Zig-Zag Ramps & the Great Pyramid of Giza
 

I first heard of Dieter Arnold's reversing ramps idea from Miguel Aguirre.
Unfortunately, either Aguirre, or Arnold had committed a major mistake
in the execution of the otherwise sound concept, and changes had to be
introduced to make the concept physically viable on the Great Pyramid.
The above diagram shows my version of how the pattern might look if
the ramps were to carry blocks no bigger than 3 tons.

In discussing the zig-zag method,  we shall inspect key quotes from Aguirre.
Aguirre says:
> The reversing ramp ascending in zigzag by on one side of the pyramid is
> the favorite answer of Arnold, and it appears not to have any major
> shortcoming. It uses the core of the pyramid as support, so no large
> supplementary work is needed to do it. It can be done with good
> materials, so the lateral walls can be vertical. It would have been
> destroyed as the finishing outer casing of the pyramid was installed,  so
> no remains to look for. In this concept the zigzag ramps rest on the
> steps of the still exposed core masonry

Yes, the concept is executable, I agree.

>To have a ramp inclination of 5 degrees, the length of each leg of ramp
> has to be 10 times (1/tangent) the height it raises. The most practical
> height of one leg is equal to one row, i.e. one horizontal layer of core
> masonry of the pyramid. The average value for this is 0.7 m. So the
> length of a single ramp leg is 7 m and it allows to raise the load by
> 0.7 m.

It is a big mistake not to realize that a ramp needs to widen
sufficiently, before starting anew, in reverse. As the ramp rises,
its lateral wall remains vertical, but its base recedes with the
pyramid's face. Thus, the ramp widens. Only when its width
becomes double, a new ramp of the original width can be started
in the opposite direction.  This principle is illustrated by the quick
sketch below.

On the Great Pyramid, when a ramp rises 0.7 meter vertically,
it widens by only 0.55 meter - not nearly enough to start another
ramp.  IMHO, these ramps should start out at least 2 meters wide.
Doubling of the ramp's width from 2 to 4 meters requires a rise of
roughly 2.54 meter (diag. below), or more than three average Great
Pyramid courses.

 

 
Perhaps, Arnold did not bother to simulate his theory on a model.
However,  there is a quick way to visualise the situation :
Take a paper ribbon. Split it lengthwise into 2 ribbons, but stop
halfway through. Bend one narrow strip upwards, one downwards.
This produces a crude model of the zig-zag ramp.
The 1 : 10 slope, which Aguirre proposes seems too steep, so I drew
CAD diagrams of the ramp using the 1 : 12 slope. I find it puzzling
that neither F.M. Barber, nor Arnold, nor Aguirre realized that they
had extended the idea of lubricating the way in front of a single sledge,
to solving the equation for a continuous procession of sledges. But, if
the way was lubricated, the pullers would slip and slide, and would
never get anywhere -  that much is rather obvious.
This omission to calculate using specs for unlubricated surfaces is
then wholy unjustified, and leads to more miscalculations. Even so,
Aguirre merely manages to squeeze his projections into the 20 year
schedule. It is evident that a drastic cutback on the projected capacity
of this reversing ramp system, will stagger Aguirre's timetable by
many years.
Here are my figures for the ramp as I see it.
At the 1 : 12 slope, for a ramp to rise 2.54 meters vertically,  one leg
of the ramp has to be 30.65 meters long.
If 45 pullers pull 66.6  kilo each (there is much doubt in my mind
that such effort can be sustained for years), they can together
pull a 3 ton block up a 1 : 12 ramp.
A team of 45 pullers on 3 ropes has 15 ranks in a column about 45 feet, or
over 13 meters. So, the sled and the pullers together take up at least
16 meters in length.  We need that much space on both wings of each
inclined leg of the ramp .   16 + 16 + 30.5  =  62.5 meters.
But because the side of the pyramid recedes 4 meters over two ramp
legs, as shown below - each inside horizontal wing of the reverse ramp
must be extended by 4 meters.
One leg of the reverse ramp   +    its wings   =    66.5 meters,
when transporting blocks up to 3 tons big.

This allows three ramps up to the level of 30.5 meter,
and two ramps up to the height of 73.8 meter, about half-way up.
One reversing ramp could rise to the height of 107 meters.

> According to Arnold page 276 the transportation was done in sledges. Not
> only the Egyptians had a hieroglyph for it, but also sledges have been
> discovered several times as archeological remains, e.g. south of the
> pyramid complex of Senwosred III at Dahshur. According to Arnold this
> method was used also to transport very large masses. In the tomb of
> Djehutihotep there is a picture of a group of 172 persons pulling over a
> flat surface a 58 tons figure of the monarch. They are using a sledge
> and wet lubrication. Being over a flat surface the workers are working
> only against the friction times the weight of 58 tons. A reasonable
> friction coefficient for a whetted surface will be 0.1.
                                                   *
Time to interrupt the flow of ideas. Note that this coefficient of friction
for a lubricated stone pavement will from now on be used by Aguirre
in calculations for unlubricated ramps.
Each of the group pictured in Djehutihotep's tomb is pulling behind him
a third of a ton, or 337.2 kilograms. As if that was not enough, Aguirre
increases the burden to be dragged up a 1: 10 slope to 400 kilograms,
while labeling this escalation a "reasonable compromise". .  _ Is it?
Imagine yourself dragging heavy wooden sled without steel runners
over cobblestones up a long hill, while the sled is occupied by the world's
heaviest sumo wrestler. For all I know, this reasonable compromise
would kill a horse, after a while.

>      This will
> produce a force of 330 Newtons per person (i.e. 33 kg force) to be able
> to pull the mass. A team of Sevilla Holy Week ‘costaleros’ can keep
> pulling with 500 Newtons for many -with short rests from time to time of
> course- hours but they are very motivated and cannot by taken as a
> typical case. Then a value around 400 N can be used as a reasonable
> compromise. This note will use this value here after.

The American engineer F.M. Barber, who was stationed in Egypt,
as a naval attache circa 1,900s came to a conclusion that 900 men
could pull a 60 ton block over a lubricated stoneway rising at
the slope of 1: 25. Each man would be pulling 66.6 kilograms.
Barber thought that such work would have been fairly easy, so
I have decided to keep the same numbers valid for the unlubricated
ramps in my own calculations. Perhaps, it gives the conservative side
an unfair advantage, since this unlubricated ramp is more than
twice as steep (1 : 12) than Barber's, but at the same time, I create
a nice safety margin. From here on, things can only get worse for
the orthodox side.

Copyright - Jiri Mruzek - Vancouver, B.C., Canada 


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