46 Lessons in Early Geometry, part 6/10 / provisional version in freestyle English / a corrected version will follow in March, April or May (hopefully) / Franz Gnaedinger / February 2003 / www.seshat.ch

Lesson 25

Isaac Newton found that the King's Chamber in the Great Pyramid at Giza measures 10 by 20 royal cubits.

Jean-Philip Lauer, who spent seventy years of his long life restoring the monumental and graceful Djoser-Complex at Saqqara, discovered an imaginary Sacred Triangle measuring 15-20-25 royal cubits in the same chamber:

diagonal of the small wall  15 royal cubits   3 x 5 rc

length of the chamber       20 royal cubits   4 x 5 rc

cubic diagonal              25 royal cubits   5 x 5 rc

The diameter of the inscribed circle measures

15 plus 20 minus 25  =  10 royal cubits

Using the square 10 by 10 royal cubits, which is contained in the double square of the King's Chamber, the inscribed circle of the same diameter, the grid 10x10 royal cubits and the Sacred Triangle 3-4-5 we have established the pi-value 22/7.

One John Taylor introduced an imaginary circle whose vertical diameter is given by the former height of the Great Pyramid.

Let us calculate the area of this circle.

The base of the GP measured 440 royal cubits. The sekad measures 5 palms 2 fingers: while ascending by 1 royal cubit or 7 palms or 28 fingers a face recedes by 22 fingers or 5 palms 2 fingers. This measurement of the slope yields a (former) height of 280 royal cubits.

The radius of the Taylor circle measures 140 royal cubits. For the area we obtain

140 rc x 140 rc x 22/7 = 61,600 square cubits

Now let us calculate the area of the cross-section:

'2 x 280 rc x 440 rc = 61,600 square cubits

The cross-section and the imaginary circle have the same area

area cross-section = area of the imaginary circle

The imaginary circle can be seen as a transformation of the pyramid's triangle, and this would have a symbolical meaning.

An Egyptian pyramid was believed to be the body of the deified king, so the above equation may be understood as Pharaoh's transformation into Ra or Re, god of nature who appeared in the sun and whose hieroglyph was a circle.

Furthermore, an Egyptian pyramid symbolized the Primeval Hill that gave rise to all thing that are, first of all to the sky and the sun.

A beautiful quote from Mark Twain, The Innocents Abroad (Signet Classic), may show that the seemingly simple shape of a pyramid is able to change, at least in the eye of a beholder, and so it might well have been able of any transformation in the mind of an ancient believer:

"At a distance of a few miles the pyramids rising above the palms looked very clean-cut, very grand and imposing, and very soft and filmy as well. They swam in a rich haze that took from them all suggestions of unfeeling stone and made them seem only the airy nothings of a dream - structures which might blossom into tiers of vague arches or ornate colonnades maybe, and change and change again into all graceful forms of architecture, while we looked, and then melt deliciously away and blend with the tremulous atmosphere."

Lesson 26

The Great Pyramid combines a pair of pyramids, namely the 'solar' pyramid according to John Taylor, and the 'golden' pyramid according to an information Herodotus received from an Egyptian priest.

Solar pyramid: the cross-section and the imaginary circle whose vertical diameter is given by the height have the same area. Base 440 royal cubits, height 280.01129... royal cubits or 5.9  centimeters more than 280 royal cubits. Using the value 22/7 for pi we obtain a height of 280 royal cubits.

Golden pyramid: the area of a face equals the squared height. The golden minor of the slope equals half the base, while the golden major of the height serves as radius of the inscribed arc or hemisphere. Base 440 royal cubits, height 279.84429... royal cubits or 8.15 centimeters less than 280 royal cubits. Using the pseudo-triple 70-55-89 we obtain a height of 4x70 = 280 rc, a base of 8x55 = 440 rc, and a slope of 4x89 = 356 rc, while the radius of the imaginary hemisphere measures 173 royal cubits, according to the golden sequence

9 + 16 = 25

16 + 25 = 41

25 + 41 = 66

41 + 66 = 107

66 + 107 = 173

107 + 173 = 280

The points wherein the hemisphere touches the four faces divide the oblique heights into 136 + 220 = 280 royal cubits according to this golden sequence:

4, 4, 8, 12, 20, 32, 52, 84, 136, 220, 356

Also the imaginary hemisphere might have a symbolical meaning: it would represent the sky once enclosed in the Primeval Hill; or (as an arc) the sky goddess Nut standing on one horizon, bending her body and placing her hands on the opposite horizon.

