newlasvegas

08-03-2007, 02:42 PM

chào các bạn ! Bạn nào có tài liệu về Pyramid thì cho mình xin ! thank các bạn nhiều ! :D (tiếng Việt càng tốt)

Xem đầy đủ chức năng : Cần tìm tài liệu về Kim tự tháp

newlasvegas

08-03-2007, 02:42 PM

chào các bạn ! Bạn nào có tài liệu về Pyramid thì cho mình xin ! thank các bạn nhiều ! :D (tiếng Việt càng tốt)

Seafoam

09-03-2007, 07:40 PM

Tài liệu bằng tiếng Anh được hoh kưng?:D

The Pyramid

Dedicated to the proposition that the Great Pyramid is a rational (in the mathematical sense) structure, 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' satellite dishes and the BART tunnel in San Francisco).

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.

CLICK HERE--> THE PYRAMID PAPER

If you don't like numbers and math, at least read my summary of insights below

--------------------------------------------------------------------------------

--------------------------------------------------------------------------------

--------------------------------------------------------------------------------

--------------------------------------------------------------------------------

--------------------------------------------------------------------------------

Some people have demanded that I send them a complete drawing (with all the lengths and angles) of what I consider was the original dimensions of the Great Pyramid. There are many interesting things that I would prefer to leave off the web and save for my book. So the pictures below is the only summary of my work I will give, for now. (They are in the new .png format, for faster loading.)

http://www.aloha.net/~hawmtn/nom2.png

To fully understand the above information you will also need the picture below. It shows some subtle details, typical to each side. Because of the equal angles many surfaces are mirror reflections .

http://www.aloha.net/~hawmtn/Cadd.png

and

Some insights since 1991 and plans for the future

(If you came from the Pyramid paper, also check the above pictures)

I am now searching for a publisher to get a book out on my research, 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 Great Pyramid!

http://www.aloha.net/~hawmtn/1$.gif

I now understand the unfinished pyramid with the eye on top on our one dollar bill. I think George Washington and other masons knew that it was a symbol for starting and maintaining a nation, but they did not know it's real mathematical meaning. The "key" for people who have read the above paper is that it is not 479 feet 11 and 15/16 inches but 479 and 191/192 feet (they are equal). Converting 191/192 to Egyptian fractions and then into hieroglyphic give the answer (Hint: 191/192 = 63/64 + 1/(128-32)). Plato seems to have understood the true meaning of all this, (see Plato, Laws 7, 819)

For a taste of how the Egyptians "used" the eye click on image.

The story about the Tower of Babel seem to be about the pharaoh climbing up the unfinished pyramid and supervising some subtle moves on the stones of the king's chamber and thus making the chamber a steep sided truncated pyramid that if raised to its full height would raise miles into the air (he shot the arrow!). But since he didn't use Egyptian fractions in this maneuver when he tried to explain it to the priest, it sounded like babble. To see how steep sided it was, study a triangle with sides of 915, 682 and 610 which I found in the king's chamber. I'm amazed by this triangle, this would make Pythagoras turn over in his grave. If anyone knows of other triangles like this please E-mail me.

I was perplexed by the fact that over 100,000 people worked on the Pyramid, yet the mathematics in the design remained a secret for so long. I now know that they did it with a tall pole mounted near the center of the unfinished Pyramid (see page 132 of Tompkin's book for an illustration). With ropes tied to this pole the regular workers could position stones accurately but have no idea of the math behind it. I believe this is where the May pole festival.started. You can find the size and position of this pole by using Cole's survey and finding the centers of 4 circles. The first drawn through the NW, NE and SE corners, the next through NE, SE and SW corners, the next through SE, SW and NW corners and finally the SW, NW and NE corners. The centers show about a 7 inch pole that the whole pyramid was based on. It is the intersection of four circles that define the corners. It also shows why the Pyramid could not be square. A mysterious line has just (3/97) been found.

The peace sign came from the pyramid (remember Bertrand Russel said it was an ancient symbol). It's a representation of the sun on the south face about 8 miles away.

And the final irony -- it's name betrays it's secret, pyro--middle --- fire in the middle --- the sun reflecting off the highly polished south face (that it was polished is well documented and remember the faces were indented) thus: http://www.aloha.net/~hawmtn/peace.gif

----the world's first solar telescope!---

- http://www.aloha.net/~hawmtn/2sun.gif

(This is my best attempt yet, at rendering just one posible view, using the Pyramid as a telescope.)

