An Egyptian Geometry Lesson - Part Two

If geometry is the representation of the Cosmos then what is meant when the King "assumes the God form", does it mean he assumes a geometrical pose? It may sound a little too literal to presume that, until we take into account the philosophical aspect of sacred geometry and then take a look at the pictures of King Thutmosis IV. (Drawings 24 -32). Yes, the King actually did strike a geometrical pose for his portrait, and that portrait alone describes a life that could only have been founded in the strict self discipline and hard study of an initiate destined to be King.                                                        

Frequently we read in the Egyptian texts that "the King became a star", what does that mean? Usually it is assumed that the King's soul was to take its place in the sky and literally " become a star ". However there is a small visualization problem that needs to be addressed before I can happily go along with that premise.

If you were asked to step outside and look at the night sky, and then asked to draw a true representation of a star in the sky, what would you draw? Let's assume that you had never before seen anyone else's version, that you are the first person to ever do this peculiar task. Be perfectly honest now, how many points would you put on your "star". The truthful answer is none, stars don't have points, they're a white shiny spot on a black background, no stretch of the imagination would take you into the area of geometrical constructs that we colloquially call "stars.

Do you see the problem? We have lived with the image for millennia, it was inherited. We see a symmetrical geometrical shape with a number of points and we call it a "star“, and assume it is representative of an astral body. Is it more likely that "stars" are depicting the God form, a mathematical rendition of the Cosmos, on the ground? The hieroglyphic symbol for the Duat is a five pointed star enclosed in a circle, a complex construction, as you are about to find out for yourself.

The Pentagram, the five pointed star, has been a geomantic symbol of protection since the beginning of history, and was especially endemic to Dynastic Egypt. In earlier times Egypt had held reverence for the heptagram, the seven-pointed star, as we can see on the Narmer Palette (note). The heptagram was also synonymous with Sumer.

The "Book of What is in the Duat " guides the initiate to" build a replica, on the ground ", of an area of "the Sky " known as " the hidden circle of the Duat “."

Taken at literal value, an architect's plan is "in the sky “, looking down on the elements of the picture in plan form. The plateau at Giza is not level, yet its measurements are those of a flat surface, so presumably the literal instructions are telling the initiate to make a map first and then draw the geometries that reveal the hidden circles and the Gateway to the Underworld. The body of the goddess Nut is always depicted arched over, balancing on toes and fingers. (See picture). The outline she forms with her belly and breasts is virtually the shape of the hill Gebel Ghibli, as seen from the Sphinx. Nut depicts the path of the stars through the night sky and she acts as a canopy over the tomb of Osiris. This same hill shape, in geomantic lore, is representative of Hermes / Mercury. (Pennick).  Shall we "build a replica“.

We have our framework, and the prime circles and vesica piscis, constructed already, but we now need to add the other "hidden circles". The "bass circle" need not be developed at this stage, it has made its statement for now, but the Rostau and Sokar circles need completing. Rostau consists, so far, of two concentric circles, one of radius "8”, our original circle, and one of radius "10”. (1600 & 2000 rc). Sokar, so far, has only one "10" radius circle. If we now draw the "8" radius circle from the Gateway / Sokar centre it creates symmetry for the "eye”, or vesica piscis, formed by the interlocking "10" radii circles centred on the G.P. and the Gateway.

Looking at the picture being generated we notice that Menkaure is "floating" between the "8" and "10" circles of "Rostau", and its centre is almost precisely halfway, or "9”. As we mentioned earlier, its precise distance is one cubit short of the distance to the Gateway from the GP, 1,788.85 rc., call it 1,789 rc (sq.rt.5 x 800 ). It seems to warrant a circle of its own, so we draw it with the G.P. as centre. Symmetry demands we duplicate that circle from the Gateway centre, and when we do just that we are led to the next hidden circle. Here's why.

That 1,789rc. (rt.5) circle circumference passes very close to the S.E. corner of G3, the S.W. corner of G2 and then the precise centre of G1. If the radius had been 11 cubits larger, 1800 rc., or "9", it would have touched the corners of both G3 and G2, or rather, the "corrected" s.w. corner of G2, which, if you remember, was the point that produces 1244. The obvious move to make is to draw the "9" radius from the Gateway and then duplicate it in the G.P. / Rostau circle. The "as built " measure to the S.W. corner of G2 from G1 is 1220 rc., so, as a result, its true position falls halfway between the 1789 and 1800 circles, just as the "uncorrected" N.W. corner is a little short of the theoretical 2000 rc from the Gateway, being 1994.9 rc. However, the designer left a lovely note behind to say " don't worry, all is in tune". Here's how he did it.