Going a step further I like to say that not only Ra/Re and Nut but the entire Ennead and Horus were symbolically present in the Great Pyramid:

RA or RE --- present in the imaginary circle whose vertical diameter is given by the pyramid's height

SHU, god of air and light --- his raised arms are symbolized by the shafts of the King's Chamber

TEFNUT, goddess of fire and moisture --- her raised arms are symbolized by the shafts of the so-called Queen's Chamber

GEB, god of the earth --- present in the naturally grown limestone hill at the pyramid's base, or in the base itself

NUT, goddess of the sky, wife of Geb, bending her starry body over the earth --- present in the imaginary hemisphere

OSIRIS, ISIS, SETH, NEPHTYS, children of Geb and Nut --- present in the eastern, southern, western, northern face respectively

HORUS --- present in the former pyramidion

Early geometry was guided by symbols (a firm belief of mine). We have the square of the base: Geb. We have the four faces: Osiris, Isis, Seth, Nephtys. And we have an imaginary hemisphere standing on the base and supporting the faces: Nut, wife of Geb, mother of Osiris and Isis, Seth and Nephtys ... A mythological family giving rise to a geometrical ensemble, so to speak.

Lesson 27

Imagine a model of the Great Pyramid whose base measures 1 royal cubit. How high is the model? 7/11 royal cubit. Let me call this measurement a Horus cubit and use it as a complementary measure:

7 royal cubits equal 11 Horus cubits

royal cubit  52.36 cm     Horus cubit  33.36 cm

Using both measures we can define several geometrical shapes quite easily and amazingly accurately:

diameter of a circle 1 Horus cubit, circumference 2 royal cubits

radius of a circle 1 Horus cubit, area 2 Horus cubits x 1 royal cubit

diameter of a sphere 1 Horus cubit, surface 2 Horus cubits x 1 royal cubit

diameter of a sphere 1 Horus cubit, volume of 3 spheres 1 Hc x 1 Hc x 1 rc

side of a square 10 Horus cubits, diagonal 9 royal cubits

side of a square 9 royal cubits, diagonal 20 Horus cubits

a length 5 royal cubits, golden minor 3 Horus cubits

The Great Pyramid combines a pair of pyramids, and so does the Grand Gallery.

The integral golden pyramid is based on the pseudo-triple 70-55-89: height 4x70 rc, half base 4x55 rc, slope 4x89 rc. Use the 89 royal cubits as the slope (ceiling) of the Grand Gallery and know that the tangent of the angle of the rise measures slightly less than 1/2. You will obtain these numbers:

rise 39 rc   run 80 rc   slope 89 rc   triple 39-80-89

The height of the Great Pyramid measures 280 royal cubits or 440 Horus cubits. Divide it by the number of the circle 22/7 and you obtain 140 Horus cubits. Use this measurement as the slope (ceiling) of the Grand Gallery and you obtain these numbers:

rise 39.2 rc   run 80 rc   slope 140 Hc

cosine  44/49 = 4rc/7Hc =  4 palms / 1 Horus cubit

Using a small unit measuring 1/55 rc = 1/35 Hc we obtain a fascinating triangle:

oblique height 44x44  rise 44x49  run 44x100  slope 49x100

based on the pseudo-triple 539-1100-1225

The ceilings of the ideal galleries measure 89 royal cubits or 46.60 meters, and 140 Horus cubits or 46.648 meters, with  an average length of 46.624 meters. The length of the gallery built measures 46.71 meters according to Rainer Stadelmann or 46.63 meters according to Karlheinz Schuessler. The difference between the ideal average length and the actual length as given by Schuessler measures only 6 millimeters.

