The architects of the pyramids were evidently very capable at using mathematics to work out what they needed by way of resources, and the architectural features of their pyramid. Akhet Khufu, and its principal architect Hemiunu, was a case in point. He knew his mathematics...
Hemiunu's Height Budget
The figure shows the architect's vertical height budget for the Great Pyramid. If the various dimensions are measured in cubits (1cubit = c.53cm), rather than in modern units, the basic architectural plan becomes apparent. The preferred major unit, horizontally, was 40 cubits. Vertically, 35 cubits seems more usual, although the overall height was 280 cubits (=7 x 40). The ratio of vertical to base for the Great Pyramid, 14 to 11, or 7 to 51/2, the latter defining the slope of the Pyramid as being 51/2 seked. The predominance of 35 cubits in the height budget may reflect that 35 divides by 7, whereas 40 cubits does not divide simply by 7…
Hemiunu's Concept?
The Giza Plateau (otherwise Rostau) was considered a sacred place, where the sacred Mound of Creation was to be found. The concept is of a revitalised Mound of Creation, incorporating and extending the original mound, with an upper, flat level set at the top of the so-called Relieving Chambers above the King's Chamber. On the same upper level are envisaged a Southern and a Northern pavilion (as homage to those in Djoser's Stepped Pyramid Complex) also with capstones over them, like those above the Relieving Chambers, but much smaller. Each of these pavilions is reached by a shaft from the Queens Chamber, QC(S)) and QC(N) respectively for southern and northern shaft. These two shafts do not pierce the outer surface of the pyramid.
This envisaged "surface" on top of the Mound of Creation, hidden to us by the contiguous pyramid built upon it, with no apparent join between mound and pyramid, might be considered as the 'real' surface upon which the pyramid is set. This hidden surface on top of the Mound of Creation could represent a conceptual map of Egypt, over which the Pharaoh's ka might roam as though over the real territory…
At the time of building, the now-hidden surface would 'show' three sets of capstones. The Southern and the Northern Pavilions, (5 x 40 =) 200 cubits apart would be reached via their respective shafts from the Queen's Chamber. And the capstones above the King's Chamber showed where a "pit burial" of the Pharaoh may have taken place, with the layers of granite stones representing some depth between the surface and the King's Chamber. (This to compensate for having to abandon the underground chamber?) The shafts from the King's Chamber may have been pre-fabricated*, and these pre-fabrications would pass through the surface on their eventual way to the outside of the Pyramid, also separated by some 200 cubits at their exit points… Note from the first figure above, that the points at which the King's Chamber shafts emerge from the pyramid are vertically above the points where the Queen's Chamber shafts pierce the Mound of Creation, at the supposed pavilions. The King's Chamber has been neatly offset from the centre line by just the right amount to make this so.
The slopes of the various shafts suggest the magic of numbers may be at play:
- The slope of the shaft to the south, KC(S), is 7 seked - a rare use of 7 seked (=45˚) - which was symmetrical and seemingly avoided for that reason. The culmination of al Nitak, the left hand star in Orion's belt coincides with 7 seked, and that star was considered sacred to Osiris - which came first, the 7-seked slope or the choice of star is not known…
- All rising and falling passageways, including the Grand Gallery, occur in the northern half only, at exactly half of 7-seked slope, i.e. 14 seked.
- The slope of the northern shaft, KC(N), is 11 seked, pointing toward the celestial pole, but also exactly half the slope of the pyramid at 51/2 seked, suggesting that the slope of the pyramid might have been chosen on this basis. Note, too, the distance between the points of emergence of the two shafts is 5 x 40 = 200 cubits.
- Meanwhile, the slopes of both southern and northern shafts from the Queen's Chamber, QC(S) and QC(N), are 81/2 seked, a second, curiously symmetrical choice until it is noticed that 51/2 + 81/2 = 14 seked, the standard slope of rising and falling internal passages, including the entrance passages and the Grand Gallery. The choice of 81/2 seked thus provides numeric "closure," with no summations left unsatisfied…
Evidently, as both diagrams show, Hemiunu was very comfortable with numbers and figures. The various slopes and locations within Akhet Khufu fit together like a Swiss watch. I cannot prove that he was instilling "number magic" into his design for the Great Pyramid, but there had to be some reason for his choices of slope and elevation, and architectural "magic" comes up first on my list!
Is the concept for the Great Pyramid as shown above valid? Again, I cannot be sure. But, it certainly makes a lot more sense than presuming that the so-called Relieving Chambers were introduced to relieve pressure on the King's Chamber. The capstones above the Relieving Chambers would have relieved pressure equally as well had they been placed immediately over the King's Chamber - again, spreading out beyond the walls of the Chamber and not resting on them. (To assume that the ancient Egyptian architects did not know this, and that they simply made a tremendously expensive mistake, would be to insult their abilities and experience. They clearly knew what they were doing.)
Similarly, the shafts leading up to the surface of the Mound of Creation from the Queen's Chamber would have been protected by capstones that did not rest on the shaft walls. And access to the Mound via these shafts would have been barred to all but the King's ka by three portcullis doors, through which his ka could pass magically. (Access to the King's Chamber was originally barred by three portcullis doors, too.) And the portcullis door found when exploring the southern shaft would, conventionally, be the first of a set of three… it was the practice at the time.) It is noteworthy that the walls of the the southern shaft were smoothed and polished when approaching the first portcullis door, an indicator of approach to a special, or sacred, place.
All of this feeds the notion of a Southern Pavilion, which could be simply notional or which might incorporate some space, even some objects. And the symmetry of the design from the Queen's Chamber about the Pyramid centre line, demands that there be a corresponding Northern Pavilion catering for the northern population of the two nations…
On the other hand, the King's Chamber is decidedly not symmetrical about the centre line — it is offset to the south. This allows for the asymmetrical shafts from the King's Chamber which lead to the outside world at 7 seked to the South and 11 seked to the North respectively. It should also be noted that the King's Chamber is above the deserted subterranean chamber shown in the diagram, similarly offset to the south of the centre line…
So, if QC(S) leads to a Southern Pavilion, and QC(N) leads to a Northern Pavilion, and The King's Chamber is more towards the Southern than to the Northern Pavilion, where does that place the King's Chamber on a notional map of Egypt? My guess would be Abydos, the sacred burial spot of the ancestral kings of Egypt.
Looking at each dimension individually tells us little. Looking at them all together and seeing how they all fit is rather more telling. There is a lot yet to emerge about the Great Pyramid…
* Why presume the shafts to be prefabricated? Well, QC(N) which rises a 81/2 seked also has to circumnavigate the Grand Gallery—which is tricky, geometrically. It is easy to overcome the difficulty if a platform is constructed with an 81/2 seked slope, so that the mason is working horizontally and he can then lay out the necessary curve to go around the Grand Gallery with relative ease… Hence the strong suspicion of pre-fabrication, here and elsewhere in the Great Pyramid.