A Vision of Early Egypt (2/4) / © 1991-2001 by Franz Gnaedinger, Zurich, fg(a)seshat.ch, fgn(a)bluemail.ch / www.seshat.ch


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Part 2: Pyramid program of the Old Kingdom / Harbor of the heavenly kha-channel (Sneferu’s pyramids) / Hemon’s masterpiece (Great Pyramid) / Long before Archimedes (calculating pi)




7) Pyramid program of the Old Kingdom  (provisional version, to be corrected)


Imhotep, ‘chief of all the works of the King,’ Grand Vizier, Architect, Chief Lecture Priest or Kheri-heb, Sage and Scribe, Astronomer, and Magician-Physician (Jamieson B. Hurry), designed the monumental and gracious Djoser Complex at Saqqara. In the center he placed a large mastaba, which he then evolved into a step pyramid, as a royal tomb and a spiritual building meant to transform the king’s ba (soul) into the god Re-Osiris.


The artificial hill on the western bank of the river Nile symbolized the Primeval Hill that rose above the Primeval Water Nu(n) and released both the sky Hathor/Nut and the sun Re. The step pyramid also symbolized a star: Heka, lambda Orionis, head of Sahu; thus linking the earthly incarnation of Osiris, the majestic river Nile, with his heavenly abode, the magnificent constellation of Sahu/Orion.


The first pyramid was followed by many more. The successors of Imhotep expanded upon his ideas and designed a ‘stream’ of pyramids along the Nile which linked the river to further stars in Orion and Taurus 




     0 River Nile, rising Nile - Osiris (Plutarch)


     0 Water system of the Nile including canals, harbors

     0 dams, dikes, basins and pyramid lakes - living Osiris


     0   Sahu/Orion - heavenly appearance of Osiris (Rolf Krauss)


     0            Djed Sneferu - Maidum pyramid

     0            left hand of Sahu - Pleiades


     0   (Lisht)  kha-channel --- band of ecliptic (Rolf Krauss)


     0        Sneferu's Southern Epiphany - Dahshur

     0        left elbow of Sahu - Epsilon Tauri (Robert Bauval)

     0        Sneferu's Northern Epiphany - Dahshur

     0        left elbow of Sahu - Aldebaran (Robert Bauval)


     0   Saqqara pyramids - head of Sahu - Heka


     0     Pyramid at Zawyet el-Aryan

     0     left shoulder of Sahu - Bellatrix (Robert Bauval)


     0   Giza Pyramids - belt of Sahu (Robert Bauval)


     0     Djedefre's pyramid at Abu Rawash

     0     left upper tigh of Sahu - Rigel (Robert Bauval)





8) Harbor of the heavenly kha-channel (Sneferu's pyramids)


Sneferu, as a young king, wished to build a pair of pyramids that surpass the Djoser Complex by Imhotep. The new pyramids should represent Orion's raised arm and be the earthly harbor of the heavenly kha channel: Hyads and Pleiades (called Golden Door by the astronomers of Babylon) 


Sneferu's architect planned a step pyramid for Meidum and a bent pyramid for Dahshur, whose peculiar shape should evoke the Hyads. Here are the numbers of the (first) bent pyramid (base, angle of lower slope and heights of entrances according to Josef Dorner; bending lines and angle of upper slope added by me):


  base                300 by 300 royal cubits

  bending lines       174 by 174 royal cubits


  lower height         98 royal cubits

  run                  63 royal cubits

  slope  practically  116 1/2 royal cubits


   sekad  18 fingers  (tangent slope 14/9)


  upper height         87 royal cubits

  run                  87 royal cubits

  slope  practically  123 royal cubits


   sekad  1 royal cubit  (tangent slope 1/1)


  entrances of gangways  10 and 50 royal cubits above base


Grid of base and bending lines:


  63+174+63  by  63+174+63  royal cubits


The grid consists of squares and rectangles whose diagonals measure practically 89, 174 and 185 royal cubits, what makes the measuring out of the base an easy task.


Sneferu approved of the plans, and the pyramids were built.


