The purpose of this paper is to construct a rational, mathematical three dimensional model of the exterior of the great Pyramid at Giza. The starting data is from a 1925 survey first proposed by Ludwig Borchart of the German Institute of Archeology in Cairo, and conducted by J.H.Cole (The first modern surveyor to un-cover the cornerstones).
The data is summarized below.
|North West||North East||South East||
|The chiseled mark at the bottom of the North face|
|115.090m from North West corner and 115.161m from the North East corner|
A major goal now is to find integer ratios associated with the data. The first question is, ratios between what?
Figure 1 shows some of the line lengths to consider. There are four
base lengths, two base diagonals, four face edges, but only one height.
An obvious choice then is to look for ratios between the height and the other line lengths.
But the actual height has been lost, and the number of ratios involved
could be quite large. There is a way to solve both of these problems at
the same time, but a little background in Egyptian fractions is necessary
The early Egyptians had a different way of handling fractions and ratios.
Whenever a fraction was involved in a math problem, they would convert
it into a sum of fractions, all with a numerator of one. Here are some
|2/7 = 1/4 + 1/28|
|2/97 = 1/56 + 1/679 + 1/776|
|2/99 = 1/66 + 1/198|
|23/40 = 1/2 + 1/14 + 1/280|
The first three are from documents written about 1700 B.C. These are conversion examples starting with the integer fractions. I'll take another approach, using the survey data and guesses about the height to form ratios, then converting the ratios to the nearest equivalent Egyptian fraction.
The procedure used for converting is essentially the same as the one
provided by J.J. Sylvester, a British mathematician (1814-1897).
Instead of one guess for the height of the Pyramid, I guess a safe low height of 140 meters (much too low!) and a high of 150 meters (much too high!). The actual height is most definitely somewhere within this range.
With that out of the way, I can form the first ratio. It is between the North base (230.253m) and the low guess for the height (140m), putting the height in the denominator.
230.253/140 = 1.644664
This decimal value will now be converted to a four term Egyptian fraction. The first term is obviously 1/1 or 1. The best choice for the second term is 1/2.
The third terms denominator ( the numerator is one, of course ) is found by adding the first two terms, then subtracting that sum from the original decimal value. Then put one over ( invert ) the value found to get the third terms denominator.
1.644664 - (1/1 + 1/2) = .1446642, 1 / .1446642 = 6.91255
230.253/140 = 1 + 1/2 + 1/6.91255
To get the fourth term, take the integer part of the denominator of the third term, add one to it to get 7 (note that this is different from rounding up). Seven becomes the new denominator of the third term. I then add the three terms and proceed as before.
1.644664 - (1 + 1/2 + 1/7) = .001807, invert .001807 to get 553.359
230.253/140 = 1 + 1/2 + 1/7 + 1/533.359
The numbers after the decimal point in the denominator of the fourth term have to be there, of course, to keep both sides of the expression equal. They also serve as a fit indicator. They give a way to indicate how well the particular Egyptian fraction fit the ratio that is being tried.
For now I will look at other ratios without going through the full procedure. The first is the South base (230.454m).
230.454/140 = 1 + 1/2 + 1/7 + 1/308.370
This fraction shows why four terms are needed for the data being used. If only three terms were used there would be no difference in the fractions for the South base (longest side) and the North base (shortest side).
Next I check the ratios of the North and South bases with the 150m height.
230.253/150 = 1 + 1/2 + 1/29 + 1/1861.361
230.454/150 = 1 + 1/2 + 1/29 + 1/532.697
So now I have tried both height guesses and have checked the longest and shortest base lengths. This gives enough information to make a general statement.
If there is a set of four Egyptian fractions that equal ratios between respective base lengths and the height, they follow this pattern:
base length/height = 1 + 1/2 + 1/X + 1/Y
With X between 7 and 29
Y would be different for each base length
but X would be the same for all four base lengths.
Equation 1 is true, of course, if we allow Y to be a decimal expression (as was shown above) but the right side of equation 1 is an Egyptian fraction. So Y is certainly an integer. From now on I'll only deal with ratios that convert to integer Egyptian fractions. I'll also start calling Egyptian fractions by their more modern name, unit fractions, and use the abbreviation UFR for them. So equation 1 becomes by some simple rearrangements:
UFR = base / height
So back to the task at hand. That task is to find a set of unit fractions that best fit the data. By set I mean four different unit fraction. Each would have the same first three term. But the fourth term would be different for each.
