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Geometrical analysis

 


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The Giza plateau slopes gently down to the valley floor from northwest to southeast and the pyramids of Khufu, Khafre, and Menkaure were built on the line of relatively level ground from northeast to southwest. The central pyramid of Khafre is built on higher ground and dominates the site, so why did Khufu not choose this prime location for the first pyramid, rather building at the northern edge of the plateau? Some say to be near to Heliopolis across the nile valley. However, the foundation ofthe pyramid of Khafre, at its south east corner, is built up using megalithic blocks -


 


- if this pyramid had been sited a short distance to the west this platform would not have been necessary, which provokes the thought that the positioning of one pyramid relative to another was of some importance. Positioning was certainly critical in the layout of Khufu eastern cemetery. Lehner showed that it was laid out with reference to the internal features of the pyramid -


 


The central shaft of the 'Replica Passages' , shown to right, is in EW alignment with the Grand Gallery junction. The replica passages are offset EW from the NS axes of the satellite pyramids by an equal amount to the offset of the passage system inside Khufu from the central axis. The cemetery limit is determined by the EW axis of the King's Chamber.
Many years ago, while studying a map of Giza, my attention was drawn to the satellite pyramids - those of Khufu had been built to the east when most pyramids had them to the south (one explanation being that Khufu's architect had decided to place his main quarry to the south). However I noticed that parallel lines could be drawn from the centres of these satellites (and from the satellites ofMenkaure) to the corners and centre of Khafre -


 


 
The alignments to Menkaure are at 60 degrees, and to Khufu 26.5 degrees (the diagonal of a double square). Notice also the blue line between the leading satellites at 45 degrees. These 3 angles express the fundamentals of simple geometry. They appear to relate the north side of Khufu and the west side of Khufu relative to the centre of Khafre as below -


Subsequently I learned of other layout propositions and these have been important in developing the following narrative. My own contention is that Giza was built to an overall plan centred on Khafre (though not in the way I originally imagined).
 
The objects of the following analysis are the three pyramids and their internal structures. Of course there are many other features on the Giza plateau, and indeed other pyramid sites, which are amenable to analysis, but they are beyond the scope of this presentation, save for references relevant to the propositions set out below.
 

Survey data


The following analyses are based primarily on the survey of WMF Petrie.
While the accuracy of pyramid orientation to the cardinal points is very good, it is not perfect. The differences are small and have been very greatly exaggerated in the diagram below.


 

The orientations of the pyramids are :

Khufu               3'43"     west of north
Khafre              5'26"     west of north
Menkaure         14'03"    east of north



In addition, Khufu is not  exactly square - whereas the northwest corner is a right angle, the northeast corner is 3'2" greater than a right angle, and this therefore gives an additional variation. Menkaure's discrepant orientation may not be due to building error (perhaps resulting from an astronomical procedure used to lay out the pyramid) but it should be noted that the entire Menkaure complex appears as if finished in haste : pavement was not laid and, although begun with some of the largest stones at Giza, temples were finished in mud brick. Two of the subsidiary pyramids were not cased but left as step pyramids.
In spite of these discepancies it would seem that the intention of the builders was to lay out pyramid bases to the cardinal points.Petrie decided that, because the orientations of  the sides, and passages, of the two large pyramids are similar, he would choose a mean between them - 5' west of north - as the axis of his survey. In the diagram above the grey rectangle is oriented to the cardinal points, while the white rectangle is oriented to Petrie's survey.
But what of Menkaure's aberrant orientation? -



 


While the discrepant orientation is small it introduces uncertainty in east west and north south dimensions at the scale of the layout. The simplest approach is to assume that Menkaure's architect intended the pyramid to be aligned to the cardinal points.

The unit of measure used by the builders was the royal cubit. This was divided in seven palms or 28 digits. In the egyptological literature its length is typically quoted as 0.536m, but could vary locally. Petrie averaged a number of values derived from features inside Khufu according to the principle of 'inductive metrogolgy' (giving weighting to those derived from the King's Chamber) producing a value about 0.5237m, used by some analysts. It is quite possible that the cubit varied locally at different sites and this should be taken into account, as must setting out and building errors. (Butler estimates building error for built structures about 1 in 2000, and for excavated parts no better than 1 in 1000. The descending passage of Khufu, the first major excavation on the Giza plateau, was however cut with incredible precision).
If Khufu's architects really did envisage a coherent plan for the whole site then they would presumably have used the cubit used to lay out the base of Khufu. This is the longest built-stone length and should therefore give the best estimate, which is 0.52355 m. These are Petrie's figures for pyramid base and apex distances oriented to his chosen axis, converting his inch measures into metres and then into cubits of the latter value -