The king was in excellent health, hoping to live much longer. His fame grew and grew. He was no longer satisfied with the small pyramids, and so he asked his architect to enlarge them. Now the architect had a highly gifted pupil, a mathematical genius by the name of Hemon, a cousin of Sneferu's son Khufu. The king's architect and Hemon studied the problem and decided to add more steps and layers to the Maidum pyramid, while they found similar solutions for an enlarged bent pyramid, which, however, based on different numbers 


Solution by Sneferu's architect (actual Bent Pyramid of Dahshur, numbers given by Josef Dorner, lower slope and upper edge added by me):


  base                 362 by 362 royal cubits

  bending lines        236 by 236 royal cubits


  lower height          90 royal cubits

  run                   63 royal cubits

  lower slope  nearly  110 royal cubits


  upper height         110 royal cubits

  run                  118 royal cubits

  upper edge   nearly  200 royal cubits


  upper height         110 royal cubits

  lower slope  nearly  110 royal cubits


  total height         200 royal cubits

  upper edge   nearly  200 royal cubits


Solution by Hemon:


  temenos             570 by 570 royal cubits

  base                360 by 360 royal cubits

  bending lines       228 by 228 royal cubits


  lower height         99 royal cubits

  run                  66 royal cubits

  slope  practically  119 royal cubits

  edge   practically  136 royal cubits


  upper height        108 royal cubits

  run                 114 royal cubits

  slope  practically  157 royal cubits

  edge   practically  194 royal cubits


Grid of temenos, base and bending lines:


  105+66+228+66+105  by  105+66+228+66+105  royal cubits


containing the triples 


  66-360-366  or   6 times 11-60-61

 175-600-625  or  25 times  7-24-25

 171-228-285  or  57 times   3-4-5


Several rectangles of the grid have diagonals that measure exactly 285, 366 and 625 royal cubits, while the key-number 57 can be used for astronomical reasons 


                                             0  15 Re marks



         57 fingers or 114 Re marks          0

                                             0  15 Re marks


This very simple device allows to measure small angles from half a degree up to 15 degrees (accordingly, the circles on the astronomical ceiling of Senenmut's tomb at Deir el-Bahari are divided into ideally 24 times 15 degrees). The same device can be used as an astronomical clock: angle of the sun and the moon 1 Re mark on 57 fingers each, 15 Re marks = half an hour, 30 Re marks = 15 fingers = 1 hour 


Sneferu: I like your plans for the enlarged Bent Pyramid. However, you pose me a difficult problem. Both solutions are excellent. Now which one shall I choose? How shall I decide? It should be a fair and wise decision, worthy of a king of my rank. Well, your numbers differ, but the drawings look the same, the differences are so small that no one would recognize them, and so I ask you, my dear architect, build your version. Now for you, Hemon. You gave me a proof of your genius, and so you shall conceive my cult pyramid south of the Bent Pyramid. If I like it, you may perhaps build a third large pyramid for me, representing Aldebaran ...


Hemon conceived this cult pyramid: base 100 royal cubits, height 49 royal cubits, slope practically 70 royal cubits, radius of inscribed hemisphere practically 35 royal cubits. The small pyramid was a symbol of the Primeval Hill, while the imaginary hemisphere symbolized the heaven once enclosed in the Primeval Hill (Nut, bending over the earth) 


Sneferu was charmed by this elegant small pyramid. He asked Hemon to plan a third large pyramid for him, representing Aldebaran. Hemon found a wonderful solution: base 420 royal cubits (Rainer Stadelmann), height 200 royal cubits (Stadelmann), slope exactly 290 royal cubits, diagonal of base practically 594 royal cubits, edge practically 358 royal cubits, radius of inscribed sphere exactly 84 royal cubits. The imaginary sphere symbolized the sun once enclosed in the Primeval Hill while the reddish core blocks remind of Aldebaran. In the high chambers were stored three gilded sun barks, allowing the soul of the deified king to go on its heavenly journey, starting from the earthly harbor of the heavenly kha-channel, uplifted by the strong arm of Osiris ... 




9) Hemon's masterpiece (Great Pyramid)


Khufu, son and successor of Sneferu, asked Hemon to plan a new pyramid, and this building was to become Hemon's masterpiece: akhet Khufu, Cheops’ horizon, standing on the former hill sanctuary at Giza 


Hemon combined the royal cubit (308 Horus marks or 52.36 centimeters) with a Horus cubit of 7/11 royal cubits (196 Horus marks or 33.32 centimeters) and thus obtained simple numbers:


  height of pyramid model     1 Horus cubit (divine measure)

  base                        1 royal cubit (human measure)


  height of cult pyramid     40 Horus cubits

  base                       40 royal cubits


  height of Great Pyramid   440 Horus cubits

  base                      440 royal cubits


  height of Great Pyramid   280 royal cubits

  half base                 220 royal cubits

  slope        practically  356 royal cubits


The Great Pyramid rising above the Nile symbolized the Primeval Hill, which rose above the Primeval Water Nu(n) and gave birth to the sky and the sun. The building was shaped in such a way as to evoke the Great Ennead of Heliopolis and further Egyptian deities and cosmological concepts:


PRIMEVAL ONE --- symbolized by the seemingly simple shape of the pyramid


NUMBER 2 --- lower or heighten the pyramid by a few fingers and you obtain a pair of most demanding pyramids based on the golden number and on the number of the circle


(Atum-)RE, supreme sun god, god of cosmos appearing in the sun, hieroglyph a circle --- present in the imaginary Taylor circle whose vertical diameter is given by the original height of the pyramid. The circle symbolizes the sun rising from the Primeval Hill. Heighten the building by a few fingers and the area of the circle equals the one of the cross-section - symbolically turning the pyramid (which was seen as the very body of the deified king) in the solar disk of Re


SHU, god of air and light, holding the starry body of Nut high above the ground --- his raised arms were symbolized by the shafts of the King's Chamber, while the five rays of an Egyptian star correspond to the five fingers of a hand


TEFNUT, goddess of warmth and moisture --- her raised arms were symbolized by the shafts of the so-called Queen's Chamber. As goddess of water and fire she was also present in the pit in the lowest chamber: as water of life rejuvenating the worthy king; as fire destroying an unworthy soul. The pit was filled with Nile water for ritual reasons and now also represented the Primeval Water Nu(n) in the Primeval Darkness Keku Semau, while the blind gangway in the same chamber deep down in the rock was a symbol of the Primeval Snake, which, coming forth and seizing its tail by its mouth, created a round universe and reversed the cycle of time, thus rejuvenating the king again. The ridges and ditches in the western part of this chamber symbolize the lowest part of the heavenly kha-channel


GEB, god of the earth --- present in the pyramid's base, in the hill of nummulite limestone at the base of the building (a former sanctuary of the Primeval Goddess and her wise and charming daughters, who live on in Nut, Isis and Nephtys)


NUT, goddess of the heaven, alter ego of Hathor, bending over the earth --- present in the imaginary hemisphere in the frame of the Golden Pyramid (which is the Great Pyramid lowered by a few fingers). A chamber on the zenith of the imaginary hemisphere symbolized the womb of the heavenly mother goddess wherein the soul of the deified king would be raised as the sun child before leaving the pyramid as sun god and traveling across the sky in a sun bark. Three gilded barks were stored in the Great Gallery; further boats were stored in the pits at the base of the Great Pyramid. The radius of the imaginary hemisphere measures practically 173 royal cubits, according to the golden sequence 3x3 = 9, 4x4 = 16, 5x5 = 25, 41, 66, 107, 173, 280 . . .


OSIRIS, ISIS, SETH, NEPHTYS, children of Geb and Nut --- present in the four pyramid faces; the chamber deep down in the rock is the chamber of Sokar (a mysterious form of Osiris)


HORUS THE ELDER, sun god, alter ego of Re, another child of Geb and Nut, also HORUS THE YOUNGER, child of Isis and Osiris --- present in the gilded pyramidion



Symbolic presence of Nut and Re in the Great Pyramid 




10) Long before Archimedes (calculating pi)


Re's hieroglyph, a small circle, symbolized the solar disk. Every property of a person was believed to be a part of his or her very being. Thus the circle was more than a symbol of the solar disk, it was Re, while drawing and measuring a circle and calculating the secret number extant in the perfect form of the ideal circle was a way to understand Re and to participate in his power.


Now let me demonstrate how ingeniously Hemon approximated the secret number of the circle. Here is the key figure of his method which makes use of the Sacred Triangle 3-4-5:



          . . . . . d . . . . .

          . . e . . . . . c . .

          . f . . . . . . . b .

          . . . . . . . . . . .

          . . . . . . . . . . .

          g . . . . + . . . . a

          . . . . . . . . . . .

          . . . . . . . . . . .

          . h . . . . . . . l .

          . . i . . . . . k . .

          . . . . . j . . . . .



The side of the square measures 10 royal cubits or 70 palms or 280 fingers. The diagonal measures practically 99 palms. The 12 points a b c d e f g h i j k l mark the circumference of a circle whose radius measures 5 royal cubits. The 4 short arcs measure practically 40 fingers each, the 8 longer arcs measure practically 90 fingers each, yielding a circumference of practically 880 fingers or 220 palms and the approximate value 22/7 or 3 1/7 or 3 '7 for the number of the circle.