Two suns are seen because, according to my model, the south indentation causes one face to reflects one half of a degree (actually 27' 50") differently than the other one. And 1/2 degree is about the same as the diameter of the sun (max 32' 30").

I believe Akanaten started the worship of the sun because of this.

Recent 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)

NEW EXTRA STUFF

I completely revised this page about "An accidental discovery while looking at a star map".

http://www.aloha.net/~hawmtn/man_p.gif

The Pyramid

Dedicated to the proposition that the Great Pyramid is a rational (in the mathematical sense) structure, 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' satellite dishes and the BART tunnel in San Francisco).

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.

CLICK HERE--> THE PYRAMID PAPER

If you don't like numbers and math, at least read my summary of insights below

--------------------------------------------------------------------------------

--------------------------------------------------------------------------------

--------------------------------------------------------------------------------

--------------------------------------------------------------------------------

--------------------------------------------------------------------------------

Some people have demanded that I send them a complete drawing (with all the lengths and angles) of what I consider was the original dimensions of the Great Pyramid. There are many interesting things that I would prefer to leave off the web and save for my book. So the pictures below is the only summary of my work I will give, for now. (They are in the new .png format, for faster loading.)

http://www.aloha.net/~hawmtn/nom2.png

To fully understand the above information you will also need the picture below. It shows some subtle details, typical to each side. Because of the equal angles many surfaces are mirror reflections .

http://www.aloha.net/~hawmtn/Cadd.png

and

Some insights since 1991 and plans for the future

(If you came from the Pyramid paper, also check the above pictures)

I am now searching for a publisher to get a book out on my research, 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 Great Pyramid!

http://www.aloha.net/~hawmtn/1$.gif

I now understand the unfinished pyramid with the eye on top on our one dollar bill. I think George Washington and other masons knew that it was a symbol for starting and maintaining a nation, but they did not know it's real mathematical meaning. The "key" for people who have read the above paper is that it is not 479 feet 11 and 15/16 inches but 479 and 191/192 feet (they are equal). Converting 191/192 to Egyptian fractions and then into hieroglyphic give the answer (Hint: 191/192 = 63/64 + 1/(128-32)). Plato seems to have understood the true meaning of all this, (see Plato, Laws 7, 819)

For a taste of how the Egyptians "used" the eye click on image.

The story about the Tower of Babel seem to be about the pharaoh climbing up the unfinished pyramid and supervising some subtle moves on the stones of the king's chamber and thus making the chamber a steep sided truncated pyramid that if raised to its full height would raise miles into the air (he shot the arrow!). But since he didn't use Egyptian fractions in this maneuver when he tried to explain it to the priest, it sounded like babble. To see how steep sided it was, study a triangle with sides of 915, 682 and 610 which I found in the king's chamber. I'm amazed by this triangle, this would make Pythagoras turn over in his grave. If anyone knows of other triangles like this please E-mail me.

I was perplexed by the fact that over 100,000 people worked on the Pyramid, yet the mathematics in the design remained a secret for so long. I now know that they did it with a tall pole mounted near the center of the unfinished Pyramid (see page 132 of Tompkin's book for an illustration). With ropes tied to this pole the regular workers could position stones accurately but have no idea of the math behind it. I believe this is where the May pole festival.started. You can find the size and position of this pole by using Cole's survey and finding the centers of 4 circles. The first drawn through the NW, NE and SE corners, the next through NE, SE and SW corners, the next through SE, SW and NW corners and finally the SW, NW and NE corners. The centers show about a 7 inch pole that the whole pyramid was based on. It is the intersection of four circles that define the corners. It also shows why the Pyramid could not be square. A mysterious line has just (3/97) been found.

The peace sign came from the pyramid (remember Bertrand Russel said it was an ancient symbol). It's a representation of the sun on the south face about 8 miles away.

And the final irony -- it's name betrays it's secret, pyro--middle --- fire in the middle --- the sun reflecting off the highly polished south face (that it was polished is well documented and remember the faces were indented) thus: http://www.aloha.net/~hawmtn/peace.gif

----the world's first solar telescope!---

- http://www.aloha.net/~hawmtn/2sun.gif

(This is my best attempt yet, at rendering just one posible view, using the Pyramid as a telescope.)

