In 1880 Petrie travelled to Giza and spent two years meticulously surveying the pyramid of Khufu. He did this to test various theories about its design that were current in his day, and found them wanting. He also made a survey of the exteriors of the other two pyramids, but not their interiors. For these, and measurements of the other structures on the Giza plateau, it is necessary to sift through a variety of excavation reports, of varying precision. Where Petrie’s results have been tested by modern surveyors they stand up well, and they form the basis of the analyses on this website.
In his analysis of Khufu interior Petrie concluded that the internal architecture had been laid out using a cubit of 0.5237 meters. His measures, referenced to floor lines, are shown below –
It is to be noted that his measurement from the north base, horizontally to pyramid centre, is 219.9 cubits and not the expected 220 cubits. It has been suggested that the architect made the slope steeper so as to reflect ‘tru Pi’ in the pyramid’s design but most scholars put this down to building error.
Legon converted Petrie’s measures (made in inches) into cubits (of value 0.52375 metres). He found that the King’s Chamber had been placed at a root two division of the height –
The level of the King’s Chamber floor is determined by a root 2 division of pyramid height – height of pyramid 280 cubits, KC floor below apex 198 cubits. Value for root 2 is 198/280, reducing to 99/70.
In consequence the diagonal at this level is the same as the base of the pyramid : 440 cubits –
Legon further found that the positions of other features were governed by this geometry –
– the south wall of the King’s Chamber is 26 cubits south of pyramid centre, or 246 cubits south of the north base, which is 3 X 82. The Queen’s Chamber floor is at half this height at 41 cubits, while the end of the Descending Passage floorline lies 205 cubits (or 5 X 41) from the north base.
The Grand Gallery is also 82 cubits long – 79 cubits to the north and 3 cubits to south, the length of the Great Step –
The 79 cubit length of The Grand Gallery is equal to the height at which the diagonal of a double square intersects the casing –
It is interesting to note that 280 – 79 = 201, or 3 X 67, and the intersection of the floorlines of the Descending and Ascending passages is about 67 north of the beginning of the Grand Gallery –
– however this relation is not precise. Legon shows the true configuration (referred to the intersection of the floorlines of the passages) –
– the angles of the Ascending passage and Grand Gallery are each less than the diagonal of the double square, but their heights are whole numbers of cubits (33 + 39 = 72). However their horizontal lengths are not, being 78.9 and 67.4. In the next diagram we see why this is –
– the sloping lengths of the passages are whole numbers of cubits, 88 and 75, which shows that these slopes were defined by height and slope and not by the SEKED. It would then appear that the floorlines of the upper passage were laid out using a module of 24 cubits.
I noticed that the floorline of the Grand Gallery, if projected, intersects the junction of the south wall and ceiling of the King’s Chamber, and the length of this projection is 29 cubits. So the total virtual sloping line is 29 + 88 + 75 = 192, or 8 X 24 – perhaps more than serendipity?
Legon explained the passage divisions using this diagram –
– if a double square of side 14 units is drawn within the pyramid it generates the segments 11 and 14, sum 25.
If these proportions are applied to Khufu, Legon showed they precisely define passage junctions. So 280 divided as 14+14:11 (in blue) produces the Grand Gallery length of 78.9, and 205 divided as 25:14 (in orange) defines the floorline junction of the passages –
Phi
Surprisingly there is another way to define passage junctions, by considering passage ceilings rather than floors. Allison found that the ceiling of the Descending Passage is divided in Phi at the base of Khufu –
He further showed that the Descending passage was laid out in the same Fibonacci terms defining the pyramid –
The overall scheme looks like this –
Alison also found that the position of the junction of the Ascending Passage floor and Descending passage ceiling is defined by a Phi construction from B. If a further Phi construction is made (from C to D) it defines the north end of the Grand Gallery at E –
There are some points of concordance between Legon’s and Alison’s figures. It seems the designers combined two parallel layouts, represented by the floors and ceilings of the passages, just as they did in design of the pyramid as a whole –
Many other proposals have been put forward to explain Khufu features, including the calculation of area and volumes, the sizes of coffers and suchlike, and who knows if such approaches will bear fruit? For the moment it is more useful to pursue the theme of duality apparently uncovered in the preceding analyses. Phi and Pi (as 22/7).
(The abormally thick 35th stone course divides the pyramid into two equal volumes. The centre of this course is about 57 cubits above base).
The King’s chamber
Whereas the base of Khufu is not a whole number (if one uses Petrie’s cubit) the prevalence of the 14/11 ratio in IVth dynasty work suggests that it is the intended basis of design – 280 height and 440 base. And this is reflected in the dimensions of the King’s Chamber. The floor plan of this chamber is 10 X 20 cubits and the chamber height is equal to half the diagonal of the plan, 11.18 cubits. The consequence of this is that a virtual 345 triangle is contained within the chamber –
Or perhaps better shown from the chamber ceiling –
– because the floor is actually inserted between the walls (the narrow pink band) the actual wall height is 11.43 cubits, or 320 fingers –
– the walls have 5 courses and each course is 64 fingers thick, so the height of the wall is 320 fingers. The King’s chamber passage is 56 fingers wide, or a tenth of the chamber width of 560 fingers (20 cubits).
Petrie noticed that these dimensions represent a scale model of the pyramid – the circuit of the wall (1760) divided by the length of the wall (560) = 22/7. The figure is developed within a square of side 560 ( 320 is 4/7 of 560) –
Drawing the full figure –
– we see the elegance of the builders in designing the King’s Chamber walls (whose height would not be known to us had not the floor been desecrated) so as to reflect the geometry of the pyramid, and to confirm that the 14/11 ratio was the basis of its design. The two designs of the chamber (with heights of 320 and 313 cubits) may reflect the earlier described Pi and Phi aspects of the pyramid.
Whatever the truth of this, the preceding analyses can succesfully explain the design of the major features of Khufu. Yet the particular manner in which the described geometric themes intertwine is difficult to understand – passage floorlines laid out according to the geometry of the pyramid while passage ceiling junctions are determined by Phi. An open question.