 

Petries figures for the interior of Khufu are shown below -

 
Geometry

Khufu and Menkaure have the same slope 14/11 (or in egyptian terms seked 5.5 or 28/22). Khafre slope is 4/3 (or seqed 5.25 or 28/21) - the pythagorean triangle. These slopes may be generated using simple geometry -

Geometric roots

 



Pentagons



Construction of the 345 triangle on the diagonal of the double square


Operations on the diagonal of the double square also generate Phi, or the division into mean and extreme ratio -


The Phi ratio, 1.618 to 1 , may be approximated in whole numbers using an additive series first described by Fibonacci. It produces increasingly precise values for Phi -
 
1 1 2 3 5 8 13 21 34 55 89 ...
 
- so, for example, 34/21 = 1.619 and 89/55 = 1.61818. A simple way to generate this series is through the addition of squares -


 
Similar contructions based on the addition of squares generate roots. The first is attributed to Theon of Smyrna -


1/1. 2/1. 3/2. 7/5. 17/12. 41/29. 99/70...


 


Generation of terms for root three -

 


Terms for root five -

 


Khufu


The agreed dimensions of Khufu are base 440 cubits and height 280 so the slope is 14/11 or 1.2727... This slope has been interpreted in different ways. Some see it merely as an expression of the cubit 28/22, but others argue that it had a geometric origin. The most popular proposition is that based onPi - the relation between circumference and diameter of a circle. It is that the perimeter of the base equals the circumference of a circle with radius equal to pyramid height. This requires a slope of 1.273 -


The alternative proposition is that the base to side ratio is as 1 : Phi , so the height becomes root Phi giving a slope of 1.272.
If the Pi relation was intended it means the architect used the well known value 22/7 for Pi in making the base 440 and height 280. But this value is not attested in egyptian mathematical writings, rather a value of 3.16 was employed for calculating the volumes of cylindrical containers and the like. Egyptian iconography contains a few illustrations of circles, theshen ring for example and of course the ubiquitous sun disk - was Re then brought down to earth translated into the form of the Pi pyramid? At any rate the principal factors 22 and 7 have their precursors in the chamber system of the Red pyramid at Dashur -
Propositions for Pi relations inside Khufu focus on, sometimes rather complex, volumetric calculations the evidence for which is rather arcane. But the fact remains that the Pi pyramid cannot be geometrically constructed and we might turn our attention to something that can - the Phi pyramid. Here Khufu profile is constructed on the diagonal of the double square -

Starting from a double square of sides 440 X 220 the Phi construction determines the side as 356 - this being divided into 220 and 136 - reducing to the Fibonacci terms 89 : 55 : 34. (The limestone-based height of Khafre was made 272 or 2 X 136). The diagonal of the double square was 'monumentalised' right at the beginning of building in Khufu's descending passage. The polar passages of other pyramids vary somewhat from from the precise value we find in Khufu.
As to the choice of dimensions for the pyramid, these derive from a simple geometric operation on the base figure, starting from unity (times one thousand) a pyramid of base 1000 and height 1272 produces the dimensions 440 and 280 -


 
(It is interesting that the14/11 ratio (1.2727...) can be closely approximated as 9/10 X root two -




Khufu Interior


The King's Chamber floor is situated at a root two division of the height -
 



The diagonal at this level equals the pyramid base side -


Dimensions are shown below. The architect employed the root 2 approximation 99/70 (as in Theon's series) -
 


 


Legon found other features defined by this geometry - the height of the floor of the Queen's Chamber passage, the south wall of the King's Chamber, and the termination of the descending passage. He also analysed passage floorline lengths showing that they had been laid out according to a vertical module of 72 cubits, divided in the ratio 39 : 33 to locate passage junctions, and divided by 3 to locate entrance height -

 



(The line of the upper passage floors, if projected to the corner of the KC roof, produces a length of 29, and a total length of 192, or 8 X 24. This may be serendipitious.)


