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This presentation offers powerful tabular evidence that the principal architect of the Great pyramid of Giza, in the era of the Old Kingdom of Egypt, chose the 'foot' as the primary unit of linear measure for the design, then retained in secret the mathematical system that was used to define the unit of measure for the pyramid. This secret is uncovered and disclosed in this paper by the author, and the chosen unit is found to be dimensionally the same as the 'foot' defined by the British Imperial system of units.
A mathematical process termed number distillation, which has been referred to in the distant past as “the casting out of nines”, is an essential part of the self-limiting, sequential, single-digit per cell tables that are developed here. The column and row sums, as well as specific numerically marked areas whose digits are cubed and summed provide overwhelming evidence that the term 'foot' is the only possible candidate for the primary pyramid design unit.
There is no evidence as to just when the unit was adopted for use in the Old Kingdom era, but the sequentially based, tabular method introduced here is mathematically and geometrically irrefutable.  It provides a rational, non-circular   argument that even hints at the proposition that the pyramid designer may have developed and been the first ever to use this particular unit of the 'foot'.

Introduction

My earlier research into Fibonacci’s mathematical series led to a method of number tabularization that would provide a rich source of what were called canonical numbers by ancient philosophers. For some mysterious reason some of the major tabular summations were almost identical to external measurements of the Great pyramid of Egypt as had been recorded by W.M. Flinders Petrie1 .
The tables2 however provided no units for these special numbers, which only caused one to assume it was self-evident, that to have any meaning, the unit for such numbers had to be one of those that would be found in the British Imperial system of units, e.g. (mile, foot. inch, etc.)   In other words, no correlation effects were apparent if summation units were assumed to be meters or cubits.
A numerical reduction process termed distillation3 by the author was used to   reduce the first 24 terms of the Fibonacci series to single digits, and it was this reduction to single digits that was responsible for the self-limiting aspect of the tables. The distillation value (dv) of the Fibonacci terms beyond the first 24 terms only duplicated the original string of distilled value digits. Distillation was a shortcut term for what had been called “the casting out of nines” in the past.

 

What is to be presented now is a similar technique of tabular construction that involves using only the sequential digits (1– 9) as the cell occupants of the first row of a table rather than the first 24 terms of the Fibonacci series.  The cells of each row thereafter will contain a single digit that is the distilled value (dv) of the product of the row number by its respective top row factor.   

e.g.     Cell_R,C  digit  =  dv(R_n× C_n)   Þ   Cell_3,4 # = 3×4 = 12 Þ 3

                Cell_R,C  digit  =  dv(R_n×C_n)   Þ   Cell_4,6 # = 4×6  = 24 Þ 6     etc.

Tables containing nine columns and eight rows will be presented in the discussion that follows. Rows and columns greater than nine would simply be repeating the original columns and rows, therefore the tables are said to be self-limiting.

 

Inherent Design Ratios & Measures of an Unknown Pyramid

From examination of Table 10-A it will be noted that column and row sums are shown in the white outside border cells, and the different grayscale to white scale inside the table pattern provide important numbers that we now examine.                                 

                           Eight Rows of Distilled Value Digits (1- 9)

Table 10-A

 

 

Note:

• First, imagine that the four 9’s cornering the dark gray rectangle in the center of the 64 cell (8x8) section represent the cornerstones of some unknown pyramid.

Table 10-B

 

 

• Next, imagine that the four white sections shown in Table 10-B with the digits (6,8,1,3) on each side of the pyramid are cubed and summed. The pyramid will then have a perimeter value of (4 x 756) = 3024 units.

• Do these numbers make it seem self-evident that we may be describing the Great Pyramid of Giza?

     Lets proceed on and examine the white cells in the central area of Table 10-C.

Table 10-C

• The twelve white cells form a double-cross of the digits (3,7,2,6).    

If the digits are cubed and summed, the total will be 4 x 594 = 2376.

     (Notice the sum number 2376 has the same digits as the double-cross).

