The Great Pyramid
of Giza and assorted towers
In Lewis Dartnell’s book,”Origins” [1], he describes the
Great Pyramid of Giza, built during the reign of the pharaoh Khufu (Cheops), which was completed around 2,560 B.C. - remained
the tallest man-made structure until the completion of the Cologne Cathedral in
1880. He adds that it was made of 2.5 million limestone blocks – each weighing
2.5 tonnes on average – arranged in 210 layers. (For residents of Delhi: the
Qutab Minar is 73 metres high).
This set me off: can I figure out the total number of
blocks? And the height of the pyramid? It turns out (naturally) that the
heights of pyramids, towers (including the Tower of Babel) and mountains, have
been studied, at length, already.
The Figure above shows a possible arrangement of blocks in a
stepped pyramid with a square base: the central block (labelled ‘1’) is at the
top of the pyramid, on top of the 3x3 layer (left) just below it (labelled ‘2’,
right), which, in turn, rests on the 5x5 layer (that should be labelled ‘3’).
Each block rests directly on one block below it (i.e. looking down from the
top, there are no overlaps).
I can estimate the number of blocks (assuming the
arrangement in the diagram above) from the sum of the series: 1 + 9 + 25 + …
with 210 terms. The formula for the sum of the squares of the odd-number series
is [2]:
S = n(2n +1)(2n-1)/3
However, this number comes out as 12.34 million – way too
many! What happened?
Well, I was assuming a straightforward square pyramid with
one block on top of another, not overlapping, which would give a pyramid angle
of 45°. The actual angle is about
51°52’ [4,5, see figure
above]. So if we reduce the number calculated by the ratio of [cos(51.5)/cos(45)]2
= 0.7626. This gives a better number: (0.7626)(12.34) = 9.41 million – but it is
still off by a factor of about 4 - clearly not right. What is clear is that there
is ‘overlap’ (if you take a top view), and, from the figure below, it is more
than that assumed by just changing the pyramid angle. (Note: the topmost block
of figure below on the extreme right should be placed at the centre,
symmetrically, as should the central 2x2 set of blocks be properly centred [6]).
Consider the sum of squares of natural numbers: 1 + 4 + 9 +…+
n2 – as indicated in the figure above [6]:
This adds up to: Sn = n(n+1)(2n+1)/6
For n = 210, this works out as: S210 =
(210)(211)(421)/3 = 3.109 x 106
The angle of this pyramid is also 45°, except right at the apex where
it is 63.5°
(i.e. tan-1(2)).
Taking the angle factor into account, this goes down to: (0.7634)(3.109
x106) = 2.373 x106.
This is fairly close to the standard estimate of 2.5 million
limestone blocks. But:
Sn – Sn-1 = n2
So the number of blocks in the lowest level is 210x210.
The weight of the pyramid is estimated to be 5.75 million
tons [5].
A further reality check: the density of medium-density
limestone is in the range 2160-2560 kg/m3 [7]. Take the average as
2360 and the weight of each block as 2,500 kgs. The volume of the pyramid
becomes: (2500)(2.5x106)/2360 = 2.648 x106 m3.
The volume of a pyramid is given by: V = Ah/3, where h is
the height and A is the base area. The height given is h = 147 metres, and the
base is b = 230 metres [5], related by the factor tan(51.843°) = h/(b/2). This gives: V
= 2.5x106 m3 which agrees with the previous estimate. According
to Wikipedia [8], the base is 230.34 metres, the original height was 146.7
metres (currently 138.8 metres), and the volume is 2.583283x106 m3.
Because of the height reduction there are only 201 layers now [9], with a
pyramid angle of 51.843°.
Note: there are various chambers inside the pyramid, but the
total volume of these (~4,000
m3) is small compared to the total [8].
The height of each block is about 147/210 = 0.7 metres. The area
of the block should be:
2500/[(0.7)(2360)] =
1.51 m2
So the side of the block is about 1.23 metres. This should
give a base value of b = 258 m – which is about 12% more than the actual value
which should be 1.097 metres (i.e. 230.34/210). This discrepancy is probably
the same as the pyramid angle factor noted above [cos(45°)/cos(51.843°) -1] = 0.144.
An absolutely fascinating analysis by Gary Meisner [10]
relates the pyramid angle mentioned above to the square root of the golden
number, f (@ 1.618) and p. That is, f1/2 = 1.272,
and tan-1(1.272) = 51.83°.
Meisner argues that the Egyptians did not know about decimals but they could
have approximated p @22/7 and f @ 196/121 (or f1/2 =14/11). He
also mentions other properties of the pyramid.
