi cubed x 1000 x 2 = 8472.14

G3 = 9069.13 / [(1 / Pi x 10) - 1) = 4154.25

Please note it is all about Phi and Pi and the square root of 3 ... Just a fluke ? Just a simple co-incidence that these three pyramids can be obtained using these simple formulae ?

I doubt it.

Cheers
DON

[Don Barone]

Okay I had this posted over at Ma'at as well but it has been deleted ... maybe too many images ... who knows with that crew over there but I will never understand why someone would not give you chance to save it before deleting it ... Well that's Ma'at for you however let us continue here.

So far we have used Phi, square root of 3 and a round about way of showing Pi and but there actually is a better way. I have claimed (as has DUNE in the past) that the height of The Great Pyramid was designed to be EXACTLY 5773.50 inches or the square root of 3 divided by 3 and then times 10,000. Okay so given our new base length for The Great Pyramid of 9069.13 inches what angle is now produced when we use 5773.50 and 9069.13. Well we use 1/2 base and 5773.50 so we divide 5773.50 / 1/2 9069.13 or 4534.565 or 5773.50 / 4534.565 and this gives us a tan of 1.273220253762 or 51.85355232521 degrees.

51 degrees 51 minutes 12.79 seconds

Let us see what the angle using Pi is. We then use 4 / Pi and this gives us 1.273239545 and this checks for ...

1.2732200375 / 1.27323954474 = 0.999985 or 99.9985 % accurate to being The Phi Angle.

Proof positive I should think that both Phi and Pi and used in the designs at Giza.

Emhotep
DON

[Charlotte]

Hi Don,

The tide seems to come in once more, but, you are aware of what I understand of your findings, so I'll say:

There is a tide in the affairs of men.
Which, taken at the flood, leads on to fortune;
Omitted, all the voyage of their life
Is bound in shallows and in miseries.
On such a full sea we are now afloat,
And we must take the current when it serves,
Or lose our ventures.

WS

[Don Barone]

Hi Charlotte no just a very small wave

here is a bit more ...

Hi all ... sitting here drinking my coffee and thinking about the simplicity of Giza.

The base of The Great Pyramids can be written with the equation ...

Phi squared (2.618034) x 1000 x square root of 3 (1.7320508) = 4534.5679 inches

4534.5679 inches x 2 = 9069.1358 inches for the base of The Great Pyramid

For The Pyramid of Khafre we have already determined that this translates into the very simple equation Phi cubed (4.236068) x 1000 x 2 = 8472.1360 inches.

Therefore for the Pyramid of Khafre since the angle is that of a 3, 4 and 5 right angled triangle the height would be 4236.068 / 3 x 4 = 5648.091 inches while the slope would be (4236.08 / 3) x 5 = 7060.114 inches.

If we keep with the square root theme for The Great Pyramid then it's height would seem to be (square root of 3 / 3) x 10,000 or 5773.5027 inches. To calculate tan of the angle we simply divide 5773.2027 / 4534.5679 and we get 1.2732200375.

For the Pi angle we divide 4 by Pi and we get 1.27323954474

So using my method and lengths for the sides of Pyramid of Khafre and The Great Pyramid we get a value of Pi for the angle checking to 1.2732200375 / 1.27323954474 = 0.999985

The angle for The Great Pyramid thus becomes ... angle that has tan of = 1.2732200375. = 51 degrees, 51 minutes, 12.8 seconds while Pi give us 51 degrees, 51 minutes, 14.3 seconds

51 degrees, 51 minutes, 12.8 seconds - mine (using Phi and square root of 3)
51 degrees, 51 minutes, 14.3 seconds - Pi
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Differences is but 1.5 seconds !

Petrie

Quote:
N. face, by air channel mouth = 51º 51' 30"



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Angles of The Great Pyramid as per Petrie: link

So if one were to assume that The Pyramid of Khafre was built first and was meant to draw attention to the fact that it is 4/11ths of the way up The Giza Plateau then logic would dictate that the base of The Pyramid of Khafre would be 411 cubits ... exactly. This would give a cubit of 20.61347 inches

Thus The Great Pyramid would be (one of it's sides) = 439.96 cubits while G3 (at 4125.25 inches would be 201.54 cubits ....

NOW SURPRISINGLY using this measurement of the cubit at 20.61347 and Petrie's distance of the north south plateau at 35713 inches we get a north to south of 1732.507 cubits ... interesting ...

Emhotep
DON

[Don Barone]

Sent this latest to John Legon and he suggested an even simpler way of doing this ... he points out that square root of 5 (2.236068) + 2 = 4.236068 which is Phi cubed ... so now we have this ...

square root of 5 + 2 = Phi cubed

Base of Pyramid of Khafre = (square root of 5 + 2) x 2 = Phi cubed x 2 = 8472.14 (inches)

Nice and simple