Coming to the famous shafts leading off from the King's and Queen's chambers, Gantenbrink confirmed that the KC shaft positions and angles derive directly from the shape of the pyramid - the south being the diagonal of the square and the north determined by the ratio 440/280 = 121/77 or the profile of the pyramid. So the pyramid is dissected like this -



The King's Chamber shafts intersect 77 cubits above base and 22 cubits south of centre. Note the black root two rectangle 280 : 198 reducing to 99/70. (The line of the upper passages intersects the projection of the pyramid side about 44 below base.)
It is usual to use floorlines to measure passage lengths but Gantenbrink found that ceilings are also significant. The ceiling of the Queen's Chamber passage is 44 cubits above base -



A double square diagonal from the centre of Khufu base intersects pyramid casing 79 above base, defining the length of the grand gallery to the Great Step. The square on height of 280 is divided into the two parts 79 and 201, or 3 X 67. The ascending/descending passage junction is then defined as 79 + 67 = 146 north of pyramid centre, and the 'virtual beginning' of the descending passage ceiling as 213 north.
Support is given to the importance of considering ceilings from Alison's observation that the ceiling of the descending passage is divided in Phi at base level -




(He adds that the ceiling of the descending passage Menkaure is also divided in Phi.)


The King's Chamber


 
The floor plan of the King's Chamber measures 10 X 20 cubits. Its height is derived from the diagonal of the floor -



As a consequence the chamber contains a virtual 345 triangle lying across the chamber. Considering Gantenbrinks observation concerning ceilings rather than floors, the geometry may be conceived as developing 'from the top down' -



Whereas the ceiling is flat the floor is uneven, being made of flat granite slabs. However, the walls descend below the floor. Each of the five wall courses is 64 digits thick, making the height of the wall 320 digits. Wall height + wall length = 320 + 560 = 880 digits. So the chamber reflects the dimensions of the pyramid at a scale of 1/28 -



Without the partial destruction of the chamber's granite paving slabs we would not know about this relation - it would seem that knowledge of it was 'reserved for the gods'. But the designers also wished to monumentalise the geometry of the double square and the 345 triangle, and so laid a floor 7 digits thick so wall height became 313 digits, or 11.18 cubits.


In sum the design of Khufu is based on the ratio 14/11 - dimensioned in terms of the Fibonacci sequence (89/55), the square root of two, and the square root of 5 - the diagonal of the double square. Dimensions derive from unity (1000 = 440 + 560).



Pyramid layout

Using Petrie's data (and a cubit of 0.5237 m) Legon put forward this plan -



The construction is part geometrical and part modular. Legon proposed that the intention  was to make an overall plan exemplyifying roots : north south dimension root 3 , east west dimension root two, unit 1000 cubits. To arrive at this, the builders first set out 5 modules of  220 cubits  to define Khufu and locate Khafre south base 1100 cubits south. A second modular arrangement  (unit 125 cubits)  locates the north and west sides of Khafre. The east west dimension containing the two pyramids is then 1065 cubits, and  Khafre is defined as 410 cubits north south. This plan was adjusted - one cubit subtracted from the east west  dimension and added to the north south dimension  to produce the figures shown above.


To explain the location and base dimension of Menkaure Legon put forward a very elegant 'circle-squared' scheme (implying the concept of Pi) beginning from a module of 500 cubits. The diagonal is 707, and the figure is then dissected as below to define Menkaure base as 201.5 cubits. The west sides of  Khafre and Menkaure are separated by a distance of  250 X root 2  =  353.5 cubits (see note 1) -



The overall dimensions of the final plan are 1732 north south and 1417.5 east west. Legon's plan need not have been designed at the outset of building but could have been developed  by  Khafre and Menkaure's architects.
 



Legon noticed that within an implied square of side 2000 cubits, the diagonal of the plan and the diagonal of the square intersect to define the south side of Khafre -

 

The ideal root rectangle figure is constructed on the diagonals of the square and double square of side 1000 cubits -


 
As mentioned above, the size and slope of Khufu may also be constructed from the unit square -


Notwithstanding Legon's elegant constructions it is very tempting to conclude that a perfect 'root plan' was intended but was not achieved because of errors in laying out Menkaure. And indeed Tedder shows how the diagonal of Khafre is determined in such a perfect plan -

 

However, the error supposedly made by Menkaure's builders seems remarkably large (cubits of 0.5236mm) -


The east west dimension 1417.5 may be derived in another way (see note 2)


Khafre

The 345 triangle defining Khafre's profile is constructed on the diagonal of the double square. The dimensions of Khafre (granite base) are height 274 , base 411 , and slope 342.5 - thus module 137 cubits. (This is the original length of the Sphinx according to Lehner). In side elevation the monument appears to have been laid out on a 2 X 7 grid -


Most descriptions of Khafre interior rely on the measures of Maragioglio and Rinaldi. But Legon published an analysis, based on his own measurements, which is at variance with the figures of M & R.


