The Solving of The Geometry of Khafre: The Tsunami continues

[Don Barone]

Hi all ... well ... I really don't know what to say other than once my brain gets into gear it simply keeps going of it's own volition. If you remember earlier in the previous thread I suggested that one could also use the Pyramid of Khafre to illustrate the 3, 4 and 5 sided right angled triangle and well ... so I said to myself ... let's do that. It seemed like a simple task but every single image I checked online DID NOT CORRESPOND TO THE CORRECT 3, 4 AND 5 SIDED RATIO. No matter what image I used it simply did not check. Strange I thought but finally found one that was close ... and here is is below ...



Now before we go on we have to remember that there are also two passageways in this pyramid. They call them the upper and the lower. They have absolutely no idea why they are are where they are but  ... well I now know and soon so shall you. Here is what is written about them ...

"The oldest of the two entrances into the subterranean depths of Khafre's pyramid is now located in the ground about thirty meters north of the pyramid. Carved completely out of the rock subsoil, it is sometimes called the "lower entrance". This portal communicates with a corridor that at first descends before running horizontally. In this horizontal leg of the corridor, a passage gives way on the west wall to a small chamber cut from the bedrock and provided with a pented roof, where part of the burial equipment was possibly stored. After the horizontal section of the entrance corridor, it finally ascends into a horizontal corridor shared by the "upper entrance". ... The second portal, known as the "upper entrance", is located in the north wall of the pyramid's face about twelve meters above ground level. It communicates with a corridor lined in pink granite that first descends before running horizontally at the base of the pyramid. At the transitional point between its descending and horizontal sections, there is a barrier made of pink granite, which in antiquity, grave robbers managed to dig around. The horizontal passage continues south after the barrier, eventually arriving at the burial chamber, which lies on the vertical axis of the pyramid. Given the location and relatively simple construction of the access corridor and the burial chamber, it is likely that the architects of this pyramid sought to avoid the complications that builders of Khufu's pyramid had encountered with their technically difficult system of passageways,barriers and chambers."

Strangely I could not find anywhere the angle of the shafts themselves but an idea was already forming in my head and I was pretty sure I already knew what the angle was ... and so I began to draw lines and circles ... which I seem to do so well

Again before I go on this was all based on my previous work of the circle and the sqaure and so I reasoned that the solving of this pyramid HAD TO BE JUST AS EASY AS THAT ORIGINAL DIAGRAM WAS ... and you know what ... turns out it was ... and so I began to try to duplicate my previous work using The Pyramid of Khafre as my template and came up with these diagrams ... You will note they are not exact but that is becasue I don't think the iamge was the correct ratio in height and width ... but onward we go ...

First I drew in our 5 unit radius circle ..



And then our "box"



And then our three unit side ...



And then our 4 unit side ...



And then our 5 unit hypotenuse side ...



Now the next line I am going to draw, from the moment I first drew it onto my original diagrams was I knew deep down inside of me the angle of all  the shafts in all the pyramids of Egypt or at least at Giza ... and as I drew it on this picture I knew I had been correct ... I was sure it was going to mark the upper passageway ... BUT IT DIDN'T !

 DRAWING THE DIAGONAL LINE AND THE KEY TO THE PUZZLE ...



Now if you look at this image the place where the diagonal exits the pyramid appears to be way too high to be the upper passageway but an idea was already forming in my head and so I drew the diagonal where the many images I had seen of this passageway seemed to be ... here is that image ... 



And so it was that using my ever inquiring mind I decided to see where and why this diagonal had been placed where it was and the result will be sure to make you smile at once again the sheer simplicity of it all ...

Meanwhile I had gone in search of somewhere where the angle of this upper passageway was mentioned and believe it or not came up totally blank but that was really not a problem because I already knew what it was ... and I will illustrate using a nice diagram I found on line ...

Going to save this so far just so I don't lose it ... stay tuned for the next post coming next ...

Cheers
Don Barone

[Don Barone]

Okay let's keep going. I searched the internet but couldn't find the angle of the upper passageway but did manage to find this iamge which I knew would come in quite handy ... and it did ... Here is the template we will use ...



Now that image was too large so I shrunk it to fit but not before trying these experiments on it ...

Well the first thing I did was measure the height and horizontal distance of the shaft and found, as I had suspected that it was 120 pixels by 60 pixels and of course that is 1 to 2 and of course yields the angle of  26.565 degrees. You will see the small black triangle in the upper left of the next diagram. The next thing I did was expand the triangle as below ...



I then of course drew an extension of the upper passageway as below ...



So next I drew a vertical line form the apex of our pyramid to where the upper passageway would have met the vertical line as in this diagram below. Now I have to admit here that I had kind of cheated and had already scaled the distance in my diagrams of the actual pyramid so I kind of had an idea where all of this was going but it was nice to have it it proved correct ... anyway the iamge ...