The half diagonal measure of G2 is 290.65 rc. and the measure to G2's centre from the Gateway is 1710 rc.. Added together they make 2000.65, restoring the harmony, and please don't forget how many other equations G2 is describing, and there are many more to be found as it unfolds its manuscript. It has just given us the "9" circle, the mnemonic for the squaring of the circle. Subtract one ninth of the diameter and square the result ; diameter 9 minus 1/9 = 8 x 8 = 64. The "9" radius circle has the same approximate area as our original 8 x 8 square and it's circumference is equal to the perimeter of the square inside the 2000 circle.

*  8 x 8 = 64  *   

*   (64 -:- Pi. r. sq.) = dia. 9.02703333+ *

There is more than one purpose to the "9" radius circle, it is not only a mnemonic for the squaring of the circle, it is the framework for something far greater. But once again we are rushing ahead too quickly, we must allow ourselves to be led down the path. All we have at the moment is the "flesh" of Rostau, we need to give it "bones". The pentagram is the symbol for the Duat and also the symbol of Man, it therefore seems appropriate that we move from the square of the Tetrad and build our "bones" from the "5" of the Pentad. The 2000 rc radius circle was our original "5" radius, and it encompassed all except G3C within its bounds. So we shall  construct our pentagram within it. (Construction method in appendix.)

If you were very good and read the appendix you now know how to divide the circumference of a circle evenly into five, it is not simple like the perfect Hexagon, yet it does have a perfection all of its own, in the Golden Section proportion. We will begin with one pentagonal construction, although this will soon become four constructs, marking out 20 points on the circle. 

Once again, the architect has given a clue, and a short cut, to it. The 622.222 rc diagonal of the G.P. is 1/20th of the circumference of the 2000 circle. Setting the compass points on the diametrically opposite corners gives the perfect division, but, for now, assume we missed that "grace note" completely, it will be repeated again later, at double forte.

If we begin at the northern cardinal point of the G.P. circle and divide the circle into five points, the first "hit" is once again G3B. The southwestern point on the circumference falls on that same second step as does the corner of the 3-4-5 triangle, only this time on the southern flank of G3B, not the western. The "bass player" has just turned up his volume a little.

When we join the northern point to the S.W.. point the line passes through the centre of Menkaure (G3) and a point on the western edge of G2, that becomes important as we develop the pentagram. By continuing to join every second point we build the first pentagram. The points which are roughly N.W and N.E. are joined and form a line parallel to the G.P's east / west meridian and north of it by exactly 618 cubits, it forms the northern wall of Khufu's valley temple. This is one factor of proportion of the Harmonic Mean, defined by its real number times 1000. (.618 x  1000 = 618). By then joining each point in order we create a pentagon around the pentagram. Notice that the "baseline" of the pentagon runs alongside the original "baseline", it is 1,618rc south of the G.P. meridian, another ratio of the Harmonic mean times one thousand.

*  1.618 x 1000 = 1618 *

To establish symmetry once again, since the pentagram is asymmetrical, we now construct three more pentagrams and pentagons, beginning from each of the other three cardinal points, South, East and West. Now we have produced 20 points on the prime circle, 4 pentagrams and 4 pentagons. Menkaure and Khafre are beginning their harmonious duet, they are now going to be locked together in beautiful synchronicity, in 5/4 time.

Remember that the diagonal of G1 was 622.222rc, and that measure divides the circumference into 20 ? This is why. There is a classical system for squaring the circle that can be achieved by drawing lines through the points on either side of the meridian points. This is a reasonably accurate system, which produces a of the value 3.1752.

As you can see in the drawing, the line drawn from west to south forms the precise diagonal of G3; it marks the edge of the square that has the same area as the 2000 rad. circle. Amazing! But as we said, this is merely a reasonably accurate squaring of the  circle, and since we have found this correlation it seems we should explore its accuracy.

