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Pi and the cubit
The most simple ratio for the pi (ã)
formula is 22/7, although today it is usually expressed decimally
as 3.1416. Using the first of these two equations it becomes
obvious that a circle with a radius of 7 units has a circumference
of 44 units. In other words a circle with a radius of 1 cubit
has a circumference of 44 palms. Using this method it is very
easy to sub-divide the circumference into halves, quarters, eighths
and sixteenths; there being 22 palms in a semi-circle, 11 palms
in a quadrant and 5.5 palms in 1/8th of circle and 2.75 palms
in 1/16th. In addition it is also very easy, through simple multiplication
or division, to arrive at length of the circumference or a circle,
or part thereof, once the radius is known. For example a radius
of 2 cubits gives a circumference of 88 palms; 1.5 cubits has
a circumference of 66 palms and so on.
There is no evidence in the mathematical
texts that the Ancient Egyptians knew the 'pi' formula as such.
In practice this would not be necessary for the formula was already
contained within their measuring system through the simple adoption
of the sevenfold division of the cubit.
Seen in this light it now becomes apparent
why the Egyptians might have adopted a seked of 5.5 [5 palms
and 2 digits] for the Great Pyramid and at least two other pyramids.
As has often been stated the distance around the base of the
Great Pyramid exactly equals the circumference of a circle whose
radius is the height of the pyramid. Through the ratio of 7:11
there is a direct relationship between the three main elements
in architecture, the square, the triangle and the circle. In
the case of the Great Pyramid the circle is unseen, which equates
with how it is often perceived in mystical tradition representing
infinity, or the element of the divine.
In further support of these arguments the
cubit measures of the Great Pyramid are generally given as base sides being
440 cubits and the height as 280 cubits, which are directly related to the
numeric ratios stated above. The two numbers 7 and 11 can also be found
woven into other elements of the pyramid's design. For example the height of
the King's Chamber is 11 cubits and there are 7 corbels to each side of the
Grand Gallery. These elements may have been coincidental being applied for
practical or aesthetic reasons. It is also possible that they may have
contained a symbolic function in relation to the numeric ratios contained
within the pyramids design.
.Seked problems
Having considered reasons why seked's of
5.25 and 5.5 might have been used, there is still the problem
of those pyramids which do not easily fit with the known seked
ratios. These are the pyramids of Menkaure (51.19°), the
Northern Stone Pyramid (43.6°) and the upper part of the
Bent Pyramid (43.35°).
In examining these anomalies I first wish
to focus on the Northern Stone Pyramid of Sneferu. Slight variations
in the angle of slope are given by different authorities. For
example I. E. S. Edwards gives an angle of 43°-36', whilst
Mark Lehner in his book The Complete Pyramids suggests 43°-22'
and Grinsell in his book Egyptian Pyramids 8gives 43°-40'.
It should be remembered that accurate determination of the angle
is difficult, although in the case of relatively intact pyramids,
of which this is one, the best assessment can be made from measuring
the length of the base and the size and height above base level
of the highest remaining course. This gives reasonably accurate
figures from which to calculate the angle of slope.
In a recent article by George Johnson,
in KMT magazine (Vol. 8, no. 3), the angle of slope of the Northern
Pyramid is given as 43°-36'-11", with an original base
of 722 feet and a height of 343 feet. If these measurements are
correct then the angle of slope can be shown to be 43° -
32' - 7". However the angle given suggests that the height
measurement may not be completely accurate. To precisely fit
an angle of 43° -36' -11" the height would need to be
343.812 feet.
This distances can be compared with the
standard Egyptian measure of a Royal cubit. The base sides can
be shown to be 420 cubits derived from a cubit length of 1.719
feet. Dividing this ratio into the height gives a figure of exactly
200 cubits (343.8 / 1.1719 = 200). The height to base ratio can
therefore be shown to be 20:21 which conforms to the ratio figure
given by Petrie. However this ratio does not fit a standard seked
giving a measure of 7.35 sekeds. As already stated it, would
not be practical, in constructional terms, to work with a seked
based on a fraction of a digit.
All authorities suggest angles that fall
around forty-three and a half degrees. This needs to be compared
with sekeds of 7.25 which gives an angle of 43.99° and 7.5
which produces 43.02°. By way of comparison a pyramid with
a base of 722 feet and a seked of 7.25 would have a height of
348.55 feet, whilst one with a seked of 7.5 would be 336.93 feet.
Of these two measures 348.55 lies closest to the presumed height
of 343.81 feet. However the discrepancy of 4.75 feet is greater
than one would expect from an accurate survey. All of this evidence
suggests that, on occasions, the Ancient Egyptians used different
ratios for calculating their seked.
This situation is further highlighted in
the Bent Pyramid which also has an upper angle of around 43.5°.
The angle of the lower half of this pyramid (54.52°) falls
within 3.6' of a seked of 5.00 (54.46°), whilst the upper
portion is nearly one third of a degree adrift from a seked of
7.5. If we assume that the measurements of this pyramid are accurate,
which the seked for the lower portion suggests, then we are forced
to the conclusion that variations for seked calculations must
have occurred, despite the lack of textual evidence in RMP.
The ratio of 20:21 can be derived if the
one cubit height is reduced by 5/7ths, which can be restated
as working to 20 palms height rather than 28 for a full cubit.
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