Made by Fleur de Roos, Ilse De Waele, Vanessa Heyndrickx
September 1998

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```FOREWORD

A. ANCIENT MATHEMATICS : EGYPT

1. Introduction
2. History
3. Approach
4. Number system and symbols
5. Operations
a. The Rhind papyrus
b. Fractions
c. Area
d. False position
6. The construction of a pyramid

B. RUINS AND PYRAMIDS : CONSTRUCTED BY MARTIANS ?

1. Introduction
2. Formations and structures
a. Inca city
b. Elysium
c. Margartifer Sinus
d. Sharonov
e. Cydonia
- The Fort
- Main Pyramid & Pyramid Town
- D&M Pyramid
3. Angle of construction
4. Location
5. Mathematical interpretations
a. Circle segments and curious numbers
b. The sarcophagus
c. Dimensions of the Cheops pyramid in reference to Mars
d. The number 27

CONCLUSION

BIBLIOGRAPHY```

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• Foreword.

We want to show you our work "Egypt
mathematics and the pyramids" in a funny,
and of course, instructive way.
The old Egypt is discussed historically and
mathematically.
will end with a lot of enclosures such as
the 3 pyramids from Gizeh, the Rosetta stone, etc.
We have done the best we could and we are
hoping that you will enjoy it !!!

Have fun,
Ilse, Fleur and Vanessa.

• A. ANCIENT MATHEMATICS : EGYPT

• 1. Introduction

We choose for this subject because the way mathematics was done in ancient times is very interesting. They succeeded to determine everything quite sure without any help from a computer, unlike we do today. They just used their mind, and that is what appealed to us.

2. History

Ancient Egypt was governed fairly peacefully and uninterruptedly by a succession of dynasties. Being essentially a theocracy it was ruled by rich and powerful bureaucrats hand-in-glove with the temple priests. Most of the manual labour was done by a large slave class which built the temples and pyramids in Egypt. There are many structures in Egypt that involved engineering problems : the Colossi of Rameses II at Abu Simbel, the Great Sphinx standing near the Great Pyramid at Gizeh, the Temple of Amon Re at Karnak. Rameses II completed the Great Hall of the temple in the 1200s B.C. ; with columns over 78 feet tall, it was the largest columned hall ever built by man. Basic surveying and engineering practices, with their concomitant mathematics, were created to assist in the design and construction of these works.

Following is a chronological list of events in the history of mathematics in ancient Egypt.

3100 B.C. In a museum at Oxford you can find a royal Egyptian mace dating from this time. On the mace are several numbers in the millions and hundred thousands, written in Egyptian hieroglyphics, recording exaggerated results of a successful military campaign.

2600 B.C. The Great Pyramid at Gizeh was erected about this time and undoubtedly involved some mathematical and engineering problems. The structure covers 13 acres and contains over 2 000 000 blocks of stone, averaging 2.5 tons in weight, carefully fitted together. Some chamber roofs were made of 54 ton granite blocks, 27 feet long and 4 feet thick, set 200 feet above the ground. The engineering skill implied by these impressive statistics is considerably diminished when we realise that the task was done by an army of 100 000 workers working for a period of 30 years.

The Great Pyramid is the largest of 3 sitting on the desert at Gizeh, a little south of present-day Cairo. Pyramids were built as royal tombs, since the Egyptians believed in an afterlife that depended upon the preservation of the body.

The Great Pyramid, originally 481 feet high, was built to house the body of Pharaoh Khufu , the other 2 were built for Khafre and Menkaure ( respectively Cheops, Chepren and Mycerinus ). There are about 80 pyramids still standing. The Great Pyramid became known as one of the Seven Wonders of the Ancient World (which are, by the way, the Great Pyramid of Egypt, the Hanging Gardens of Babylon, the Statue of Zeus at Olympia, the Temple of Diana at Ephesus, the Mausoleum at Halicarnassus, the Colossus of Rhodes, and the Pharos Lighthouse at Alexandria) and of these Seven Wonders, only the Great Pyramid is still standing.