Rhett Allain asks a different question [11]: how high could
we make a pyramid or ‘does its slope matter?’ He attempts to find the limiting
height by using the compressive strength of limestone (40 MPa) and the weight
of the pyramid, and derives some results, but is rather unhappy because the
limiting height is much larger than that observed. He estimates h as:
H = s/(rg)
Where s
is the compressive strength of limestone (40 MPa), r is the density (2,500 kg/m3)
and g = 9.8 m/s2. This gives a number of 3,200 metres!
Allain [11] also plots data for the height h vs the base b
for many (~20) pyramids
and a rough inspection suggests a roughly (empirical) linear relation: h @ 0.6b.
Allain [11] also calculates the pressure exerted at each
cross-section of the pyramid as a function of height, in order to calculate the
maximum height of the pyramid – but, apart from the fun in the exercise, he
need not have bothered. As far as the maximum height is concerned, a pyramid
(consisting of its central blocks only) is the same as a tower; the advantage a
pyramid has is not its height (which is limited by s, as for the case of the tower),
but its superior stability.
The Tower of Babel was presumably built later than the Great
Pyramid, and is also supposed to have been a ziggurat (square-based pyramid),
but there are various reported figures of its height, some quite incredible [12]:
a a) 2,484 metres (The Book of Jubilees)
b b) 211 m (The Third Apocalypse of Baruch)
c c) 91 m (Gregory of Tours)
d) 13,000 m (John Mandeville)
e) 7,600 m (Verstegan)
But, assuming it was
made of brick, its maximum height was calculated by Prof.J.E.Gordon [13] as:
H = 40x106/(2000x9.8) = 2,040 metres
Dartnell [1b] points out that skyscrapers in New York went
so high because they are built on hard metamorphic rocks (schist) that serve as
a perfect foundation, and areas that do not have schist have smaller buildings.
This requirement of a strong foundation may answer Allain’s question [11]: why
are the pyramids so small compared to his theoretical estimate? Allain [11]
suggests that finances and resources may be the actual constraints – but it may
be down to sandy foundations…
This is somewhat similar to the question answered by
Weisskopf [14] many years ago: how high can a mountain be? But he takes into
account the energy of liquefaction EL (0.136 eV) of the material under the
mountain (metals/minerals) to calculate its maximum height:
h = EL/(mpAMSg)
where mp is the mass of a
proton, AMS is the mass number of SiO2 (silica):
Numerically, this becomes:
h = [(0.136)(1.6x10-19)]/[(1.67x10-27)(60)(9.8)]
= 21.6 kms
The height would be reduced if one assumed that the material
under the mountain does not get turn liquid under the weight of the mountain,
but merely undergoes plastic flow (at lower stress values than for liquefaction).
A more recent estimate [15] – using the same formula as
Allain [11] - yields 10 kms, but a lot
depends on the parameters assumed… The question is: could one, in principle
(never mind the logistics or the finances!), build a pyramid-shaped mountain 10
kms high?
Auerbach and Wan [16] have communicated an analysis of how
the number of skyscrapers (defined by them as buildings taller than 150 metres)
has increased at a rate of 8% per year since 1950, to the current number of
3,251 in 258 cities all over the world. They estimate there will be 41,000
skyscrapers by the year 2050, with a 9% chance of the world’s tallest building
being 1 mile high. This statistical analysis assumes that current trends
continue. But we probably will not be building any mountains any time in the
near future!
I remember reading about a proposal to build a tower at the
South Pole that would be tall enough that it would intercept sunlight the whole
year around, providing solar power even in winter. This would need be about 570
kms high (to take care of the 23.45°
tilt of the Earth’s axis) [17].
cos(23.45) = R/(R+h)
where R is the radius of the Earth.
Anyway, it is way above Weisskopf’s limit [14] !
1 1. “Origins: How the Earth made us” Lewis Dartnell
(The Bodley Head, 2018) a) p.127 b) p.154
13. J.E.Gordon, “
Structures, or why things don’t fall down” (Pelican, 1978)
1 14. Victor Weisskopf.”Of
Atoms, Mountains and Stars: a Study in Qualitative Physics” Science 187 (1975)
605-12, and: http://www.hk-phy.org/articles/mount_high/mount_high_e.html
1 16. J.Auerbach
and P.Wan,”Forecasting the urban skyline with extreme value theory” arXiv
29Oct2018 1808.01514v2
1 17. http://theworld.com/~reinhold/lunarpolar.html:
Arnold G.Reinhold,”A solar powered station at a lunar pole”.
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