Looking at passage angles, quoted figures for the upper descending vary from 26 degrees 28 minutes to 26d 46m with a mean of 26d 35m – the diagonal of the double square. Figures for the lower passages vary from 21d 22m to 22d 40m (!) but with a mean representing the diagonal of a 2 X 5 rectangle, (also adopted for the plans of Khafre sarcophagus and Menkaure King's Chamber). These two angles are shown in the diagram below -

With base 411 the sloping side of the pyramid measures 685 cubits. Simple constructions within the square of side 685 produce the angles of the passages 1 X 2 and 2 X 5. Quoted heights of entrance over pavement vary from 24 to 24.8 cubits (the construction gives 24.9) and appear to account for the general positions of the passages -


However the scheme does not correlate well with M & R survey data, particularly with regard to passage levels, and so a better explanation of positions must be sought.

Legon put forward a very elegant scheme, expressed in cubits, palms, and digits. Salient aspects are shown below -



Legon's scheme is persuasive because it reflects the spacing between the two large pyramids 440c + 250c = 690c, and 500d + 880d = 1380d in the passages.


According to this scheme, the distance from pyramid centre to lower passage junction is 146.35c. It is interesting to note that this distance is comparable to the distance separating Khufu centre to the AP/DP junction - in each case the distance is marked by a gabled chamber (Khufu QC and Khafre Lower Chamber).


Possibly the distance from pyramid centre to junction was intended as 146 = 2 X 73. Khafre KC measures 27 X 9.5 cubits, so circuit of walls is 73. (Alison noted that 411 X root two = 581.24. Taking this as the diameter of a circle the circumference measures = 1826. This is 5 X 365.2 or the number of days in the year.)

In this case the distance NS from Khafre south side to Khufu passage junction becomes 1027.5 = 2.5 X 411 -

These grids - 1100/440 - 625/250 - 1027.5/411 - defining Khafre from Khufu, emphasize the importance of the 2 X 5 rectangle. It is the plan of Khafre's sarcophagus (other proportions are numbered) -


 


Menkaure

 

Classical authors report that the lower part of the pyramid was cased in granite and the top in limestone, but today most of the casing has been stripped. Petrie believed that granite extended to the top of the 16th course, partly because no granite fragments were found higher than this level. However, Butler proposes that the granite/limestone junction is indicated by a change from thick to thin masonry courses. This division is visible in the core blocks -
 

 


Here is Petrie's table of course heights (in inches) -


 

And here the course heights drawn to scale and showing the break between thick and thin courses -


 
It has already been mentioned that there are many aspects of Menkaure's pyramid complex which show it was not constructed with the same care lavished on the two large pyramids. For one thing a platform was not cut before building but rather stones were laid directly on the sloping rock of the plateau (much as the satellite pyramids were built). Also the baseline has not yet been fully excavated on the west while preliminary reports suggest the pyramid might not be square but rectangular! Nevertheless the pyramid was obviously intended to be square but executed poorly, and Petrie concluded the intended base was 201.5 cubits. As to the pyramid slope, older estimates have been revised - Menkaure has the same 7/11 slope as Khufu. So the resulting height is 128 cubits.

Subtracting 39 for the granite casing gives 89 cubits for the limestone-cased part. This is the same factor found in the design of Khufu (slope 356 = 4 X 89 and in the diagram defining Khufu base (as 356 X 2 = 712) - notionally, Khufu slope = Menkaure height -

A 7/11 pyramid with a height of 89 will have a base of 140 , or 560/4. Khufu and Menkaure are linked.

 


The limestone bases

 

Since the pyramid of Menkaure is divided in two according to an underlying geometry what then of the pyramid of Khafre? Butler noticed that subtracting the 2 cubit high granite course, the limestone base became 408 cubits, and this altered the spacing between the pyramids (cubits of 523.6mm) -


On the right (in blue) the spacing according to the granite plan.

On the left (in red) the limestone bases produce the sequence 440 : 252 : 408 , or the Fibonacci terms 55 / 21 / 34.

With limestone base 408 Khafre height is 272, or 2 X 136.   Khufu slope = 356 = 220 + 136.

