Built 4500 years ago (2400 or 2800 BC), the pyramids at Giza, near the apex of the Nile delta, have impressed everyone who has ever seen them. The pyramid of Cheops is forty stories tall, and is reputed to have taken 30 years to build. It is the largest, the oldest (of monumental blocks), the furthest North. The proportions remained the model for another thousand pyramids for the next ten centuries.
Secrets of the Giza Pyramids
the Arcana of Universal Wisdom
New Mathematical Conclusions
Secrets of the Giant Pyramids
of South Bend, Indiana
- revealed -
The gentlemen who inhabited the 18th and 19th century, still believing in "the wisdom of the ancients," were determined that the pyramids held meaning beyond being grave markers. Newton waited for measurements before declaring the theory of gravity, for he was convinced that the pyramid of Cheops held the secret to the diameter of the earth.
Pyramids on the Danube at Budapest, 1993
These images were hand scanned -- years ago -- which explains their strange proportions. I stole them from a number of sources, and used them to illustrate the Giant Pyramids of South Bend, Indiana.
Goats grazing along the lower level of blocks at Cheops' pyramid.
Some of the casing blocks on the right.
The Pyramids, as with most grave markers in ancient Egypt, were endowed for their upkeep. Lands were set aside for the priesthood, and the revenue paid for cleaning, repairs, and fresh flowers. In Roman times the pyramids at Giza were still being maintained by its janitors and priests. But after 600 AD Egypt started to fall apart. The outer layer of Cheops' pyramid was stripped of its limestone to build a mosque at Giza.
Using a 3-4-5 triangle to check the casing stones.
The secret of the pyramids is that they are grave markers, and hold no secrets other than the use of the simplest construction methods. The Egyptians were familiar with all the utilitarian mathematics still in use today by carpenters and masons. To face off the outer layer of limestone (the inner layers are rough cut local sandstone) a 3-4-5 triangle was used.
You can readily construct a hundred 3-4-5 triangles, and they will all be accurate. You can have a hundred workmen all chopping away at the outer face of limestone blocks, and expect that all of them will come to the same angle. The "3" side simply has to be compared against the horizon, or a plumbbob can be compared against the "4" side.
Additionally, the stacked up rough steps of the pyramid can be tested at any point against a 3-4-5 triangle visually -- as could the slope of the outer face blocks.
The pyramids have a crossection of two 3-4-5 triangles, with the "3" sides as the base, and the two "4" sides butted up against each other. Thus the pyramids are 4 units high, and 6 (twice 3) units across at the base.
The mathematics now starts to add up very curiously... Consider that the sum of 3 + 4 + 5 is 12 -- certainly an auspicious number, for the Egyptians recognized the 12 houses of the zodiac, and the ratio of the longest side to the shortest side of the 3-4-5 triangle is 1.66 - close to the golden mean (1.62), a measure frequently used in Egyptian monumental construction. (I will reveal the source of the golden mean in another contribution to the Arcana of Universal Wisdom, later.)
- The crossectional area of the pyramid is then the sum of 3, 4, and 5. I have no idea what this would be used for in determining some element of the construction.
- The base perimeter of the pyramid is twice the sum of 3, 4, and 5. This is the total line of exterior finished casing blocks used at the bottom.
- The area of the base of the pyramid is three times the sum of 3, 4, and 5. This is the area which has to be leveled before construction starts.
- The volume of the pyramids is four times the sum of 3, 4, and 5. This is the total volume, approximately, of blocks used as fill. Divide by the average block volume to find the number of blocks required for fill.
- The area on the sides of the pyramid is five times the sum of 3, 4, and 5. This is the total area of finished limestone blocks used for the casing. Divide by the outer surface area of the blocks to find the number of required blocks.
If nothing else, these measures would come in handy for the contractor; from a single measure (say the heigth) all the materials could be ordered: the size of the base (which was leveled before construction started), the volume of rough cut rock needed to fill the inside, the amount of face stone needed for the exterior.
No other triangular relationship gives the same results.
I should add a note on the "angle" of the pyramids. The Pyramids of Kafre and Menkaure have enough casing left to measure the angle. It happens to be 53.1 for Kafre, 51.3 degrees for Menkaure (recent published results I have seen somewhere). I have also seen: Khufu (Cheops) at both 51 degrees 52 minutes, and at 51 degrees and 50 minutes; Khafre at 52 degrees and 20 minutes as well as 53 degrees and 8 minutes; and for Menkaure 50 degrees and 47 minutes and just 51 degrees.
None of the casing blocks of the Khufu pyramids are extent. Supposedly there are samples in the British Museum, but I have also seen photographs of bottom casing blocks still intact. The original thefts (extending from 600 AD) were hampered by six feet of sand, so that there were still some in place in the 19th century. The top of the pyramid is also missing, as if someone removed it, or started to take down the Khufu pyramid. At any rate, the published angle (currently) is 51.5 degrees.
A 3-4-5 triangle has an angle of 53.1 degrees between the 3 and 5 side. Close enough for monumental government construction, if you ask me.
The remainder below are a series of scanned images ranging from the stupid to the bizarre.
There are thousands of these base bearing stones found all over Egypt. The are often identified as mill stones, but they just don't work that way. One source's suggestion for their use as part of cranes makes much more sense. The Egytians moved enormous masses of materials up and down the Nile. The wind is always blowing South, the current carries the same vessels North.
For the interior enclosure of the Giza pyramids, two hundred ton blocks of stone were transported by ship for a distance of 200 miles.
If you have ever worked with a roofing square to mark the cuts on hip rafters (or whatever) you will realize the utility of such a simple device
Thales (b 640 BC) used this method to figure the height of the Pyramids at Giza (Plutarch notes).
But the Sears tower is three times as tall as Cheops' pyramid.
In case you fail to read this correctly, the man on the sled is pouring milk in front as a lubricant.
From Napolian's notebooks.
See also a small collection of [books] at the CPL.
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