Well now with this diagram in front of me it seemed only logical that I measure or calculate what the green distance was and what do you suppose it equalled ?



And lo and behold ... keeping in with our "simple is as simple does" the vertical distance obviously had to equal ...




Now interestingly I automatically assumed that the lower passageway was the same angle ... BUT I WAS MISTAKEN it was different ... as shown below ...



They were converging and when ever I see two converging lines I have to find out where they are going to meet AND CAME UP WITH THIS DIAGRAM ...



And this diagram followed naturally ...



And then the elegant proof of my assertions ... drawing circle with radius of 5 units ...



And the meeting of the two shaft angles seem to be directly under the left hand side of the pyramid ... I am about to calculate that distance ...

I mean really how can it be this easy and be missed for over 4500 years ? I mean really in my opinion geometry should be a prerequisite for Egyptology !!!!!!!!!!!!!!

Anyway I have shrunk the image so it is a bit clearer ... enjoy ...



Cheers until next time ...

Don Barone








 

[Don Barone]

And then as I was just staring at this image the final beautiful conclusion exploded in my brain for this is where it all leads I suppose ...

this is beautiful !!!

AND SIMPLE !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!







The Master Builders strike again ... and as I have said from the beginning ... It is all about the geometry!!!!!

Absolutely beautiful to behold ...

Another even more satisfying day !!!!

Cheers and what the devil will be next ?

Don Barone

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[Don Barone]

The moving finger writes and having writ it moves on ... Leaving the underground chambers, which seemed to vibrate at the core of the Third Pyramid like the convoluted, multi-valved heart of some slumbering Leviathan, we made our way along the narrow entrance corridor and into the open air.

Our objective now was the Second Pyramid. We walked along its western flank (just under 708 feet in length), turned right and eventually came to the point on its north side, about 40 feet east of the main north-south axis, where the principal entrances were located. One of these was carved directly into the bedrock at ground level about 30 feet in front of the monument; the other was cut into the northern face at a height of just under 50 feet.

From the latter a corridor sloped downwards at an angle of 25° 55’.10 From the former, by which we now entered the pyramid, another descending corridor led deeply underground then levelled off for a short distance, giving access to a subterranean chamber, then ascended steeply and finally levelled off again into a long horizontal passageway, heading due south (into which also fed the upper corridor that sloped down from the entrance in the north face).

High enough to stand up in, and lined at first with granite and then with smoothly polished limestone, the horizontal passageway was almost at ground level, that is, it lay directly beneath the pyramid’s lowest course of masonry. It was also extremely long, running dead straight for a further 200 feet until it debouched in the single ‘burial chamber’ at the heart of the monument.

As we have already noted, no mummy had ever been found in this latter chamber, nor any inscriptions, with the result that the so-called Pyramid of Khafre was wholly anonymous. Latter-day adventurers had, however, carved their names on to its walls—notably the former circus strongman Giovanni Battista Belzoni (1778-1823) who had forced his way into the monument in 1818. His huge and flamboyant graffito, daubed in black paint high on the south side of the chamber, was a reminder of basic human nature: the desire that all of us feel to be recognized and remembered.



It was clear that Khafre himself had been far from immune from this ambition, since repeated references to him (as well as a number of flattering statues) appeared in the surrounding funerary complex.11



11 Traveller’s Key to Ancient Egypt, p. 123.

[Don Barone]

Finding the numbers ... starting from a square and it's enclosed circle ... no measuring tools need apply.



Cheers
Don Barone

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[Don Barone]

Now where have we seen this before ?

[Don Barone]

But now what is the angle of the second passageway ... the lower and what does it signify ?

To get that answer we find that according to reports the lower passageway was some 30 feet from the face of the pyramid. 30 feet is of course 360 inches and if added to the base length of 8472 inches we get 8832 inches. Now to compete the triangle and to solve the problem we have to use 2.5 units where 1 unit is 8472 / 6 or 1412 inches. 2.5 units then becomes obviously 2.5 x 1412 or 3530 inches as in the diagram below ...



Using simple trigonometry we find that the tan is 3530 / 8832 or 0.39968 which is more than likely meant to be exactly 0.4 ... And of course if we take the reciprocal of 0.4 we get 2.5 which just happens to be the unit length of the opposite side ...  Co-incidence ? I would doubt it very much ...

Angles of this triangle are 21.80 and 68.20

Cheers
Don Barone

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[Don Barone]

Hi just a very quick update. It has occurred to me that ANY SQUARE ENCLOSING ANY CIRCLE ... will yield the results of the 3, 4 and 5 sided right angled triangle ... The ratios and distances involved will always be in the 3, 4 and 5 ratios. Very nice discovery. Thanks Osiris, thanks Isis, thanks Horus Oh yet and thanks Khafre

Cheers
Don Barone

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