Butler interprets the limestone plan like this -


   

How does this plan compare with Legon's dissection?  According to Petrie's survey the granite base of  Khafre is 1101.47 cubits south of Khufu north side. ( Legon employed a cubit of 0.5237 m. thus making the distance 1101.1 ). Subtracting the granite base (height 2 width 1.5) from Butler's figure gives almost 1100 cubits . Other spacings are similar to Legon's and the small differences might be put down to setting out error, although the departure from Petrie's survey (638 for 638.74)  seems excessive. Nevertheless the limestone plan produces some surprising results.

Butler noticed that Giza pyramid chambers had not been planned on an ad hoc basis but shared common dimensions -



Plans of the main pyramid chambers are shown above to the same scale and orientation. From top left to bottom right they are in order :
 
GI SC..... Khufu subterranean chamber (unfinished)
GI QC.....Khufu Queen's chamber
GI KC.... Khufu King's sarcophagus chamber
GII KC....Khafre King's sarcophagus chamber
GII FC.....Khafre first chamber
GIII AC...Menkaure ante-chamber - leads to sarcophagus chamber.
GIII NC...Menkaure niche-chamber (unfinished)
GIII KC...Menkaure King's sarcophagus chamber


Not shown is the portcullis chamber of Khufu (often called the 'ante-chamber' , perhaps having its analogue in the vestibule of Menkaure), or the Grand Gallery of Khufu which, although its corbelled ceiling has precursors in the chamber designs of the Dashur group, is usually considered part of the passage system. Common dimensions have been employed in the designs of these floor plans as well as similar positioning of exits, and may be drawn together like this -

Butler proposed  that  these chambers were  proportioned to correspond to limestone layout dimensions when scaled up. For instance the Queen's chamber -



 

Khufu Queens chamber floor has dimensions 11 X 10 (proportion 1.1). This appears to correspond to the layout rectangle 1387 X  1262 (proportion  1.099) formed between the corners of Khufu and Khafre.

The sarcophagus chambers of Khufu and Menkaure were lined in granite. At layout scale they are symmetrically disposed with respect to the pyramid of Khafre -


 


Khufu Kings chamber has dimensions 20 X 10 cubits (proportion 2 to 1).  Layout  measures 1317 X 660  (proportion 1.995 to 1). Menkaure Kings chamber  measures 12.5 X 5 cubits  (proportion 2.5). Layout  measures 1482 X 593  (proportion 2.499).

 

Two chambers are 27 cubits long - Khafre sarcophagus chamber and Menkaure Antechamber, and having the same 22 cubit length to the entrance -


 

Menkaure antechamber is 27 X 7.3 (proportion 3.014). Layout rectangle is 1387/462 (proportion 3.021).

Khafre Kings chamber is 27 X 9,5 (proportion 2.316). Layout rectangle 1387 X 602 ( proportion 2.304).

(The perimeter of Khafre's chamber is 27 + 9.5 X 2 = 73. The total width of the limestone plan is 1387 = 19 X 73 cubits. To the west of Khafre's pyramid are two sets of  'barracks' - Butler concludes 19 east west and 73 north south. It may also be significant that a circle enclosing Khafre granite base (diameter 411 X root 2) has a circumference of 1825 = 5 X 365 = 25 X 73. )

The dimensions of the chambers do not relate only to the limestone plan. At site scale they extend 315 east of Khufu. Using a cubit of 0.5236mm the north south dimension of the granite plan is 1732.5 (factorising to 2 3 7 and 11) and east west 1417.5. So the plan has the proportions 11 : 9, confirmed by the chamber extensions -


It is possible that other chambers  have layout analogues. Khafre antechamber  measures 20 X 6 cubits (proportion  3.333)  and possibly corresponds to the rectangle 1387 X 408  (proportion  3.4). Khufu subterranean chamber  measures 26.87 (27?) X 15.89 (16?) cubits  giving the proportion 1.69 (1.688), at an outside stretch corresponding to the layout rectangle 1387 X 806 (proportion 1.72). But these are not essential to the conclusion that the planning of chamber dimensions could not have been an afterthought as each pyramid was being worked on but planned at the outset of building.


The site reflection

According to the best modern estimates the granite base of Menkaure is just over 24 cubits higher than Khufu . If the two pyramids are drawn together a sensible plan emerges -


The apex of the 'virtual' Khufu is 256 above Menkaure pavement level, or twice the built height of the pyramid. So it would seem that the dimensions of Menkaure were decided with reference to those of Khufu -


Supporting the idea that the two 7/11 pyramids are related in other ways, B. A. Hokum noted that the levels of the entrances in both Khufu and Menkaure appear to be the same. This does indeed appear to be the case -

 

In this diagram Menkaure passage system is superimposed on that of Khufu. Red passage is common to both pyramids.