1850 B.C. This is the approximate date of the Moscow papyrus, a mathematical text containing twenty-five problems. The oldest extant astronomical instrument, a combination plumb line and sight rod, dates from this time and is preserved in the Berlin Museum.

1650 B.C. This is the approximate date of the Rhind papyrus, a mathematical text in the form of a practical handbook, which contains 87 problems.(see further)

1500 B.C. The largest existing obelisk was quarried about 1500 B.C. It was erected before the Temple of the Sun at Thebes. It is 105 feet long, has a square base of 10 feet to the side and weighs 430 tons.

1500 B.C. The Berlin Museum has an Egyptian sundial dating from this time. It is the oldest sundial extant.

1350 B.C. The Rollin papyrus, preserved in the Louvre, contains elaborate bread accounts showing the practical use of large numbers at the time.

1167 B.C. The time of the Harris papyrus. It was prepared by Rameses IV when he ascended the throne. It lists the ample wealth of that time and gives us the best example of practical accounting in ancient Egypt.

 Egypt was long the richest field for ancient historical research. The reasons for this lies in the veneration that the Egyptians had for their dead and in the unusually dry climate of the region. The former led to the erection of long-lasting tombs and temples with richly inscribed walls, the latter preserved papyri and objects that would otherwise have perished. The ability to read Egyptian hieroglyphics resulted from the successful decipherment by Jean François Champollion of inscriptions of the Rosetta Stone. The stone measures 3 feet 7 inches by 2 feet 6 inches, and the inscriptions on it give a common message in Egyptian hieroglyphic, Egyptian demotic and Greek. It was engraved in 196 B.C., and as part of the treaty of capitulation when the French surrendered to the British, it went to England, to the British Museum.

3. Approach

Egyptian mathematics was dominated by arithmetic, with an emphasis on measurement and calculation in geometry. Unlike the Greeks who thought abstractly about mathematical ideas, the Egyptians were only concerned with practical applications. In Egyptian maths we don't find any trace of later concepts such as axioms or proofs.

They used their knowledge to solve practical problems. They calculated the correct areas of triangles, rectangles, and trapezoids and the volumes of bricks, cylinders and of course pyramids.

Theory was not really involved in their maths, when it does appear, it seems that theory was used as a way to aid technique, not understanding. All their areas of mathematics were centred around addition. There were no plus, minus, multiplication, or division signs. No square root signs, zeros, or decimal points were used. If they wanted to indicate a sign, they had to write it out in hieroglyphics. Using the system of sums of unit fractions, supplemented by the fraction 2/3, the Egyptians were able to solve all problems of arithmetic that involved fractions, as well as some elementary problems in algebra. Although they didn't have mathematical notation, there twice times table and two thirds table enabled them to do a significant amount of mathematical problems.

4. Number system and symbols

 The earliest Egyptian texts, composed about 1800 B.C., reveal a decimal number system with separate symbols for the successive powers of 10 ( 1, 10, 100 and so forth ), just as in the system used by the Romans. Numbers were represented by writing down the symbol for 1, 10, 100, and so on as many times as the unit was in a given number.The hieroglyphic system of writing was a pictorial script where each character represented an object. Egyptian arithmetic lacks a sign for zero and has no place-system.

 The symbol for 1 was written 4 times to represent the number 4 and evenso the symbol for 10 was written 4 times to represent the number 40. Together with 2 times the symbol for 100 and 3 times the symbol for 1000 the number 3244 was represented. Any number could be written using the right symbols additively as described above. The symbols for plus and minus were represented by a pair of legs walking from left to right, or from right to left.

