Numerical Coincidences and Pyramidology

Bob Forrest

I first came across Charles Piazzi Smyth about ten years ago when I read Leonard Cottrell’s book Mountains of Pharaoh.  Then, about four years ago, having more or less forgotten about Smyth completely, I was writing an essay on probability theory and coincidences, and I was looking for something to illustrate the apparent fact that one could find curious coincidences galore simply by going out and looking for them.  It was then that I recalled Smyth’s theory about the Great Pyramid.  He had found curious (numerical) coincidences galore, simply by going out and looking for them. 

I suppose that I still really believe that most pyramid theories are coincidence hunting of varying degrees of ingenuity.  Smyth’s theory was beautifully put together, even if it was based largely on erroneous figures, and Davidson and Aldersmith’s work, it cannot be denied, is a veritable monument to ingenuity in coincidence hunting. 

The numerical coincidence argument has ever been levelled at pyramidologists, and the present contribution is devoted to them. 

The coincidence-hunting theory of pyramidology is borne out by the evidence, side by side, of so many divergent theories about the Great Pyramid – the fact that the same monument can be used to back up theories from Biblical Fundamentalism, via circle squaring and Ancient Mysteries, to Extra-terrestrial theories. 

However, let us concentrate for the moment on looking at numerical coincidence. 

1.  I personally strongly suspect that the occurrence of π in the Great Pyramid is a coincidence.  I therefore reasoned that if π could coincidentally arise, then it ought to be possible to find almost any other universal constant ‘hidden’ in the Great Pyramid, but by accident rather than design.  I settled on trying to discover e, the base of Napierean logarithms, and sure enough I found it, thus:

the relationship being accurate to within 1 part in 1000. 

2.  A purely numerical coincidence here:

3.  Numerical coincidences concerning π: the commonest, of course, is that π is close to 22/7 in value.  Here are two others:

4.  Numerical coincidences concerning e:

5.  The Pyramid of Sahure’s Queen at Abusir has a semi-base-perimeter to height ratio equal to e, a fact originally noticed by Borchardt and put forward as a purely fortuitous relationship. 

6.  The Northerly Pyramid at Bashur has an angle of slope of 43° 36′, according to I.E.S. Edwards Pyramids of Egypt, p. 109.  This would make its semi-base-perimeter to height ratio equal to 4.200, which is equal to the mechanical equivalent of heat in joules/calorie. 

7.  The pyramid of Neferirkare (Edwards ibid. p. 186) has a semi-base-perimeter to height ratio = 3.158, which is very close to √10.  Further,

8.  The Pyramid of Sahure (Edwards p. 185) has base side 257 feet and height 162 feet.  This makes the tangent of the angle of slope to the horizontal of its faces equal to 1.2607, which, to within 1 part in 1000, is the cube root of 2.  Are we therefore to believe that Sahure’s pyramid is a monument to the duplication of the cube – or are we to believe the thing is simply a coincidence?  I believe the latter. 

9.  There are two curious relationships – both ‘coincidental’ – between π and φ (the Golden Mean or Golden Ratio) and they are as follows:

The former relationship has been used as a basis for several methods of approximately squaring the circle, since φ is a constructible quantity, whereas π is not.  Both relationships have been noted by John Ivimy in his book The Sphinx and the Megaliths, p. 121 ff. 

10.  The following is taken from a letter from Henry JamesNot the novelist to the Athenaeum of May 19th 1860:

“The length of a solar year is 365.242 days.  The length of a degree of longitude at the equator, taken from the printed Geodetical tables of the Ordnance Survey, is 365,234 feet; so that if the length of a degree at the equator is divided by the number of days in the year, it will give 1000 feet, or more accurately 999.977 feet, which would give the foot to within the 1/4000th part of an inch, a quantity which cannot be seen. 

Again, the length of a degree of latitude at the central point to the British Islands (which is near Stranraer in Wigtonshire), on the parallel of 54° 55′, which runs near Londonderry, Stranraer, the far-famed Gretna Green and the mouth of the River Tyne, is 365,242 feet; and the length of a degree of latitude, measured on that parallel, divided by the number of days in the year, gives exactly 1000 feet. 