One way to approach it is to try them all! Try all unit fractions from:
1 + 1/2 + 1/7 + 1/? to 1 + 1/2 + 1/29 + 1/?
With the procedure I've been using this amounts to over 40,000 unit fractions to check. Actually it would be four times that number, because I am looking for a set of four to fit the four base lengths.
The numbers are so large because of the 140 to 150 meters guess done earlier. It's time for a more realistic approximation for the height.
That range will now be changed to 145 to 148 meters ( still a safe guess ). The associated range of unit fractions that results is:
1 + 1/2 + 1/13 + 1/100 to 1 + 1/2 + 1/16 + 1/1000
I won't go into reasons behind the choice for the fourth terms. For now accept that the values 1/100 to 1/1000 for the fourth term are appropriate for our data and these unit fractions.
Now I'll make the first "set" of unit fractions. Beginning with:
UFR_north = 1 + 1/2 + 1/13 + 1/100 = 1.586923
The important starting assumption is that UFR_north represents the length of the of the north base. This lets me find UFR_east by the using following proportion:
230.391 (east base) / 230.253 (north base) = UFR_east / UFR_north
Then solving for UFR_east gives:
UFR_east = UFR_north x 230.391/230.253
Changing to decimal:
UFR_east = 1.586923 x 230.391/230.253
UFR_east = 1.587874
Finally convert this value to the nearest unit faction.
UFR_east = 1 + 1/2 + 1/13 + 1/91
For UFR_south I set up a proportion as before:
230.454 / 230.253 = UFR_south / UFR_north
Solve for UFR_south and convert to nearest unit fraction:
UFR_south = 1 + 1/2 +1/13 + 1/88
The west is figured in a similar way. The full set is summarized below.
|Base||unit fraction||decimal equiv.|
|UFR_north||1 + 1/2 + 1/13 + 1/100||1.586923|
|UFR_east||1 + 1/2 + 1/13 + 1/91||1.587912|
|UFR_south||1 + 1/2 + 1/13 + 1/88||1.588286|
|UFR_west||1 + 1/2 + 1/13 + 1/93||1.587675|
This is the first set. The next set starts with UFR_north = 1 + 1/2 + 1/13 + 1/101 The one after that is 1 + 1/2 + 1/13 + 1/102 and so on until 1 + 1/2 + 1/13 + 1/1000. Then do the same for the unit fractions containing 1/14,1/15 and 1/16 .
The thing to find now is a way to rank the sets according to how well they fit the data. The corner angles in the data from Table 1 can help here. I begin by splitting the base into two triangles. (see fig. 3)
The values of UFR_north UFR_east UFR_south and UFR_west are taken from the set (from above) and combined with the corner angles in trig formulas to find the values of Diag.1 and Diag.2 .
Example using trig. cosign law to find diagonal:
If the triangle's sides were originally laid out with the this unit fraction set and these corner angles, then Diag.1 would equal Diag.2.
In fact, the quotient between Diag.1 and Diag.2 (it should equal 1) is what can be used for ranking. The sets whose quotient is closer to one is ranked higher than those whose quotient are further from one.
But there are two more triangles that can be made. Just split the base in the other direction (see fig.) and use the other two corner angles to find two new diagonals Diag.3 and Diag.4
So if I compute all four diagonals and combine them right, I can come up with an "error of fit" value for the set. Error of fit means that if the set of unit fractions perfectly fit the data then the "error of fit" value would equal zero. Anything less than perfect fit would give a value other than zero.
One way to figure it is to use the following equation:
In words, subtract Diag.3 divided by Diag.4 from Diag.1 divided by Diag.2, take the absolute value and then the log of this difference. This value is called the "error of fit" for the set of unit fractions. Note that because of the log function, the set with the largest negative value would indicate the best fit, -5 would indicate a better fit than -4.
So the calculations; (1) figuring the set of unit fractions, (2) the four diagonals and (3) the error of fit, were entered into a computer spreadsheet (Lotus 123) in 3600 rows (one row for each set).