5. Operations

a. The Rhind papyrus

•  The Rhind Papyrus is named after the Scottish Egyptologist A. Henry Rhind, who purchased it in Luxor in 1858. The papyrus, a scroll about 6 metres long and 1/3 of a metre wide, was written around 1650 BC by the scribe Ahmes who was copying a document which is 200 years older. When it arrived at the British Museum, it was shorter and in 2 pieces with a central part missing. The eighty-seven problems on the Rhind deal with a large variety of subjects including methods of multiplying and dividing, the use of unit fractions, simple equations, the use of false position, calculations of areas and volumes, progressions and many other applications. When describing the Rhind, it is important to note that is not merely a collection of maths problems or even an ancient maths book with explanations, general information and tables. It encompasses both of these but with its own added style. The Egyptians' use of exclusively unit fractions (fractions with 1 in the numerator) was one of the document's trade marks, and this necessitated a table in the first section of 2 divided by odd numbers ranging from 3 to 101. Multiplications and divisions were performed by a succession of doubling operations, based on the fact that any number can be represented as a sum of powers of 2.
• The papyrus recommends that multiplications are done in the following way. Assume that we want to multiply 41 by 59. Take 59 and add it to itself, then add the answer to itself and continue.

```            41          59
______________
1          59
2         118
4         236
8         472
16         944
32        1888
______________```

Since 64 > 41, there is no need to go beyond the 32 entry. Now go through a number of subtractions

41 - 32 = 9, 9 - 8 = 1, 1 - 1 = 0

to see that 41 = 32 + 8 + 1. Next check the numbers in the right hand column corresponding to 32, 8, 1 and add them.

```                        59
______________
1          59     X
2         118
4         236
8         472     X
16         944
32        1888     X
______________
2419```