If we calculate the distance along the S.W. diagonal from the G.P.'s centre to the midpoint of that side of this new square, where it is bisected by Menkaure's diagonal, we get a nice surprise. Using this system, and its value for Pi of 3.1752 , that measure should be 1781.90909+, which is the square root of  3.1752 x 1000. (See 18a detail) If G3's diagonal marks the perimeter of a truly accurate square then that measure would be the sq.rt. of true x 1000, which is 1772.453851. Using Petrie's measurements, as we have done so far, we now find another wonderful message. The centre of G3 is 1411rc south and 1096.765.rc west. Subtract 1096.765.. from 1411 and we have a horizontal displacement from the S.W. G.P. diagonal of 314.2345 r.c., which is x 100. With a little trigonometry we can calculate G3's parallel displacement away from the G.P.'s  S.W. diagonal as 222.2 rc., and then calculate that the intersection measure, which, for absolute precision, should be 1772.4538, is actually 1773.25 rc. A difference from perfection of 16 inches, if Petrie's measurements were correct, but did you see the mnemonics. 

* The centre of G3 is 411+1,000 south of G1*

*  G2's base = 411 x 411.*

* G1's  volume = 4411 x 64 x 64 * * Diag. of G1 = 622.2  x 2 = 1244 *

* 1411 minus 1096.765 = 314.2345 (Pi x 100)*

*  sq.rt. 314.2345 x 100 = 1772.666 *

* 1244 squared  = 1411 x 1096.765 *

* 1244  is the measure from G1 to the S.W. corner of G2's "box" and also G3's height x 10 *

Before we continue, there is something else to examine about this square constructed through the points of the pentagrams and its   ratio of 3.1752. The length of the side of that square is 1781.909 x 2 = 3563.818, which makes its diagonal measure exactly 5040, the ultimate magic number. For those of you who have no idea what that means, here is a clue. 5040 and its half value, 2520, are both divisible by all the numbers of the Decad, and there is more, as usual. 

It appears to be that Giza is the place where that magic number was first used, and from which the pentagram square can be constructed, but this is merely the opening of the Menkaure and Khafre duet, there is a lot more happening in the internal geometries of the four pentagrams and pentagons. The whole system replicates itself inside on a smaller scale, (drawings 15-19) and produces another 20 points where the four 'stars' intersect. One of these points falls on the western edge of G2 and another falls on the 'box' built onto the southern flank of the Sphinx.

Having drawn a circle through all 20 points, we now construct another square through the points of the internal pentagons, which now defines the diagonal of G2. Notice also that each of the points used for the internal square construction fall on the '5' (1000rc) grid lines. This enables the calculation of the radius of this internal circle and its area.

Now watch this equation ;-

* 3481956 -:-20.6066 ( p" in 1 r.c.) = 411.063 squared (G2 base) *

There is a direct mathematical relationship to the 1866 square's area in both G1 and G2, and since G3 is the volume of G1 divided   by 11.11111., we can check to see if it also bears a relationship. The base area of G3 is 39200 sq.rc. (196 x 200), so ;-

Here's how the "bridge" is built. We are heading for the "box" intersection on the Sphinx, because it is obviously a "logical" starting point. But we didn't begin our construction from there, we used the 3 x 3 square in the 3 x 5  "golden mean" rectangle. Here's the anomaly and how the composer solved it. Because the "3 square" is close to the beginning of the Fibonacci sequence, to work our way back to the "start point" we can only pass through the next three lower squares, one of which measures 2 x 2 and then two which are both 1 x 1.

As you can see, we have lost the proportion, but it is still leading us towards the Sphinx, we need a fine-tuning to complete a conclusive proportional spiral. The solution is to increase the scale as we approach the Sphinx. The higher numbers of the Fibonacci series provide the true spiral ratio of 1 : 1.618 and they run thus ;- 55, 89, 144, 233, 377, 610, 987, 1597.

The Song of the Sphinx

The spiral that begins at the "hole in the box" is left handed, it rotates anti- clockwise, as does water spiral down the drain in the northern hemisphere, this is probably not a coincidence, more likely it is another aspect of the composer's knowledge of the Earth and Cosmos. The Sphinx spiral takes us on a pathway through the elements of the composition. Leaving the Sphinx it passes through, in sequence, Khafre's valley temple   the Sphinx temple   Khentkawe's mastaba   the western doorway of Menkaure's valley temple and then the Gateway. This is the first part of the spiral sequence and we have been led from the Sphinx to the Gateway via the Valley Temples. The "bass groove" circle took a more direct route, but that's bass players for you!