The ideal galleries rise by 39 and 39.2 on 80 royal cubits, with an average rise of 39.1 on 80 royal cubits, yielding an angle of 26 degrees 2 minutes 49.46 seconds. According to Stadelmann and Schüssler the actual angle measures 26 degrees 2 minutes 30 seconds. The difference between the ideal angle of the combined gallery and the actual angle of the gallery built is less than 20 seconds or 1/3 minute or 1/180 degree.

Lesson 28

I ascribe my method of calculating the circle to the school of Imhotep, more precisely to Hemon, a cousin of Khufu's.

We have only a tiny ivory figurine of Khufu (Cairo Museum), while a statue of Hemon is kept at the Egyptological Museum of Heidelberg.

Hemon, I assume, designed the former cult pyramid to the Bent Pyramid at Dahshur South, the Red Pyramid at Dahshur North, and his masterwork the Great Pyramid at Giza.

Former cult pyramid at Dahshur South: base 100 royal cubits (Rainer Stadelmann), probable height 49 royal cubits, slope practically 70 royal cubits, edge practically 86 royal cubits, radius of the inscribed hemisphere practically 35 royal cubits.

The imaginary hemisphere symbolizes the sky once enclosed in the Primeval Hill, or (as an arc) the sky goddess Nut standing on one horizon, bending her body over the pyramid base (symbol of the earth, her husband Geb) and placing her hands on the opposite horizon.

Red Pyramid: base 420 rc (Rainer Stadelmann), height 200 rc (Stadelmann), slope 290 rc (triple 20-21-29), edge practically 358 rc, diagonal of the base practically 594 rc, radius of the inscribed sphere 84 rc (triple 20-21-29). The imaginary sphere would symbolize the sun once enclosed in the Primeval Hill.

The center of the imaginary sphere, 84 royal cubits above the base, may hold a statue of Sneferu, Khufu's father. The height of the floor of the King's Chamber in Khufu's pyramid measures again 84 royal cubits, in my opinion a reference to Sneferu.

There may also be some statues inside of the Great Pyramid.

At the upper end of each lower shaft I expect 10 blocks and 10 purification chambers for the Pharaoh's ba (soul) with a total length of 10 royal cubits, followed by a stone chest containing a 7 Horus cubits or 233.32 cm tall statue of Khufu facing Cygnus, Deneb and Draco in the North-East and Sirius in the South.

A further statue of Khufu may be found on top of the imaginary hemisphere, where I expect a Sun Chamber of these measurements: length 20 Horus cubits, width 16 Horus cubits, lateral height 12 Horus cubits, central height 15 royal cubits. Height of the floor above the base 272 Horus cubits (according to the golden sequence 8, 8, 16, 24, 40, 64, 104, 168, 272, 440) or about  90.63 meters. The statue would show Khufu as the Sun Child, again 7 Horus cubits or 233.32 cm tall.

Horus was a falcon. The Horus cubit measures 33.32 cm and corresponds to the length of a kestrel or windhover. The seven Horus cubits would be a special measurement: combining a sacred number, namely 7, with a holy measure, namely the Horus cubit.

Let me repeat my belief that early geometry was guided by symbols.

Lesson 29

A single pyramid standing on the western bank of the river Nile symbolized the Primeval Hill rising above the Primeval Water Nu(n) and releasing the sky Hathor/Nut and the sun Ra or Re, while the 'stream' of pyramids along the river linked the earthly presence of Osiris, namely the Nile, whith his heavenly abode, namely Sahu/Orion:

O Nile, earthly presence of Osiris (Plutarch)

O

O

O                        raised left hand of Sahu

O

O (Lisht) ---- kha-channel --- ecliptic (Rolf Krauss)

O

O               Bent Pyramid, Hyads (Robert Bauval)

O               Red Pyramid, Aldebaran (Robert Bauval)

O               left elbow of Sahu

O

O Saqqara pyramids, Heka

O

O     Zawyet el-Aryan, Bellatrix (Robert Bauval)

O     left shoulder of Sahu

O

O

O Giza pyramids, Orion belt (Robert Bauval)

O belt of Sahu

O

O

O     Abu Rawash, Rigel (Robert Bauval)

O     left upper tigh of Sahu

O

Pharaoh hoped to be reborn as Re-Osiris, watching over his people during day as the sun, during night as Orion.