 

Symmetrical elements

The pyramid of Khafre appears to mediate an architectural symmetry between the pyramids of Khufu and Menkaure, as demonstrated by the following features -


- the slopes of both Khufu (GI) and Menkaure (GIII) are defined by the gradient 14/11, while that of Khafre (GII) is 4/3.
- the faces of GI and GIII are indented while those of GII are flat.
- GI and GIII each contain three chambers and have one entrance, while GII has two chambers and two entrances.
- GI and GIII each have three satellite pyramids while GII has one.
- also, GII may be said to have two 'valley temples', inasmuch as the valley temple proper and adjoining Sphinx temple share architectural features found in the mortuary temple, though rotated through 90 degrees.

Consider also the granite tomb chambers of Khufu and Menkaure, of similar construction. At layout scale they are symmetrical with respect to Khafre. Khufu, on the right, is oriented east west. Menkaure, on the left, is oriented north south -
 


 

Temples appear to reflect this symmetry. The mortuary temples of Khufu and Menkaure (in yellow) are broadly similar in their design and contain open courts of similar proportions. Khafre mortuary temple shares elements from both the valley and Sphinx temples - (red) open courts of similar design, (blue) massive structures with T-shaped halls -


 

Inside the pyramids symmetry appears in portcullis arrangements -


 

Sometimes approximate relations are suggestive. Ross showed that the east west spacings between the pyramids are approximately one third of the corresponding spaces between pyramid centres -


 

There are further symmetrical relations which can be derived from the plan. Tedder found two Phi rectangles link the centres of Khufu and Menkaure wrt the centre of Khafre -


 

The northern rectangle has proportion  1.623 to 1, suggesting the ratio 13/8  (1.625). Southern rectangle is 1.605 suggesting the ratio 8/5 (1.6). In a similar way, divisions suggestive of Fibonacci proportions are found in Butler's plan -




These Phi proportions do not correspond to whole numbers of cubits but are perhaps something more than serendipitious. However, Fibonacci proportions generated by the limestone base of Khafre are quite precise -

Khafre 408 factor 34 - Khafre to Khufu 252 factor 21 - Khufu 440 factor 55.

Khafre 408 factor 8 - West side Khafre to centre Menkaure 255 factor 5.

 

There are also correspondences revealed by a purely geometrical approach. I noticed that symmetrical arcs could be drawn from the corners of Khufu and Menkaure to the corners of Khafre -


Similar symmetrical arcs can be constructed as here -

 

These arcs are fairly precise. Is this more serendipity or an indication of a deeper aspect to the plan?

 


The Giza diagonal

 


The diagram above shows the relation between the granite bases of Khafre and Menkaure along the diagonal. This is twice the distance from the centre of Khufu to the northeast corner of Khafre, which in turn is equal to the eastwest distance between the centres of the two large pyramids.

The developed figure is shown below. Diagonals from the northwest corner of Khufu and the southwest corner of Menkaure (granite base) define a rectangle (shown in red) with proportions 8 : 21. This rectangle is bisected by the centre of Khafre -


 

The design figures become 338 . 676 . 845 . 929.5 , all multiples of 169 = 13 X 13. Distance north/south Khufu centre to Khafre centre = 676 = 13 X 26. Centre Khufu to south wall of KC = 26. The intention seems to have been to achieve similar dimensions to the plan along cardinal axes.

 

The geometry at Giza begins with the square and its diagonal. It is well known that the Giza group was aligned to the cult centre of the pyramid age, Heliopolis across the valley, along the diagonal -



 

The recent survey of Djedefre (the pyramid of Khufu's son at Abu Rawash) established dimensions similar to Menkaure (also traces of granite casing). It was built on higher ground, possibly with aim of making its summit at the same level as Khafre. It also lies on a 45 degree alignment from Khufu -

 

 

Tedder has pointed out that the most precise azimuth from Djedefre passes through the threshold of Khufu mortuary temple. Considering the difficult terrain between Giza and Djedfefre one might not expect an exact relation - if for example the alignment was actually intended between Khufu and Djedefre centres. It may even be that a large scale plan was envisioned for the valley. The distances along the diagonal from Khufu to Djedfefre and to Heliopolis relate as 1 : 2 X root two -



 

 


Gebel Ghibli
 

 