Two suns are seen because, according to my model, the south indentation causes one face to reflects one half of a degree (actually 27' 50") differently than the other one. And 1/2 degree is about the same as the diameter of the sun (max 32' 30").

I believe Akanaten started the worship of the sun because of this.

Recent 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)

NEW EXTRA STUFF

I completely revised this page about "An accidental discovery while looking at a star map".

http://www.aloha.net/~hawmtn/man_p.gif

Seafoam

09-03-2007, 07:51 PM

THE PYRAMID PAPER © 1986-1996 , Terrance G. Nevin

CONTENTS

------This Page

Data

Egyptian fractions

A safe guess

The first set

-------PYR-2nd

The best set

A model height

The inch connection

Unit fractions to normal fraction

Just what did Herodotus say anyway?

-------PYR-3rd

The indent

Done

Areas

A simpler model

Back to insights

Purpose

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.

--------------------------------------------------------------------------------

Table 1 Base Length and their direction

North 230.253 meters 2' 28" South of due West

East 230.391 meters 5' 30" West of due North

South 230.454 meters 1' 57" South of due West

West 230.357 meters 2' 30" West of due North

The 4 corner angles North West North East South East South West

89 59' 58" 90 3' 2" 89 56' 27" 90 0' 33"

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.

Figure 1

http://www.aloha.net/~hawmtn/sim-lin.gif

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 first.

--------------------------------------------------------------------------------

Egyptian fractions

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 examples:

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).

--------------------------------------------------------------------------------

A Safe Guess

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.

Thus:

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.

So:

1.644664 - (1/1 + 1/2) = .1446642, 1 / .1446642 = 6.91255

Thus:

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.

So:

1.644664 - (1 + 1/2 + 1/7) = .001807, invert .001807 to get 553.359

Finally:

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).

Hence:

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.

North

230.253/150 = 1 + 1/2 + 1/29 + 1/1861.361

South

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:

Equation 1

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:

Equation 2

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.

--------------------------------------------------------------------------------

The first set

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)

http://www.aloha.net/~hawmtn/fig3.gif

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:

Equation 3

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).

CONTINUE to Pyr 2nd

updated 6/22/99

It's not really clear ...any more information ....please search more on internet :D

CONTENTS

------This Page

Data

Egyptian fractions

A safe guess

The first set

-------PYR-2nd

The best set

A model height

The inch connection

Unit fractions to normal fraction

Just what did Herodotus say anyway?

-------PYR-3rd

The indent

Done

Areas

A simpler model

Back to insights

Purpose

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.

--------------------------------------------------------------------------------

Table 1 Base Length and their direction

North 230.253 meters 2' 28" South of due West

East 230.391 meters 5' 30" West of due North

South 230.454 meters 1' 57" South of due West

West 230.357 meters 2' 30" West of due North

The 4 corner angles North West North East South East South West

89 59' 58" 90 3' 2" 89 56' 27" 90 0' 33"

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.

Figure 1

http://www.aloha.net/~hawmtn/sim-lin.gif

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 first.

--------------------------------------------------------------------------------

Egyptian fractions

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 examples:

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).

--------------------------------------------------------------------------------

A Safe Guess

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.

Thus:

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.

So:

1.644664 - (1/1 + 1/2) = .1446642, 1 / .1446642 = 6.91255

Thus:

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.

So:

1.644664 - (1 + 1/2 + 1/7) = .001807, invert .001807 to get 553.359

Finally:

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).

Hence:

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.

North

230.253/150 = 1 + 1/2 + 1/29 + 1/1861.361

South

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:

Equation 1

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:

Equation 2

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.

--------------------------------------------------------------------------------

The first set

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)

http://www.aloha.net/~hawmtn/fig3.gif

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:

Equation 3

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).

CONTINUE to Pyr 2nd

updated 6/22/99

It's not really clear ...any more information ....please search more on internet :D

newlasvegas

09-03-2007, 08:44 PM

thank Seafoam nhiều ! new cần mấy cái tài liệu nói về secret inside the pyramids or who bulit the pyramids? something like that ! chứ mấy cái bài toán này chịu thôi ! anyway thank a lot ! :D à chị có ảnh nào đẹp đẹp về The pyramids thì post lên giùm ! :D

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