The Fifth Division

The six-pointed hexagon was intrinsic to our initial 8 x 8 square, its geometries forming some of the base grid lines. We could have subdivided our first square and circle by utilising its perfect harmony. But we are now constructing our "stars" on the "10" radius circle and the Hexad is not needed in building the fundamental framework anymore. So shall we assume it has a statement of its own to make instead ?

Six is perfection, the ultimate divider, a really cool sixophonist. He has a very  short, but precise, solo to play. He doesn't need to go on and on like the Pentad, he'll impress you anyway. We need only two hexagons and hexagrams to make the statement, and we begin in the East. Using the 2000,1800 and 1600 radii we divide the perimeter of the circles into six, from the eastern meridian point, and join the dots to form a series of concentric hexagons. (drawings 10 &11)

The process is repeated from the north and this time the southern "crossbar" in the '10' circle forms the southern wall of Khafre's valley temple, one thousand cubits south of the G.P. centre and an internal square 2000 x 2000rc., which interlocks with the internal pentagram points that fall on the "5" (1000rc.) grid lines. Five and Six have joined in harmony. The inner hexagrams delineate the Sphinx temple and the boundary between the Sphinx and Khafre's valley temples. A diagonal of the '9' circle intersects G2's centre and Campbell’s tomb obelisk is bisected.

The twelve hours of the Duat place the entrance to Sokar, the Splendid Place of the Beginning of Time, in the fifth hour. The Hexad was a brief soloist, as we said he would be, but the theme established by him will not be forgotten.

We have now explored the first six movements of the Decad, Monad to Hexad and, by association, have encountered the Ogdoad (2 x 4 = 8) in the Tetrad.  The Ennead appeared in the 9 divisions of the Pentads internal 1866 square and once again in the "9" circle, which is equal in area to the 8 x 8 square. Essentially we have harmonized all the Decad, but I assume that you have noticed that the "virgin"  Heptad is missing, c'est la vie, seven has to be searched for, but you can bet that she will obey the rules of the game.

The Web of Athena

To misquote an old phrase, the fat virgin is about to sing, and the show's not over until she's finished. She has been sitting at the front of the orchestra from the start, in full view, but completely unnoticed until now. Subconsciously, we all knew she had to stand up eventually, her name is written in stone on the Giza billboard, seven has a page to herself in the symphony program. We'll pretend that she's a little shy until she begins to sing, but she has the voice of the Diva and knows it. She can bewitch all who hear her aria.

So you want to know where she was sitting that she should be so unnoticed ? She sat quietly between the square around the "9" circle and the "10" circle itself. Where the edge of the "9" square bisects the "10" circle a chord is produced which has the beauty of being one seventh of the circumference of the "10" radius circle. We can set the compass points on both intersections of the chord and produce seven points around the circle, each line of the heptagon being tangent to the "9" circle.

It is the legendary  "seven lengthed circle", the Ennead embraced by the Virgin. It ties together the Ogdoad "8" square, the Ennead "9" circle and the Decad, "10", Cosmos framework. Remember the mnemonic ; 7 x 8 x 9 x 10 = 5040. The magic seven has just begun to sing.

She knows more than one variation on the theme of constructing the Heptagram, we have heard the opening lines and the first word was Seven. She came straight out with it, not shy at all, but she was also very secretive about the depth of her virtuosity. The first seventh division was accomplished by using the compasses to measure the "chord", and then stepping around the circumference to create seven points, not an entirely accurate method, but it sets the mood. The secret of the Virgin is that she can provide all seven points with only the aid of the base grid, in fact, she can produce 28 points (4 x 7) by merely joining the dots within the framework. Don't fall asleep now folks ! This is where it starts to get really exciting.

We shall listen to the Diva sing the opening lines again, with the understanding that she was merely teasing us with her innocence when she began. The Khafre - Menkaure duet was an intricate piece and, for now, they will remain silent, we don't need their geometry until the Finale, when the whole ensemble joins in again. We shall return to the rhythm section, the base grid and circles, and watch Athena weave her web.

We begin at the eastern cardinal point (E), and we know that on the opposite western side of the circle, the vertical "9" grid line bisects the "10" circle to produce two points, one seventh of the circumference apart. We now draw the line C - C, which becomes the baseline of our eastern pointing heptagon, and connect each intersection with the '10' circle to point E  to form the eastern pointing triangle. We still need to locate four more points of the seven, and it is extremely easy. From point 'E' we draw two lines that intersect the G.P. north - south meridian  at the "8" points (A)north and south. We extend these lines to intersect the "10" circle and create points K & K.