South of the Sphinx there is a large cliff called Gebel Ghibl (there are various other spellings). Lehner remarked that it would have been an ideal spot for the builders to oversee their work and set up his own survey station there. Some years ago Hale noticed that Gebel Ghibli lay at the apex of an equitateral triangle draw from Khufu and Menkaure centres. This the view from the apex -


 


Investigating further Hale found two further constructions that appeared to define this spot -

 

- one is a construction based on the north south spacing between Khufu and Khafre. The other is a 345 triangle alignment to the centre of Khafre. At close scale these constructions appear to converge, but not with the apex of the triangle -

 

 

The reference point would then appear to be 1314.5 east from Khafre centre, and 985.9 south -


 

The east west distance 1314.5 = root 2 X 929.5 (the diagonal distance between Khafre and Khufu) according to the 'Theon' approximation 239/169. The profile of Khafre (red triangle) is given by the 345 alignment from Gebel Ghibli. It is interesting that the profile of Khufu (blue triangle) arises from the location of Menkaure -

The pyramids of Khufu and Menkaure are positioned according to simple dissection of a square of side 2629 , centred on the pyramid of Khafre.

 


The pyramids of Sneferu


Sneferu, Khufu's father, is associated with 3 large pyramids - Meidum, far to the south, and the pair at Dashur - the Bent and the Red. Sneferu could only have been 'assigned' one pyramid to be buried in, so any other monument attributed to him must have performed different functions. Whatever these may have been, with the coming of the IVth dynasty religious architecture became severe and geometric. At the same time the 'classic' pyramid complex came into being - pyramids facing the cardinal points, passages pointing towards the pole, temples facing east, and so forth. But there is some evidence that the mathematical ideas which commanded their design arose in earlier dynasties.
In a series of papers, Miatello presented data from architectural dimensions, texts, and iconography to suggest, among other things, that certain numerical sets formed the basis of pyramid design -


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Regarding the step pyramid of Zoser he lists the dimensions of successive enlargments of mastabas in the first stage of building as follows -
120 X 120. 136 X 136. 136 X 152.
- and of successive steps of the pyramid -
209 X 231. 198 X 220. 165 X 187. 132 X 154.
These terms can be factorised into simple prime numbers, just as is seen at Giza - eg. Khufu 280/440 = 7/11. And some appear at Dashur and Meidum.

Legon showed that the pyramid of Meidum was explicitly dimensioned according to the 7/11 ratio, a 5/8 model of Khufu -

The factors 7 and 11 are also found in the plan of the chambers of the Red pyramid -

Sneferu probably did not commence building at Meidum It is generally thought that he started the Bent, then the Red, and converted Meidum from a step into a true pyramid. As to why the pyramid is 'bent', the common view is that emerging weaknesses in the structure caused the builders to lower the intended slope - the lower part of the bent contains about 8 million cubic cubits and 3 million more would be required to project the sides to an apex. (The present upper part adds 0.5 million cu.cubits). The builders also laid stones horizontally in the upper part (rather than sloping as in the lower part) and horizontally in the Red and this taken as evidence of a change to a more reliable building technique. Nevertheless there is evidence that there is geometrical meaning in the division of the height.

In recent years there have been some revisions of estimates of Dashur pyramid slopes and dimensions. While it is accepted that the slope of the Bent satellite pyramid is 45 degrees, the angles for the upper part of the Bent and the Red have been thought to be something less, but now it seems they also were intended at 45 degrees. The lower slope of the Bent is more difficult to assess because it is not flat but slightly curved but is obviously intended to represent root two. If this slope is continued to an apex, the resulting pyramid will be octahedral - a diamond, each face an equilateral triangle, so base and corner slope (or 'arris') are of equal length. But which approximation for root two was used at Dashur? The 'Theon' approximation for root 2, 99/70 , was used at Khufu. The Bent base is 360 on the pavement, divisible by 12. So the appropriate Theon term is 17/12, making the virtual height 255 -



The placement of the satellite pyramid can also be explained using the 17/12 ratio. The virtual height of 255 is 'bent' in the ratio 6 : 11 (90 + 165). The apothegm of the upper slope also then measures 165 = 3 x 55 cubits. It appears that rather than planning a pyramid with a whole number height, the builders intended a whole number lower height (90) and whole number upper slope (165).

The design of the Red appears simpler. According to Miatello (2010. quoting Dorner) the base and height of the Red pyramid are both 209 cubits = 11 X 19 -

It is interesting that the lower chamber is roofed with 11 corbels and the upper with 14. The slope of the descending passage appears laid out on the diagonal of the double square with origin 88 above northbase.