Notice that the multiplication is achieved with only additions, notice also that this is a very early use of binary arithmetic. Reversing the factors we have

```             59          41
______________
1           41     X
2           82     X
4           16
8          328     X
16          656     X
32         1312     X
_______________
2419 ```
• b. Fractions
• The Egyptians endeavoured to avoid some of the computational difficulties encountered with fractions by representing all fractions , except 2/3, as the sum of so-called unit fractions, fractions with unit numerators. This reduction was made possible by tables representing fractions of the form 2/n, the only tool necessary because of the additive character of Egyptian multiplication. Such a table can be found in the Rhind papyrus, one for all odd n from 5 to 101. We find 2/7 be expressed as 1/4 + 1/28 ; 2/97 as 1/56 + 1/679 + 1/ 776 ; 2/99 as 1/66 + 1/198. Unit fractions were denoted by placing an elliptical symbol above the denominator number. A special symbol was used for 2/3, sometimes another symbol was used for 1/2.

• c. Area
• To find the area of a circle, they used the square of 8/9 of the diameter of the circle, in fact using a value of about 3.16 - close to the value of the ratio pi, which is about 3.14. Recent investigation shows that they knew that the area of any triangle is given by half the product of base and altitude.

• d. False position
• Many problems in the Rhind papyrus require more than a linear equation and are generally solved by the method which is known as the rule of false position.

Try to solve x + x/7 = 24. Assume any convenient value for x, say x = 7. Then x + x/7 = 8 instead of 24. Since 8 must be multiplied with 3 to give 24, the correct x must be 3*7, or 21.

There are also some theoretical problems. A papyrus found at Kahun contains this problem :

« A given surface of 100 units of area shall be represented at the sum of 2 squares whose sides are to each other as 1 : 3/4 »

We have : x² + y² = 100 and x = 3y/4

We can solve this problem by false position. Take y = 4, x = 3 and x² + y² = 25 instead of 100. We have to double the initial values of x and y, obtaining x = 6, y = 8.

•     6. The construction of a pyramid

• The Egyptians were fascinated by death. They believed that when someone died, their soul, or ba, continued to live on Earth, resting with the body at night. A spiritual duplicate of the dead individual, or ka travelled back and forth between the Earth and the "other world". The pharaohs had pyramids built to house their ba and all they would need for their after life. These huge royal tombs took over a lifetime to build. The four triangular sides, spread below a single peak, supposedly represented the rays of the sun shining down on the dead pharaoh, linking him to the great sun god Re. All the pyramids were built on the west bank of the Nile because the setting sun was connected, in the Egyptian mind, to death and the "other world".

The pyramid was built on a level piece of land, with the shape of a square. Canals were dug over the entire surface and then filled with water. The water level was marked on the stone, and then everything above that water level was cut away to smooth the surface. Finally the canals were filled up again to finish the surface.

An example of the accuracy of this method is the fact that the south-east corner of the Cheops pyramid is only 2 centimetres higher than the north-west corner.

Next they determined where the north was. In the middle of the construction site was built a wall. The top had to be completely level ( because it was the artificial horizon ). In the middle of the circle shaped wall a high priest would await the rise of Venus. The place where Venus showed up above the wall was marked, as well as the place where it disappeared again. Those 2 points formed an angle of which the bisector pointed directly north. This method was very accurate : the north-direction at Cheops deviates only 1/30th of a degree of the 'real' north.

Hereafter the square in which the pyramid had to be constructed was set out. Ropes with at regular distances knots were used to do this. With these ropes, they could measure angles of 90º.

The stones they used were very heavy. Most had a weight of about 2.5 tons, but some could weigh up to 200 tons. The limestones were polished carefully so that they would fit perfectly onto each other.

There was another problem they faced during the building of the pyramid. The 4 sides had to meet exactly in the top. Mistakes that were eventually made were irretrievable. A deviation of 2 degrees at Cheops would have caused that the top of this side would fall approximately 15 metres next to the intersection of the other three. To prevent mistakes like this, they would put a bar in the middle of the pyramid. The top indicated where the 4 sides had to come together. After the bar was put there, they continued building outwardly. Almost no mistakes were made using this method.

• B. RUINS AND PYRAMIDS : CONSTRUCTED BY MARTIANS ?

• 1. Introduction

There seem to be some strange parallels between the Egyptian pyramids and extraordinary stone structures on the red planet : Mars. Those abnormalities will be discussed in this chapter.

2. Formations and structures

