The results are graphed below. The vertical axis is error of fit and the horizontal axis is height.
Graph 1 shows that there are indeed some sets that have a close fit with the data. The first unit fraction of the four best sets are shown on the graph.
But there is only one set that has the best fit. This is the one that I'll investigate in detail.
The best set
|Base||Unit fraction||Decimal equiv.|
This set has some very interesting features that I will get into shortly
but first some words on modeling.
Now I come to the question of the height of the Pyramid. When I made the first "set" of four unit fractions an important assumption was that the four base lengths are proportional to the four unit fractions. Because of this, the sets themselves can be considered as part of the specifications of a model.
The best set of unit fractions from graph 1 can thus be thought of as actual base lengths on a pyramid model. The height of this new model is one. This can be shown by rearranging equation 2 to solve for height:
height = base_a / UFR_a
But since the unit fractions are considered as actual baselengths, UFR_a would equal base_a length so:
height = UFR_a / UFR_a = 1
So from here on when the model is mentioned it is assumed that the model is one high. One what? The interesting answer is one unit of measurement. Interesting because it suggests the possibility that the basic unit of measurement of the Pyramid is the height itself!
The Pyramid's actual height can also be found by using equation 4 and
the data. But the following summary shows that they do not give just one
|base_a||/ UFR_a||= height||mm diff.|
For now the height of the Pyramid will be considered to be 146.3024 meter and be abbreviated hP
The small differences in the height comes from the survey data. The data has only three decimal digit accuracy, while the unit fractions values are known exactly.
This situation means that the data that lead me to this best set can
be left for a while and a new set of model data can be used. But first
a little about inches.
A very interesting thing happens when the heights from above are changed from meters to feet and inches. The old conversion formula is:
Length in Meters / .30479974 = Length in Feet
First the average height:
146.302417/.30479974 = 479.9952159
= 479 feet 11 and 15.08/16 inches
Then the rest of them:
|North||479 feet 11 and 15.09/16 inches|
|East||479 feet 11 and 14.89/16 inches|
|South||479 feet 11 and 15.15/16 inches|
|West||479 feet 11 and 15.18/16 inches|
I did not believe this result at first. I ended up rechecking the calculations and trying three other variations of the "error of fit" equation (equation 3). For me in the end, it came down to one of the following two conclusions:
1. The data from table one is a hoax perpetrated by J.H. Cole or
2. The Egyptian used feet and inches.
On a trip to Washington D.C. I went to the Congressional Library to try and investigate both of the above. All I could find about J.H. Cole's survey was that he was criticized because his data on the north side mark (115.090 + 115.161) did not match the data for the north base (230.253).
The Egyptians used normal fraction also (2/3,8/7 etc.). But when ever they had to do a calculation with a normal fraction they would change it to a unit fraction first. There turn out to be a simple formula to change the unit fractions (from the best set) to normal fractions (or vice versa).
I'll start by converting the unit fraction 1+1/2+1/14+1/X to a normal fraction.
1+1/2+1/14+1/X = 11/7+ 1/X = [7+(11*X)]/7X
hence the unit fraction is equal the normal fraction:
UFR_a = [7 + (11*X)] / 7*X
This means that each of the denominators of the fourth term in the unit
fractions of the "best set" can be substituted in turn for X in
equation 4 to find the normal fractions that equals it. These calculations
are summarized in Table 3
|unit fraction||calculation||normal fraction|
|1+1/2+1/14+1/419 =||(7+11*419)/(7*419) =||
|1+1/2+1/14+1/323 =||(7+11*323)/(7*323) =||
Before I go further I will explore and expand on a relationship first
put forward in English by John Taylor (1781-1864). He said he got it from
reading Herodotus (484-425 B.C.) a Greek historian who visited the Pyramid
about 440 B.C.
He said something roughly like this 'The faces rise as ... peak ... two'. Taylor interpreted this as, the area of a face of the Pyramid equals the square of the height of the Pyramid (some Egyptologist and math historians say Herodotus never said anything that would translate to anything near what Taylor said). I reserve judgment. Putting it into equation form:
Area face=height ^2
A Pyramid face is a triangle so its area is:
Areaface=1/2 * base * altitude
The altitude of a face is called the apothem. Combine the two equations to eliminate area:
height^2 = 1/2 * base * apothem
But I am working with the model dimensions now so the height is equal one and the four base lengths are the fractions from table 3. So:
1^2 = 1 = 1/2 * base * apothem
Solving for apothem:
apothem = 2 / base
Using the fractions from table 3 and equation 7, I found the four apothems.
|North ap.||East ap.||South ap.||West ap.|
Furthermore, the South normal fraction can be reduced
2933/1862 = (7 * 419)/(7 * 266) = 419/266
UFR_north = 1 + 1/2 + 1/14 + 1/419
UFR_south = 1 + 1/2 + 1/14 + 1/266
This ties the South and North unit fractions together. With the North being derived from the South (this matches how the nation of Egypt was set up at the time of the pyramid).
So here is what it comes down to. Using cole's data and my error of fit analysis produces some very pleasing results (for me). But Taylor's height squared relationship really seems to tie everything together. I believe J.H. Cole could easily have fudged his survey to make the 480 foot connection. But he could not have simultaneously made the mathematics so interesting.
Now back to the task of making a mathematical model. Unfortunately even knowing mathematical expressions for the bases, apothems and the exact height does not give me enough information to even start constructing a model.
One critical piece of information is missing. And that is, what point on the area of the base is the peak of the Pyramid directly over? Until this is known with the same exactness as the bases and the height are now known, no model can be built.