Only a couple of mathematical papyrii survive, from the middle kingdom, and they reveal about Khufu design, that at that time, pyramid slopes were defined as a ratio, with height of 1 cubit (or 28 fingers of the cubit) and half base measured in palms and fingers, called the SEKED –
In the example above the SEKED is 5 palms 2 fingers, so the ratio is 28/22, or 14/11 – the profile of Khufu. Making the half base equal to 1000 (‘1’) the height becomes 1272.72. Khufu’s profile reduces to height 7 base 11.
Within this figure a simple construction gives the dimensions of Khufu – base 440 and height 280 –
That the numbers 11 and 14 were important to the Egyptians is supported by Hamilton’s finding that the chambers of the Red pyramid at Dashur were laid out using this ratio –
From this it can be taken that the fundamental figure for Khufu is base 440 and height 280 cubits.
While the 14/11 relation cannot be geometrically constructed it may be closely approximated –
– 9/10 ths of root 2 = 1.2728 (compare 14/11 = 1.2727…)
There have been many proposals explaining why the ratio 14/11 was chosen and they usually involve Pi and Phi, and are usually rejected by scholars as coincidences. Most remember that Pi is the distance around a circle with radius 1 – 3.1415926. Many will have heard of the term ‘squaring the circle’, but this has two meanings. Squaring the circle for area means finding a square that has the same area as the circle. The Egyptians had the need to do this often in their daily lives. They used a simple formula for this – square 8/9ths of a circle’s diameter. This gives a rather crude result for Pi of 3.16 –
‘Squaring the circle’ means drawing a square which has the same perimeter as the circle’s circumference. In other words defining Pi –
– in the case of Khufu with base perimeter 1760 and height 280, Pi is represented as 22/7.
The second claim is that the pyramid profile is governed by Phi –
– the double square with sides of 440 X 220. A phi division on the diagonal produces the figure 356. 356/220 = Phi.
There are thus two definitions of Khufu of slope which are very near the actual based on the simple profile 14/11. But the Pi slope is too steep (1.273 : 1) while the Phi slope is too gentle (1.272 : 1) –
Because the dimensions of the pyramid are Fibonacci numbers (356/ 220 = 89/55) and harmonize with an expression for Pi (220/70) it would seem that the two approximations are intentional –
One of the most arresting features at Giza is the Descending passage of Khufu – it is so straight that it deviates from a central axis by only 6mm from side to side and about half this vertically, and this is generally agreed to have been constructed at the beginning of building. It was laid out on the diagonal of the double square at 26.5 degrees. But where does the passage begin?
I propose that the architect was quite aware of the circle-squared relation. The origin of the axis of the Descending Passage is the point where the square meets the circle –
From this point the line of the passage will intersect the north face of the pyramid 33.4 cubits above base –
(Theoretically, with a slope of 1 to 2, the passage axis should intersect pyramid side 33.6 cubits above base, but the built part of the passage has a shallower angle).
The level below base of the end of the Descending passage ceiling marks the point of origin of the line of the Ascending passage ceiling –
The angle of the Grand Gallery is steeper than the Ascending passage. Many scholars ignore this and assume that the upper passages were laid out with imprecision on the diagonal of the double square. This is not the case.
I do not see how the builders could have laid out the Descending passage without reference to the circle-squared diagram. In daily life it was necessary to provide a simple routine for calculating the area of circles (and hence the volume of containers) effectively producing a value for Pi as 3.16. The circumference of circles had few such practical applications but this doesn’t mean that the priests would not have deeply pondered the question in both mathematical and symbolic terms. In fact they came up with a most elegant definition of Pi as 22/7, simple yet fairly accurate.
(1) Cooper found that the area of a circle can be much more closely approximated using a construction on the double square diagonal –
– the circle has a radius of 1 and an area of 3.14126 (or Pi). A double square diagonal from a to b is cut by an arc of radius 1 at c. bc is bisected at d. This forms a square of 1.4472 (in blue). This is multiplied by 1.5 to give 2.1708. 1.4472 X 2.1708 = 3.14158. But Cooper doubts the egyptians would have known of this.
(2) The squared circle cannot be constructed geometrically, but we can get quite close – A circle of radius 1 is drawn (in red). Then Phi is constructed to give 0.618 (in blue). This dimension is halved to give 0.309 (black). Intersection of the double square diagonals defines a square (in red) whose perimeter equals the circumference of the circle –