Not only the curious faces of stone made the suspicion rise that there were indeed some artificial structures on Mars. The pyramids and unexplained ruins contribute to that suspicion.

• a. Inca city
• The article 'DID NASA PHOTOGRAPH RUINS OF AN ANCIENT CITY ON MARS' said 'Mysterious rectangles on the South Pole of Mars, photographed by Mariner 9. Each of these rectangles is 2.5 to 3 miles wide.' There are 10 or 12 of these objects who stand out against the mountains. Mariner photographed them on February 12 1977, from a height of 2937 kilometres. The sun was about 15 degrees above the surface then. The exact location was determined ( by NASA ) to be 80,82° WL and 64,43° NL., right in the area of Mare Australe.

Geologist John McCauley ( from Mariner 9 team ) said ' The mountain ridge is uninterrupted, doesn't show any fractures and rises above the surrounding landscape. Origin of this network is undeterminable.

The structure also amazed dr. Harold Masursky from the US Geological Survey Centre. According to him the formations are too symmetric and precise for not being a sign from an intelligent life form. However, they could be a rare geological structure. 'But those 90° angles are very strange', said Dr. Masursky.

Dr. Jim Cutts, astronomer and part of the Viking team, and Dr . Larry Soderblom, also working for the US GSC, gave this formation the name Inca City, because of the resemblance with the walls of the Inca. Dr.Cutts : ' Lots of geological formations have a whimsical character, but in these figures we can see an certain order.' Further he also said 'If there is life on Mars, it would be very likely to be found in the polar areas.' Dr. Masursky noticed : 'If Inca City was built by living creatures, I would date it in the past 200 million years.

All the Mars probes showed that the most strange structures can be found on Mars and because Mars isn't covered with vegetation the structures are very visible.

• b. Elysium
• In 1972 some three-sided structures narrowing near the top, much alike pyramids, were discovered in Elysium. Their size is huge : bases measure over 3 km and heights vary to 1 km. James Hurtak, professor on the California State University of Northridge, and Executive Director of the Academy for Future Science in California said, when he saw those pyramids 'If there are pyramids on Mars, we better start looking for the Sfinx.' Later, the discovery of the so-called Martian Face affirmed this declaration..

• c. Margartifer Sinus
• Comparable to Inca City, we see more strange stone formations in Margartifer Sinus, near the Mars-equator. It are partially branches, partially rectangular or triangular shapes lines. It looks like a prehistoric settlement. It seems to fit in, but at the same time it is to different from Mars to be made by nature.

• d. Sharonov
• On a similar photograph, taken from Viking I, another structure could be seen. It is situated near crater Sharonov ( this crater has a diameter of approximately 150 km, while the structure has probably about the same size ). It seemed to be almost square, almost like Sakkara. Right next to this structure there is a pit, in which uplift shapes like terraces, comparable with the diggup up from mines on Earth. This area ( situated on 58.5° WL and 27° NL ) is surrounded with permafrost clouds.

•  e. Cydonia
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• - The Fort
• The most astonishing ruins can be found in the area of Cydonia near the Face of Mars. One of these formations rises up in a rectangular shape and seems to be located on a square court. The structure exists out of 4 walls, of which one has probably collapsed, because it seems to slant a little. This structure looks like the old Sumerian ziggurats. Again the sizes are enormous : the base length is possibly 1 km, the height is 500 metres. This leads to a wall thickness of 200 metres. This place is called 'Fort'.

• - Main Pyramid & Pyramid Town
• Next to it there is another pyramid. This one is called the 'Main Pyramid'. There are at least 4 of them, and together they form a 'Pyramid Town'.

• - D&M Pyramid
• In 1980 Vincent DiPietro discovered the most astonishing structure . That base of this 'pyramid' seems to have 5 sides and has the shape of a pentagon. Through history, the pentagon has a mystic character and has a special place in geometry. As we know, the Greek mathematician Pythagoras, who had been for years a priest in an Egyptian temple, could find from the pentagon the special angle values of 18°, 36°, 54°, and 72°. And Leanardo da Vinci located the anatomy of the human body in a pentagon circumcribed by a circle. One of the sides of this 'pyramid' looks like a triangle, much like the Egyptian pyramids. The base length is 1500 metres. All the other sides vary from 1100 to 1800 metres. It has a height between 800 and 1000 metres. This pyramid is called the 'D&M Pyramid', named after its discoverers DiPietro and Molenaar. The visible triangular shaped surface's top points directly at the Martian North Pole. The angle between that side and the base is 54°, on both sides.

3. Angle of construction

The first Egyptian pyramid without steps, Meidoem, was constructed in an angle of 54°, by pharaoh Hoeni. Before it was finished it collapsed. In that same period, Hoeni's son Snofroe, assisted the construction of his father's pyramid and another one in Dahsjoer also constructed in an angle of 54°. When his fathers pyramid collapsed, Snofroe ordered at once to change the angle to an angle of 43°. This explains the crack in the Dahsjoer pyramid.