There is no connection between the number of days in a year, and the number of feet in a degree of latitude or longitude; but after the lapse of a few thousand years, the scientific traveller from New Zealand may pay us the same compliment which some of our scientific travellers are now paying the Egyptians, and attribute to scientific refinement which is simply a curious, accidental agreement in the numbers.”

The above numerical coincidences, along with some others, can also be found in the Proceedings of the Royal Society of Edinburgh, Session 1867–68, in an article entitled “On the Great Pyramid of Gizeh, and Professor C.P. Smyth’s Views Concerning it”, by A.D. Wackerbath. 

The Great Pyramid holds a multiplicity of lengths, angles and ratios for the pyramidologist to juggle with.  Indeed, the sceptic might say, history has shown that the Great Pyramid has more than enough such measures to furnish a dozen divergent, and even contradictory, theories.  Any pyramidologist, therefore, must be prepared to demonstrate to the sceptic that his particular theory of the Great Pyramid is founded on more than simple coincidences of numbers.  This, I think, should be one of the purposes of the Folio – to compare the relative merits of the numerical sides to various pyramid theories, and to weigh these against the ‘number juggling’ argument. 

Robert Forrest.  Sept. 1976. 

Note on the occurrence of π in the Great Pyramid: I said in note 1 that I was not convinced that the builders of the Great Pyramid intentionally made its semi-base-perimeter to height ratio equal to π.  To begin with, the Egyptian equivalent of our π, at least according to the Rhind Papyrus, was 3.1605.  Therefore, if the Egyptians had intended to figure π in the Great Pyramid they would have made the semi-base-perimeter to height ratio equal to 3.1605, and NOT 3.142 which is our modern approximation for π, and which value was apparently unknown to the Egyptians. 

It has been said that, for the coffer in the Kings’ Chamber:

The actual value for this ratio, however, appears to be about 3.11, which is a pretty poor estimate for either Egyptian or Modern π. 

Finally, Smyth’s geometric construction for the Pyramid cross section and its passages claims that the angle of slope, to the horizontal, of the pyramid’s passages was intended to be sin⁻¹√π/4 = 26° 18′.  According to Edwards, p. 120, the actual slopes are 26° 31′ 23″ for the descending passage and 26° 2′ 30″ for the ascending passage – figures which do not lend very convincing support to Smyth’s contention.  A much simpler hypothesis than Smyth’s – and one which appears to carry as much if not more weight – is that the corridors were designed to rise 1 unit horizontally to every 2 units vertically – making the intended angle 26° 34′. 

Addendum to numerical coincidences

Continuing the coincidence numbering begun earlier:

11.  Using the Greek letter φ, as in coincidence 9, to represent the Golden Ratio, we have that 1 kilometre = 1/φ miles, or equivalently that φ kilometres = 1 mile.  Both relationships are accurate to about 0.5%. 

Related to the above, we have the common approximate connection between the kilometre and mile: 1 km = 5/8 mile, this being accurate to about 0.6%.  That the relationship between these two units of measurement should be so simply expressible in whole numbers is itself a coincidence.  That both the numbers involved – i.e. 5 and 8 – should be members of the Fibonacci Sequence is another. 

12.  The diameter of the sun in miles, divided by the number of seconds in a mean solar day, equals almost exactly 10 (accuracy about 0.2%). 

13.  De Morgan, in his Budget of Paradoxes, p. 291 ff, analyses the digits in Shanks’s 608 decimal place expansion of π.  The results are as follows:

The most frequent digit is 3, and the least frequent, 7, both numbers holding great significance for the numerologically inclined.  A more curious coincidence is that the digits 3 and 7 play a part in the widely known approximation to π, 3 1/7. 

14.  Martin Gardner in his book More Mathematical Puzzles and Diversions, p. 78, reports that in the first 500 decimal places of the expansion of π occurs the curious sequence of digits 177111777. 

15.  This coincidence is well-known, and despite attempts by astronomers to explain it causally, it is still regarded by most, I think, as fortuitous.  It is Bode’s Law: Start with the geometric series 0, 3, 6, 12, etc, add 4 to each term, divide by 10, and you have the distances, in Astronomical Units, of each of the planets of the solar system with tolerable accuracy (except Pluto):