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After visiting Giza in the seventies, like most I was greatly impressed with the geometry, scale and precision of the site. But then something caught my eye. Looking at plans of Giza I noticed a pattern related to the central pyramid – the prominent satellite pyramids of Khufu and Menkaure being so placed that they appeared to form parallel alignments to the centre and corners of Khafre –

Perhaps this was just chance? However, the angles of these alignments have fundamental geometric meaning – the root of the square (blue), of the double square to Khufu (red), and of the equilateral triangle to Menkaure (red). Ideally their layout would look like this –

Whether or not these alignments are coincidental, one must ask why Khufu’s satellites are placed to the east instead of the usual south, and indeed why Khufu was positioned at the northern edge of the plateau. Some scholars say that there wasn’t enough room for satellites to the south (because of quarrying for instance) but as we can see from the topographical map this would not seem to be the case –

The builders made very efficient use of the site, placing the pyramids on the relatively level land running northeast to southwest. But why did Khufu not choose the best real estate where Khafre now is? Rossi opines – “When Khafra started his pyramid at Giza, his aim must have been to maintain the level of Khufu’s pyramid and, at the same time, to spare material and to improve the impulse towards the sky. He chose, therefore, a shorter side-length, 410 instead of 440 cubits, and increased the steepness of the slope by 1 finger, from 5 palms + 2 fingers to 5 palms + 1 finger.” This is simply incorrect – the base of Khafre actually measures 411 cubits, or 3 times the length of the Sphinx fronting this pyramid.

Today scholars are more open to the idea that Khufu intentionally placed his pyramid at the northern edge of the plateau. We might equally consider that the pyramids were positioned with care. An enormous foundation was built to support the south-east corner of Khafre –

Further evidence of careful planning is shown by Lehner’s analysis of Khufu’s Eastern Cemetery –

to the right is an enlarged section of the ‘replica passages’, and on the left a plan of the eastern side of Khufu. The axis of Khufu’s passage system is displaced from the north/south plane of the pyramid at A. The trial passages are displaced from the north/south axis of the satellite pyramid centres (and a trench) by the same amount at B. The centre of the King’s Chamber D delimits the northern limit of the Eastern Cemetery to E. The beginning of the Grand Gallery C aligns east/west with the vertical shaft in the Trial Passages at F.

The satellite alignments suggest that if there is a ‘language’ of the pyramids it is geometric. Cultures worldwide have used geometry to plan constructions (for instance in the classical world, or the Maya in central America) and Egypt falls in this global context. Geometry may have been used in different ways but its fundamental building blocks remain the same, as illustrated in this figure (sometimes called the Vesica Piscis) –

– two interlocking circles within a 2 X 3 rectangle, and the roots of 2, 3, and 5 – the same as reflected in the satellite alignments.

These are the root values –

The profile of Khafre is a triangle with sides 3, 4, and 5 units long, usually called the first ‘Pythagorean’ triangle. It is constructed on the diagonal of the double square –

The angle of the diagonal of the double square is 26.5 degrees, as in the northern satellite alignments. Doubling this to 53 degrees gives the base angle of the 3.4.5 triangle –

The diagonal of the double square is also the basis of construction of Phi (often called the ‘golden section’) –

Phi is a relation between two quantities such that the ratio between the smaller and the larger is the same as the ratio between the larger and the combined whole. Or A to B is the same as B to A + B.

Or numerically : the root 5 diagonal measures 2.236, so in a square of side 1 unit the root 5 diagonal measures 2.236/2 = 1.118. This then is taken as the radius of a circle producing the number 1.618 or Phi – so 1.618 = 1 + 0.618, and 1/1.618 = 1.618 – 1.

The construction of Phi within a 1 X 2 rectangle is shown below –

Much has been written about the supposed ubiquity of Phi (often mistaken) while arguments can get quite heated over the supposed presence of Phi in Egyptian architecture. Here it is taken as a simple geometric relation.

On a side note, Egyptian tomb ceilings were often covered in five-pointed stars which implies they were familiar with the construction of the pentagonal figure and hence the geometrical construction for Phi ^{(1)}.

Ancient peoples did not use decimals. Instead they approximated irrationals using numerical ratios. It is unknown if they arrived at certain ratios through trial and error or whether those we can identify arose through the use of a system ^{(2)}.

## Notes

(1) Construction of the pentagon.

– once again the construction is based on the diagonal of the double square. Using AB as radius, an arc is swung to C. Then with BC as radius, a second arc is swung to D – thus defining the side of the pentagon.

An alternative construction is shown below –

-if the side of square measures 2 units then the line OX measures 1.618. Using this as the radius of an arc to D gives the side of the pentagon.

(2) Ancient peoples did not use decimals. Instead they approximated irrationals using numerical ratios. It is unknown if they arrived at certain ratios through trial and error or whether those we can identify arose through some system.

Ratios are generated from additive series, like the one associated with the Renaissance scholar Fibonacci –

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

two consecutive numbers are added to produce the next. The ratio between two consecutive numbers equals Phi at infinity. For instance 34/21 = 1.619 and 89/55 = 1.61818

It was Theon of Smyrna who first drew attention to such series. This series approximates root 2 –

1 2 3 7 17 41 99 …

1 1 2 5 12 29 70 …

the sum of the numerator and denominator of each ratio is equal to the denominator of the succeeding term. Thus 41 + 29 = 70 and 70 + 29 = 99.

No-one knows to what extent the Egyptians studied number but it is not impossible that they new of such series. They could for example have experimented with pebbles or by adding squares. So for the Fibonacci series we just add squares –

For root two we add two squares –

Terms for root 3 –

Terms for root 5 –

Some of the above terms appear at Giza but might as well be the result of trial and error as following an algorithm. For example 19/17 gives a reasonable value for root 5 yet does not appear in the above series.

Rather than searching out ever more precise values for irrationals the Egyptians seem to have been more interested in selecting values which harmonise with one another, as we shall see.

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