The slope of the Bent pyramid descending passage (which has segments with slightly different slopes) approximates the diagonal of the double square. -


Lacking detailed survey results the above passage analyses remain tentative but do suggest that common terms appear in the two pyramids. For the Red -
7.5 15 30 / 7 14 / 22 44 16.5 33
And for the Bent -
30 90 180 75 150 / 280 / 165 330


The key point is that the Bent and Red complement one another - the slope, or 'apothegm', of the lower Bent is root 2, so the corner or 'arris' slope is 45 degrees - which is the same as the apothegm of the Red, the upper Bent, and the satellite pyramid. Perhaps an allied concept was applied in the planning of the King's chamber floor level in Khufu, or the dimensions of the 7/11 pyramids -

The layout itself presents an orthogonal plan which is rotated through 45 degrees. These examples perhaps tell us something of the designer's mindset - geometry interpreted as metaphor for rebirth.

 

Conclusions

 

The evidence presented above supports the conclusion that Giza was designed as a whole. The Giza plan does not present precise root values and would amount merely to a very expensive school textbook if it did, nor would it explain many architectural features. Rather we find curious dimensions which are multiples of prime numbers. These numbers are themselves geometrical in many cases, some being  whole number approximations to root ratios . (Others may be calendrical, as in the overall east west dimension of the limestone plan = 1387 =  19 X 73 ). The result is a set of prime factors which would seem to have had sacred significance.

But did the designers really believe that this geometrical symphony had supernatural properties capable of launching a king to the stars? Perhaps - Egyptians were steeped in superstitious belief and various ritual objects have been found at Giza. In pyramids containing the Pyramid Texts, potentially dangerous animal hieroglyphs are cut in two, or ritually 'killed'. Even so, the sophistication of the Giza project implies that its architects were rather more philosophically minded. Perhaps fascination with number and geometry became an end in itself to the priests responsible for design, evolving into exercises of geometrical symbolism.

If Giza was a project of Khufu's architects, rather than added to by a succession of kings, it does not change egyptian history - it simply means that the priesthood had much greater influence than hitherto believed. The scale of the layout project and the peerless ambition required to build it are without precedent. And it is sobering to realise that, despite much speculation, we do not know how pyramids were built.

Notes


1. Legon's circle squared scheme can be 'expanded' to the scale of the layout. The diagram below shows how  a layout based on a module of 1000 cubits and septenary dimensions relate --




- with centre (a) circle is drawn with radius 1400 cubits, or 5 times the height of Khufu, shown on the left. Next centre (b) and circle of 1120 radius. Line from (b) at 60 degrees intersects circle to produce the approximation to root three  970/560.
The  line from (a) to (d) produces the quantity 440, and that from (a) to (e) completes a rectangle measuring 1000 X 2000.  In this diagram the northern bases of Khafre and Menkaure are seperated by 840 cubits. (Note. Legon has discussed the proportion 970/560 in the layout but constructs it in a different way).

It is possible  to derive the dimension 840 using Phi ratio proportions (derived from the additive series 1.4.9.14.23.37.60.97.157.254.411, and employing Legon's elegant circle-squared scheme for Menkaure 'expanded' to site scale -

 


In this diagram the square at 45 degrees has a side of 2000 cubits and is contained in a circle of radius 1414 cubits. If this circle is squared it produces the dimension 1110, which may be divided in the phi proportions  37 : 23 + 14 (690 + 420) and  60 = 37 + 23 (851 + 1369).  A pretty diagram but probably not based on reality. Nevertheless some of the dimensions that emerge are rather interesting -




It can be seen that these dimensions share a common factor of 19. There are other interesting factors at Giza seemingly not related to Legon's plan, for instance in mortuary temple layouts -





To the left is Khufu.  The peribolus wall is not regular but on its eastern side indicates a square of 490. The temple measures 100 cubits north south and 77 cubits east west outside the peribolus wall. This produces the rectangle 102 X 170 (factor 17). On the right is Khafre. The peribolus wall measures 462 (6 X 77).  The Mortuary temple is 88 north south, producing the dimension 187 (11 X 17).

 

Note 2. If the figure 1417.5 was actually intended then a possible solution to its construction is shown below -



- the construction is made within a square of 2500 cubits  and highlights the root 3 rectangle  250 X 433  (1000 X 1732 divided by 4) seperating the bases of Khufu and Khafre.


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