Sneferu's pyramids (Maidum, Dahshur) may be seen as the earthly harbor of the heavenly kha-channel (band of the ecliptic, Rolf Krauss), where the deified king went on his heavenly journey, uplifted by the strong arm of Osiris.

Horus the Elder was an alter ego of Re and went along with him in Re-Harakhty, while Horus the Younger was the son of Osiris and Isis, who protected Osiris during his transformation from the king of the Nile to Sahu/Orion ruler of the night sky and the duat (lower sky, below the ecliptic, Rolf Krauss).

Horus in the pyramidion and in the Horus cubit had the same task: protecting the king during the dangerous time of his transformation from the earthly king of Lower and Upper Egypt to the king of the Lower and Upper Sky.

Hemon, I believe, invented not only one but seven Horus cubits of about the same length: A 7/11, B 12/19, C 13/20, D 18/29, E 22/35, F 23/36, G 41/66 royal cubits.

Lesson 30

Hemon, I believe, invented seven Horus cubits, which allowed him to solve several problems that involve irrational numbers, and which, moreover, evoke the presence of the falcon god Horus, whose task it was to protected Pharaoh during his precarious transformation from the king of the Nile valley to Re-Osiris, ruler of the heavens.

The seven Horus cubits measure A 7/11, B 12/19, C 13/20, D 18/29, E 22/35, F 23/36, G 41/66 royal cubits.

Let me explain what we can do with the new Horus cubits:

if a rectangle measures 1 by 3 Horus cubits B, the diagonal measures 2 royal cubits

if a rectangle measures 1 by 3 royal cubits, the diagonal measures 5 Horus cubits B

if the diameter of a circle measures 6 royal cubits,

the circumference measures 29 Horus cubits C

if the diameter of a circle measures 20 Horus cubits D,

the circumference measures 39 royal cubits

if the diameter of a circle measures 1 royal cubit,

the circumference measures 5 Horus cubits E

if the side of a square measures 4 royal cubits, the diagonal measures 9 Horus cubits E

if the side of a square measures 9 Horus cubits E, the diagonal measures 8 royal cubits

if a rectangle measures 2 by 4 royal cubits, the diagonal measures 7 Horus cubits F

if a rectangle measures 7 by 14 Horus cubits F, the diagonal measures 10 royal cubits

if a rectangle measures 5 by 10 royal cubits, the diagonal mesures 18 Horus cubits G

if a rectangle measures 18 by 36 Horus cubits G, the diagonal measures 25 royal cubits

The King's Chamber measures 10 by 20 royal cubits, the diagonal measures 36 Horus cubits G, and the height 18 Horus cubits G.

Imagine a sphere holding the King's Chamber. Its diameter measures 25 royal cubits, while the circumference of the large circle measures 125 Horus cubits E.

Imagine a sphere holding the hypothetical Sun Chamber. Its diagonal measures 18 royal cubits, while the circumference of the large circle measures 87 Horus cubits C, with a tiny mistake of less than two millimeters.

