Cairo

Cairo lies in Egypt at the apex of the Nile delta and is by far the largest city in Africa and the Middle East. With a population fast approaching 17 000 000, it is an amazing jumble of buildings and streets, bazaars, mosques and museums, where east and west, old and new, ancient and modern co-exist in a vibrant if hot and polluted atmosphere. One focal point for most tourists in Cairo is the huge Museum of Egyptian Antiquities, which contains the immensely fascinating Tutankhamun Galleries: it takes days to see it all.

Other attractions are of course the Pyramids of Giza and the Sphinx, best viewed in the late afternoon followed by a sunset performance of the Son et Lumière (Sound and Light), which provides a colourful account of their history.    

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The pyramids (I)

Although pyramids may be found in other parts of the world, notably in Central America, it is this remarkable collection of pyramids, particularly the Great Pyramid, at Giza which has most captured the world’s imagination. The Great Pyramid of Cheops (or Khufu) has been described as ‘the most enduring monument ever built’ and ‘the greatest single building ever erected’. It was counted as one of the Seven Wonders of the Ancient World, and even today is considered to be one of the world’s great tourist attractions. It was built by the pharaoh Cheops as a funeral monument in the 27th century B.C., but it was not the first pyramid in Egypt. That distinction goes to the Step Pyramid at Saqqara just south of Cairo, which was commissioned by King Zoser around 2700 B.C. and designed by Imhotep, the world’s first great architect.

The Great Pyramid is massive: it was originally 146.5 m tall (now 137.2 m), covering a nearly perfectly square base of 53 hectares and containing almost 2 500 000 precision-cut stones each of 2.5 tonnes. Its close neighbour, the pyramid of Chefren (or Khafre) is only slightly smaller, originally 143.5 m tall (now 136.5 m).































The pyramids (II)

The extraordinary geometric precision of its internal and external construction, its accurate orientation to the four points of the compass and its location relative to the Tropic of Cancer and to the Nile Valley all suggest that the Great Pyramid was more than just a tomb. There is growing evidence that it had great significance for astronomical and earth measuring purposes as well. Some people believe that there was something rather more mystical embodied in its structure. One of the best known icons reflecting this kind of view is the representation of a pyramid on the the Great Seal of the United States of America, and reproduced on the U.S. dollar bill.





















The Great Pyramid and phi

The Ancient Greek historian Herodotus was told by Egyptian temple priests that the Great Pyramid was constructed in such a way that the area of each of its four faces is equal to the square of its height. Let the height of the pyramid be x units, and the length of the base edges be 2 units. Then the square of the height is x, and the slant height (or apothem) of the pyramid must be x units, as this gives the area of the face to be x.

 Investigate Verify that the area of a face is in fact x, and show that x2 = x + 1.

This equation is the defining relation for 1.618033 ... , the golden section. In other words, for a square pyramid of base length 2, the height of the pyramid is and the slant height is .

The actual dimensions of the pyramid in royal cubits were: height 280, apothem 356, and base length 440, where 1 royal cubit = 0.525 metres, the ‘standard’ length of a forearm from elbow to finger.

 Investigate
Verify that the above dimensions are in the ratio x : x : 2 .

 






















The Great Pyramid and pi

At right is another relationship apparent in the Great Pyramid:


“the perimeter of the square base is equal to the circumference of a circle having radius equal to the height of the pyramid.”

 Investigate
Given that the base length of the Pyramid is 2, and the height, as in the previous slide is , show that 4/. Check the accuracy of this with your calculator.

 Investigate  Now substitute the royal cubit measures of the base perimeter and the height of the Pyramid (given above) in this relationship, to show 22/7.

 Investigate  Yet another relationship discovered elsewhere in Ancient Egyptian monuments is the approximate result: 6/5.2. Check this using your calculator.

Good approximations to are given by the ratios of successive terms of the Fibonacci series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... . We can get get a good approximation to in terms of these. With a circle of diameter 21 units, the circumference is
             21 21.6/52 = 21.6/5.. 21.6/5.34/21.55/34 = 66.

This gives an approximate value of as 66/21 = 22/7.

 




























The greatest Egyptian pyramid

In the Moscow papyrus (c. 1850 B.C.) we find:

You are told: a truncated pyramid of 6 cubits for vertical height by 4 cubits on the base by 2 cubits on the top. Square this 4: result 16. Square this 2: result 4. Take 4 twice: result 8. Add together this 16, this 8 and this 4: result 28. Take one third of 6: result 2. Take 28 twice: result 56. Behold it is 56!

This is equivalent to the formula: V = .h(a2 + ab + b2), for the volume of a truncated square pyramid of height h, having base of edge length a, and top of edge length b. (What happens when we set b = 0 here?)

This problem has been referred to as ‘the greatest Egyptian pyramid’, viewed as the highest achievement of Egyptian geometry. How did the Egyptians obtain this formula? Two explanations: (a) it was derived from a knowledge of the volume of the step pyramid using square numbers; (b) it came from considering the volume left when a smaller pyramid is removed from the top of a larger pyramid.

 Investigate  Show that the above Moscow papyrus problem equates to the given modern formula. Obtain the volume formula by following explanation (b) above, using square pyramids having base lengths a and b.

The pyramid on the U.S. dollar note is actually a truncated pyramid. Such a pyramid may have been used as a platform for astronomical observations by the Egyptian priests.

 


























The 3 – 4 – 5 triangle

It is said that the ancient Egyptian surveyors or ‘rope stretchers’ made use of a length of rope which was divided into 12 equal parts by 11 knots. Forming a triangle with sides in the ratio 3 : 4 : 5 created a right angle, which was used extensively in surveying and building.

The use of the 3 – 4 – 5 triangle does not prove that the Egyptians were aware of the Pythagorean relationship. Indeed it is possible to show that such a triangle is right-angled without using Pythagoras’ Theorem at all.

Investigate  Show how to use the knotted rope to determine a right angle.

Can you show that a 3 – 4 – 5 triangle is right-angled without using Pythagoras’ theorem? The given diagram, which is of ancient Chinese origin, may help.

 Project  The early Egyptians made many notable contributions to the study of mathematics, only a few of which are mentioned here. Many of their discoveries were elementary, interesting and elegant, and have been recorded on old papyri. Use your library or the Internet to find out more about Egyptian mathematics.