The above key-figure can be developed into a systematic method for calculating pi.


Please imagine a grid measuring 10 x 10, 50 x 50, 250 x 250, 1250 x 1250 ... ever smaller units. The radius of the inscribed circle measures 5, 25, 125, 625 ... units. The circumference passes the 4 ending points of the axes, furthermore 8, 16, 24, 32 ... inner points of the grid whose distances from the axes and from the center of the grid are defined by the following triples:


  3-4-5   15-20-25   75-100-125   375-500-625   ...

           7-24-25   35-120-125   175-600-625   ...

                     44-117-125   220-585-625   ...

                                  336-527-625   ...



If you know a triple a-b-c and wish to find the next one you may calculate these terms:


  +- 4b +- 3a     +- 3b +- 4a     5c


The first terms provide four results each. Use the positive numbers that end on 1, 2, 3, 4, 6, 7, 8, 9 (neither 0 nor 5).


By connecting the 12, 20, 28, 36 ... points of the grid you will obtain a sequence of polygons. Their side lengths are whole number multiples of the square roots of 2 or 5 or 2x5.


The square roots of 2 and 5 are easily approximated by means of the following number patterns. Add a pair of numbers and you obtain the number below, double the first number of a line and you obtain the last number:


    1       1       2

         2       3       4

             5       7      10

                12      17      24

                    29      41      58

                        70      99     140

                           ...     ...     ...


If the side of a square measures 7 palms, the diagonal measures about 10 palms. If the side of a square measures 10 royal cubits  or 70 palms, the diagonal measures practically 99 palms. - Multiply the first number by a factor of 5. If possible divide all numbers of a line by 2:


  1       1       5

      2       6      10

      1       3       5

          4       8      20

          2       4      10

          1       2       5

              3       7      15

                 10      22      50

                  5      11      25

                     16      36      80

                      8      18      40

                      4       9      20

                         13      29      65

                             42      94     210

                             21      47     105

                                 68     152     340

                                 34      76     170

                                 17      38      85

                                     55     123     275

                                        178     398     890

                                         89     199     445

                                            288     644    1440

                                            144     322     720

                                             72     161     360


If a double square measures 4 by 8 palms, the diagonals measure nearly 9 palms, and if a double square measures 72 by 144 palms, the diagonals measure practically 161 palms. By the way, the above numbers contain two golden sequences, namely the so-called Fibonacci sequence and the so-called Lucas sequence:


  Ls   1  3  4  7  11  18  29  47  76  123  199  322  ...

  Fs   1  1  2  3   5   8  13  21  34   55   89  144  ...


The sides of a polygon are slightly smaller than the arcs of the circumscribed circle. We may counterbalance this by using ratios for the square roots of 2 and 5 that are slightly greater than their values. For example 10/7, 17/12 and 9/4:


  10/7   x  10/7   =  100/49    a little more than 2

  17/12  x  17/12  =  289/144   a little more than 2

   9/4   x   9/4   =   81/16    a little more than 5


Calculate the circumference of the first polygon by means of the values 10/7 and 9/4, and that of the second polygon using the ratios 17/12 and again 9/4. You will obtain simple numbers and the very fine approximate values 22/7 and 157/50 for the number of the circle. Their average is about 311/99. Using these numbers you can generate a sequence of many more values. Write 3 above 1 and add continuously 22 above 7:


  3  (plus 22)  25  47  69  ...  157  ...  311  333  355  377

  1  (plus  7)   8  15  22  ...   50  ...   99  106  113  120


By the way: the sun god Re had many names, but no one knew his true name …


Key figure of a systematic method for calculating π, which I discovered or rediscovered in February 1994 and which I ascribe to the school of Imhotep, to Hemon in particular. The mathematical correctness of that method was kindly confirmed by Dr. Christoph Pöppe from the University of Heidelberg 


A sequence of irregular polygons, based on the Sacred Triangle 3-4-5 and a sequence of Imhotepean triples (7-24-25, 44-117-125, 336-527-625) 


Paleolithic patterns; according to Marie E.P. König, the round forms symbolized heaven, while the crosses represented the axes East-West and South-North, and the grids the Houses of Heaven (free drawings after photographs in the books by Marie E.P. König) 





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