Lesson 31

Today I like to show you that I derived the seven hypothetical Horus cubits from the sarcophagus tub of rose granite in the King's Chamber of the Great Pyramid. Measurements in centimeters given by Rainer Stadelmann:

outer length      227.6 cm     royal cubit rc = 52.36 cm

7 Horus cubits G  227.687 cm   HcG = 41/66 rc

7 Horus cubits D  227.495 cm   HcD = 18/29 rc

3 Horus cubits E   98.736 cm   HcE = 22/35 rc

outer height         105.1 cm

22/7 Horus cubits F  105.136 cm   HcF = 23/36 rc

hypothetical outer height of the missing lid  6/7 HcF

hypothetical height ot the closed sarcophagus  4 HcF

inner length      198.3 cm     royal cubit rc 52.36 cm

6 Horus cubits B  198.417 cm   HcB = 12/19 rc

width              68.1 cm

2 Horus cubits C   68.068 cm   HcC = 13/20 rc

inner height         87.4 cm

21/8 Horus cubits A  87.465 cm   HcA = 7/11 rc

hypothetical inner height of the missing lid  3/8 HcA

hypothetical inner height of the sarcophagus  3 HcA

Outer and inner measurements of the closed sarcophagus:

3 Horus cubits E x 4 Horus cubits F x 7 Horus cubits G (D)

2 Horus cubits C x 3 Horus cubits A x 6 Horus cubits B

or simply

3 by 4 by 7 Horus cubits

2 by 3 by 6 Horus cubits  cubic diagonal 7 Horus cubits

If my reconstruction of the missing lid is correct, and if we consider the Horus cubits as a single but varying measure belonging to the god Horus, the closed sarcophagus measured 3 by 4 by 7 Horus cubits, the cavity measured 2 by 3 by 6 Horus cubits, and the cubic diagonal of the cavity measured 7 Horus cubits, according to the quadruple 2-3-6-7.

The chamber door is too small for the sarcophagus, which must have been placed in the King's Chamber while the pyramid was under construction. The careful dimensioning of the sarcophagus shows that the Great Pyramid actually was a tomb.

Lesson 32

The Italian measures can easily be combined as follows (I = Imperium Romanum, R = Rome of the Renaissance, F = Florence of the Renaissance):

I mille passuum       140,000 e   148,552.727... cm

I stadion              17,500 e    18,569.090... cm

I actus                 3,360 e     3,565.265... cm

I pertica decempeda       280 e       297.105... cm

R canna Romana            210 e       222.829... cm

I passus                  140 e       148.552... cm

I gradus                   70 e        74.276... cm

F braccio Fiorentino ----- 55 e ------ 58.36 cm  gauge measure

I cubitus                  42 e        44.565... cm

I palmipes                 35 e        37.138... cm

I pes                      28 e        29.710... cm

R palmo Romano             21 e        22.282... cm

I palmus                    7 e         7.427... cm

I digitus                 7/4 e         1.856... cm

e         1.061090909... cm

The combined measures allow practical formulas:

diameter of a circle 1 palmipes, circumference 2 bracci Fiorentini

side of a square 10 palmipedes, diagonal 9 bracci Fiorentini

golden minor of 1 braccio Fiorentino = 1 palmo Romano

side of an equilateral triangle 49 palmipedes, height 27 bracci Fiorentini

radius of a circle 1 braccio, side of the inscribed decagon 1 braccio minus 1 palmo Romano

radius of a circle 11 bracci Fiorentini, side of the inscribed heptagon 15 palmipedes

Using the combined measures, the above formulas, and the polygons provided by the triples 3-4-5, 7-24-25, 44-117-125 … one can quite easily reconstruct the virtual buildings in the Renaissance book Hypnerotomachia Poliphili.

Leon Battista Alberti, Tempio Malatestiano: Rimini 1 / Rimini 2 / Rimini 3 / Rimini 4 / Rimini 5 / Rimini 6 / Rimini 7 / Rimini 8 / Rimini 9 / Rimini 10 / Rimini 11 / Rimini 12 / Rimini 13 / Rimini 14 // Alberti, Basilica Sant’ Andrea, Mantua:

Tomorrow we shall move on to the famous Egyptian series of the Horus Eye and enter the realm of down under algebra, another wonderland of number patterns, which may well provide an answer to a philosophical question of Ancient Egypt: How did the Many  rise from the One?