Athena now hits her first high note, the line drawn from K to K defines the eastern edge of Khafre's pyramid  and is 440rc west of the G.P. meridian and parallel to it. The angle generated by E - A - G1 centre is 51*20.2' (tan. 1.25); this is only 5.5 minutes of arc short of the true seventh division angle of 51*25.7' (360*-:- 7), an accuracy of one in four thousand.

On the ground at Giza that amounts to only 3.2 cubits error in the placement of the point on the circle with a circumference of 12566rc. A man with his arms outstretched would cover the actual discrepancy over the 4 mile / 6.5 Km circumference Remembering that on our human drawing scale the pencil line is 7 cubits wide. The N.E. corner of G3C, however, marks the exact seventh division ratio on the 2000 circle. Can you see how precise is the tuning of the irrational Diva ? So far she has sung five notes ; A,C, D, E & F., she has two more to sing to complete her soprano's octave.

We follow the same formula to find the remaining two points. From the G.P. centre we move north and south to the edges of the "10" square, then we move 8 units east along the perimeter to the points 1600rc east x 2000rc north and south, points B - B, and repeat the same 4 : 5 ratio. A line drawn to these two points from the G.P. centre bisects the circle to complete the seven points of the Heptagram. The octave is complete and now we can proceed to create the Web of Athena and test the virgin's expertise.

First we connect each point in sequence, to create the heptagon, wrapping around the "9" circle. There are two types of heptagram stars, one formed by connecting every second point  and the other by joining every third point.

Once the points are in place it is possible, with practice, to construct the web with one continuous line. The soprano can sing the first verse in one breath. She has created an inner heptagon with an inner circle diameter of 880rc and an inner heptagram with an outer diameter circle of 1414.21rc, in other words, the sq.rt.2 x 1000. What a voice!

A short digression is required at this stage.  1414.21 has been integral to the picture from the beginning, as it is one quarter of the diagonal of the 4000 x 4000 square, 5656.854 (sq.rt.32,000,000) The square inside the 2000 radius circle has sides measuring 2828.427 ( sq.rt.8,000,000), and consequently the half base is 1414.21 (sq.rt. 2,000,000). The distance from the G.P. centre to the southern side of Menkaure's causeway is 1414rc, the causeway therefore defines the edge of the square inside the 2000 circle on its south side, and the 1400 ("7") grid line on its north side. The 1414 diameter or 707rc radius circle has a circumference of 4444rc, another nice mnemonic. The same measure also appears between the G.P. n.-s. meridian and the western boundary wall of G3.

From the Gateway to the centre of G2 is a precise angle of 32.7272.. degrees, one eleventh of 360. Eleven is the master number, one beyond the Decad. G2 is triangulated precisely to the Gateway. From the northern cardinal point to G2 is 1 radian, that section of arc equals the radius of 1710 rc. . Multiply that number by 11 and it gives the circumference in Egyptian feet (1rc-:-7 x 4 = 1 foot of 11.7752 ins.). That circumference measures 18810 ft, the Grand Gallery measures 1881 inches. The measure to G3a, 1900rc., multiplied by one eighth of G3's perimeter (99) = 188100. The length of the year between spring equinoxes (365.2424 days) x 1.881 = 687, the length of the Martian year. Is this the music of the spheres or what!

When the 11 geometry is drawn in the Rostau circle once again the architect has left his mark and shown how to construct eleven by sacred geometry. The intersections of the 5 x 7 rt. triangle and the 5 x 4 rt. triangle, basis of the Heptagon, provide the points and the two horizontal connections box G3 with precision. The connecting line between the intersections of the construction triangles bisects the centre of G2. (See Drawings)

At this very moment, the Tympanist, G3C, is stubbing out his cigarette, he has just heard his cue. See, there he is, creeping in behind the cellist and the bass player, his measure is coming. The sun is beginning its descent into the west, the shadow of the cellist, G3A, has begun to play her closing theme.She has another secret to disclose before the equinox sunset.

You'll have to travel a long way before you will ever hear that sweet melody played. Stonehenge is exactly 1/17th. north of Giza, which is 1/12th north of the Equator. I'll see you there later, just as soon as I can get the band out of the pub. And you thought it was all over,  ha !!  I hope you have a sense of humour, it's only just beginning, remember, this is HELL.

Everything!

Copyright 2001 ã David Alan Ritchie