 

• A circle with a 2376 unit perimeter will fit exactly into a square with a    width of our pyramid. This fact provides us with the definition of Pi that some have termed as Pyramid Pi.   i.e.        p_p = 2376 / 756 = 22/7

• The ratio of the perimeter of the square, to the perimeter of the circle,  was probably one of the most valuable tools in the ancient architects  toolbox, and as you may see, it applies to many things in nature.   i.e.

      Ratio = 3024 units / 2376 units = 14/11  =  4/p_p = 1.272727

 

•  Peter Tompkins4 discusses the facts brought out in Tons Brune’s   writings about the Egyptian people’s strong belief in the sacredness of the circle, the square and the cross.   These are the same three subjects we discussed here as we defined the ‘secret method’ of numerical extraction  from  Tables 10-A, B & C

  

• Height of the unknown pyramid  =  (14/11)×(Total sum of  Table–10A)

          “        “          “             =  (14/11)×(378 units) = 481.1 units

 

• Pyramid slope = invtan (378/297) = 51.8428° @ 51°  51'

 

It is the author’s contention that he has now extracted sufficient numerical evidence from Table-10A to say for certain that the unknown pyramid has the same exterior dimensions as the Great Pyramid of Giza, as measured in British Imperial feet, and that a similar table had to have been used by the architect to secretly select the unit he would use for his design of the pyramid. Therefore we can say with some certainty now that the “foot” was born of the Great pyramid and thus became the primary pyramid unit.   The Royal cubit may have been created as a construction unit, for it would have been an easier unit for the builders to work with. e.g.

• GP design base half-width =   378 feet  =    220 RC
• GP design base width        =    756 feet  =   440 RC
• GP design height               =  481.1 feet =   280 RC
• GP design base perimeter  = 3024 feet  = 1760 RC

Other Ratios Extracted from Table 10-A

 

The useful approximation for the value of the Golden Ratio found in our pyramid extractions differs slightly from the precise value defined in the realm of pure mathematics, but it does appear in nature and was shown above joined with the pyramid Pi ratio 22/7. It has been termed Phi-sub-one (j₁) by this author for convenience.

• Golden Ratio₁ = j₁ = (from the tabular values) = (3024/2376 )² = (14/11)² =

     (4/p_p)² = 1.619835   (% error 0.11)         Pure math value Phi j = 1.618034 

                                                      

• Golden Ratio₁  (using Earth-Moon data) = ((Earth + Moon) _Radii / Earth _Radius))²  

      ((3960 +1080) mi / 3960 mi))²  =  (14/11)² = 1.619835
      Sq.rt. Golden Ratio₁ = Öj₁ = 4/p_p = 14/11 = 1.272727      

 

•  Earth: Moon scale factor =  (using Table 10-A column sums) =    (72+117+108)/(81) = 297/81 = 11/3 = 3.666

Conclusion 

 

 As the author of this work I am aware of the possible extent of its significance and I wonder in awe as to what impact it may have on the historical record. I feel as though I have found the true mathematical source of “reasoning” for a unit of measure that has come to be known as the English ‘foot’. This chosen unit is found to be dimensionally the same as this ‘foot’ defined by the British Parliament in 1592 that was used by W.M. Flinders Petrie5 in his research for his now famous book The Pyramids and Temples of Gizeh.   The source of the architect’s secret has been revealed but the time of its discovery and first application is still uncertain.

 

1   The Pyramids and Temples of Gizeh - Chapter 6-Section 21 – Published London 1883

4 Author of ‘Secrets of the Great Pyramid’ – ISBN:0-88365-957-3 – p.261 – discusses author Tons Brune’s book, ‘The Secrets of Ancient Geometry’, and his belief in how the Egyptian people held strong feelings for the sacred value of the circle, square, and the cross.         

5   The Pyramids and Temples of Gizeh – W.M.Flinders Petrie 1883 

©2005 Joseph Turbeville 

 

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