Dedicated to the proposition that the Great Pyramid
is a rational structure (in the mathematical sense),
designed and built by normal people.
This is a radical statement about the Pyramid, especially on the internet because all web pages that I have been able to find that deal with the Pyramid, maintain that it was built and/or inspired by either God or space aliens. Most don't even consider that it could be a rational structure designed and built by normal people.Background
In 1985 I retired from the Canada France Hawaii Telescope Corporation. I had worked at the Mauna Kea observatory doing troubleshooting, programming and instrument repair. I had worked on big science project.my whole career ( three telescope on Mauna Kea, secret Air Force radars, 100' COMSAT satellite dishes, the BART tunnel in San Francisco and AT&T in Oakland).
I became bored and restless in retirement. I needed a big project to be involved in. I happened on Peter Tompkins "Secrets Of The Great Pyramid" one day and I was hooked. I spent four frustrating years figuring away in Basic and Lotus 123 spread sheets on an IBM XT computer. I would fool myself that I was making progress in figuring out the mathematics behind the Pyramid but I never had any thing that I could prove.
The break through finally came after I was able to spend some time at the Library of Congress in the History of.Mathematics section. This is where I began to learn about the ways the ancient Egyptian did mathematics. This knowledge and computer spread sheets finally gave me results that I could prove (to my satisfaction at least) and others could use my methods to verify my result.
I was asked by a friendly professor in 1991 to deliver a paper on my work to his descriptive geometry class at College of the Redwoods in northern California. I wrote up my paper using Ventura Publishing and delivered it, and got an enthusiastic response from the students.
At the urging of other I have finally translated the paper from Ventura to HTML. Some of the illustrations got a little fuzzy in the translation and I will correct that in the future. The main point I want to make is,in the five years since I wrote the paper I have been able to substantiate all the results, on much more powerful computers and software (especially Mathcad 4.0)
You could read the paper and just accept my numbers or take a calculator and a blank spread sheet to verify as you go along.
(2009 update) Its been about ten years since I put this picture on the web. All the lengths, I got the hard way, I had to calculate each one. At first, I was reluctant to just give them away. But I remembered how hard it was to get any good info on lengths when I started all this.
The picture and precision was my compromise. I well only give picture of the numbers, and not a table. A table could just be cut and pasted into someone elses work. And precision ... I now know the Pyramid is a rational structure, so the lengths can be determined exactly! When I started, all I had was the Cole survey, which measured to the nearest milimeter. To check the survey you would need lengths to the nearest tenth of a milimeter, so that is how precise the lengths are in the picture.
This picture is now all over the web, but mostly in university math, engineering and architecture, web sites. I was delighted to even find it at three sites in Egypt. Most don't give me credit, but that is what I expected. A spanish mathematician took the lengths and loaded them into hisTurboCad program and correctly drived all the angles, areas and volumns. But what pleased me the most, was that he saw thru the subtleties of the indented faces and found the Cole survey numbers. You can see it , half way down his web page at http://jamrb.galeon.com/oro_pi.htm
Click here for a way, to use the picture, to build an ultra simple 6 foot pyranid.
Most of the email I get is about someone wanting to do there own physical model, with paper, cardboard, sticks, or even sugar cubes. Save the picture at right and print it out, as big as you can, then cut out and fold as indicated. (the approx. scale of 1,844 is right, if you print "Fit to page" for 8 1/2 x 11 inch paper)
You can also use the X,Y table to scale the illustration to any size you want.
Speaking of models, I rarely used lengths in my math work on the Pyramid model. Rather I used X,Y points for calculations. So here are the points if you would ever like to lay-out a full size model of the Great Pyramid.
Start by making sure the positive Y axis points due North. Probably the hardest part would be to find an, absolutely flat, square area of 14 acres.
The volume of the Pyramid as accurately as my program will give me is 2,574,464.31314276 cubic meters. That is equivalent to a cube with sides of 137.055234 meters. The picture shows such a cube superimposed on the Pyramid.
If you accept an old Egyptian length called a Pyk Belady (pkb) as .5774799 meters, then the cube’s sides are 237 1/3 pkb in length. I believe that the ancient quarrymen knew this number so that they knew just how much rock they needed to cut.
All the 64 angles of the Pyramid base to full program accuracy.
(in the picture the first number is the row the second is the column)
Note that column 0 consists of equal pairs. These angles have to be equal so that the left and right slopes of the sides are equal.
Something that surprised me was that opposite side angles are almost also equal.
This is a so called imaginary work sheet most like how I did Pyramid math. It uses the complex number “i”. I was using x and y coordinates when I started but I eventually learned that x=51 y=84 and 51,84 and 51+84i are all saying (for my purpose) the same thing. And my math program did not care. The only thing wrong, was when I tell anyone I am using imaginary numbers it usually scares them away.
This is the set of pictures that is used to explore the indent, that each side has.
Fig. 3 little arrows, show that the indent could be positioned almost anywhere (and still fit the Model).
First I looked at four possibilities.
Fig. 7 was chosen because, having the left and right slopes equal, it would make each side much easier to build.
insights since 1991 and plans for the
I am now searching for a publisher to get a book out on my research (still no book), so I will not give detailed explanation of any of the insights. But there is enough information here and in the paper for anybody into mathematics to fully reconstruct my work. Anyone with math questions e-mail me. I will answer promptly. Other questions will be answered slowly.
The millennium has
come. It's about time
we understood the
|The translation goes something
There was this unfinished pyramid whose height was made to be 479 times the length of Kufu's foot, high. Then 63/64 of a Kufu foot plus 1/96 of a Kufu foot was added to the top, to finish it.
I found this when I converted 191/192 to egyptian fractions. The conversion is, 191/192=1/2+1/4+1/8+1/16+1/32+1/64+1/96 . I realized that no scribe would write out 1/2+1/4+1/8+1/16+1/32+1/64, rather he would just draw the Horus Eye instead.
findings on the cubit
Information on the Royal Cubit I found in my work (length 523.33mm)
There is an interesting relationship between the Royal Cubit and the foot. I found that 7 Royal Cubits is essentialy equal to 12 feet. (it is actually 12 feet plus .2238 inches )
So here is what it comes down to. I put forth proof that:
Four 'unit fraction expressions' match the surveyed lengths and angles better than the 14,400 others I have tied. I then use those 4 expressions to develop a mathematical model that has some very interesting features. This is my best guess at what the original design of the Great Pyramid was.
What I expected when I put it all on the web was:
1. A debate would start
2. Others would come forward with other mathematical models.
3. Eventually every body would agree on what the true model of the Great Pyramid was.
All I ever wanted was to be part of that debate. But what I have found, is that most of the people who even know what a mathematical model is, have an aversion to anything about the Great Pyramid.
So I am in the perplexing position of saying, I have the best model for the Great Pyramid in the world, just because nobody has come forward with another. (another that is, that matches Cole's survey)