On Earth a pyramid with an angle of 54° just can't exist, because of our gravity, but on Mars a pyramid constructed under that angle can exist ( lower gravity ! ). So it is hard to understand why the Egyptians wanted to build pyramids with a 54° angle and failed to do that.

Difficult to understand is why during the reign of Cheops, all of a sudden, was choosen for an angle of 52° for the great pyramid of Gizeh . This pyramid is located exactly North-South and has an angle wherin also Pi is captured. Between the original height and half the original base-perimeter we see the relation 1 : pi. We can't find any models or precursors for this characteristic,only imitations. Calculating the angle of the Egyptian Cheops section from the ratio between the slant height and half the side of the base (this ratio happens to be the golden ratio f with f = (1 + sqrt(5)/2 = 1.61803...) gives us from cos a = 1/f the angle a = 51° 49' 38"

 4. Location   The older pyramids were good enough as graves, but now it is accepted that they may not have been built as graves, but to serve a higher purpose. How else can we explain that Snofroe had 3 huge pyramids constructed during the 24 years he ruled ? The German engineer Willy Kross also found out that the distances between the three Mars volcanos Arsia Mons, Pavonis Mons and Ascraeus Mons show resemblance with the distances between the three pyramids in Gizeh. The pyramids stand, in the Egyptian desert sand, exactly like the volcanos in the red coloured desert Tharsis.

5. Mathematical interpretations

• a. Circle segments and curious numbers
•  Willy Kross put the pyramid of Cheops in a circle divided in 7 segments. The 3 corners of the pyramid correspond approximately with 3 corners of the formed heptagon. Doing this, Kross came to the following conclusion : the reproduction seems to correspond with the constellation Mars-Earth. The lowest three of the seven parts stand for the 'third planet' from the Sun, i.e. the Earth. The circle segment included by the three parts symbolises the Moon. The upper four of the seven parts stand for Mars, i.e. the 'fourth planet' and the corresponding two circle segments for Phobos and Deimos (moons of the red planet). The ratio between the 7 parts, in which the circle is divided, to the planetary number of the Earth gives 2.33. The planetary number of Mars (4) divided by the planetary number of the Earth (3) results in 1,33. And the position of the great pyramid at 30° NL to his position towards the equator gives us 0,33...

• b. The sarcophagus
•  If we take a look at the sarcophagus in the pyramid of Cheops, in the King's room, we find some interesting measurements. The internal volume is 1.12 cubic metres, the external volume 2.24 cubic metres. The ratio of those numbers is almost the same as the ratio of the distances of Mars - Sun and Earth - Sun.The total length of the sarcophagus ( 227.6348 cm ) corresponds with the average distance between Mars and the sun in million km. The angle between the external long base and its diagonal equals almost the imaginary axis through the poles of the Earth ( 23.45° ). The diagonal angle in the eastern external side equals the angle between the trajectory of Mars and the imaginary axis through his poles ( 24.94° ). c. Dimensions of the Cheops pyramid in reference to Mars The Cheops pyramid is 150 metres high, and has a base of 230 metres with. 150 could be the average distance between Earth and sun, 230 could be the average distance between Mars and the sun. In the Cheops pyramid, the Queens' room is exactly in the centre of the to Mars referring heptagon. d. The number 27 In Sakkara, Meidoem and Dahsjoer, the angle of the corridor to the Queen's room is 27°. In the big gallery at Cheops there are 27 rectangular openings. The rotation of the sun is 27 days and we find the number 27 many times on the map of Cydonia !

• CONCLUSION

• If we look at all those regular stone structures and ruins on Mars, it looks that the "Martians" studied and practised geometry. We have to admit that something radical must have happened some time ago, something that is the origin of the strange mysterious looking surface of the red planet.

And why were the pyramids constructed ?

The pyramids might have been built by aliens coming from planets from the farthest reaches of outer space. Those aliens would have made the pyramids to give us 'messages in stone'.

The pyramids would be the result of a secret science. If we could discover the rules of that science, we would be able to predict all future events by just using the pyramid of Cheops.

The Egyptians would have put a curse on their mummies as a protection against grave robbers. This curse would have worked already for over 5000 years and because of that we should never be able to unlock the secrets of the pyramids.

•  BIBLIOGRAPHY

- Das Marsgesicht , Walter Hain, 1990, HERBIG.

- Met het oog op Mars , Gerard Bodifée , 1997, SCOOP.

- Het Marsavontuur, van fiction tot science , A.J. Wanders , 1979, TIRION

- An introduction to the history of mathematics, Howard Eves, Saunders Series

- The Crest of the Peacock, George Gheverghese Joseph, Penguin books

- The Story of Numbers, John McLeish, Fawcett Columbine