A Vision of Early Egypt (2/4) / © 1991-2001 by Franz
Gnaedinger,
Egypt 1 / Egypt 2 / Egypt 3 / Egypt 4
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
0 River
Nile, rising Nile - Osiris (Plutarch)
0
0 Water
system of the Nile including canals, harbors
0 dams,
dikes, basins and pyramid lakes - living Osiris
0
0 Sahu/Orion - heavenly appearance of Osiris
(Rolf Krauss)
0
0 Djed Sneferu - Maidum pyramid
0 left hand of Sahu - Pleiades
0
0 (Lisht)
kha-channel --- band of ecliptic (Rolf Krauss)
0
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
0 Saqqara pyramids - head of Sahu - Heka
0
0 Pyramid at Zawyet el-Aryan
0 left shoulder of Sahu - Bellatrix (Robert
Bauval)
0
0 Giza Pyramids - belt of Sahu (Robert Bauval)
0
0 Djedefre's pyramid at Abu Rawash
0 left upper tigh of Sahu - Rigel (Robert
Bauval)
0
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
171-228-285
or 57 times
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
0
oooooooooooooooooooooooooooooooooooooooooooo
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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