The First Old Kingdom Pyramids:
Working with stone became an Egyptian specialty early in the dynastic period. The First Dynasty Pharaoh Menes is
thought to have united upper and lower Egypt about 3100 BC and the capital was established at Memphis a short
distance south of Cairo. A huge stone wall was erected around the city to protect it from the annual flood and this
must have resulted in skills working with stone, using nothing more sophisticated than copper tools.
The first major pyramid was engineered by Imhotep for the Third Dynasty Pharaoh Djoser. An extensive underground
funerary complex was hewn out of bedrock first and then the pyramid was erected over it. It is believed to have
started as a stone mastaba structure above but was progressively extended in size and upward in six steps until it
became the massive Step Pyramid reaching a height of 60 meters.
Djoser’s successor Sekhemkhet left an unfinished step pyramid complex nearby. It may be that six steps were
significant because six was regarded as a perfect number. It is both the sum and the product of its factors. We inherit
our measure of circles and the cycles of time from the Mesopotamian “soss” for 60. Thus we have 60 seconds in a
minute, 60 minutes in an hour, 6x60 degrees in a circle and so on. The numbers 6 and 10 were seen as interrelated
in Egypt as well as in Mesopotamia, but it took on a different flavor in Egypt as we shall see.
Another step pyramid at Meidum was covered in a casing to give it a smooth pyramidal shape, which later fell away.
It is usually attributed to Huni, the last king of the Third Dynasty but it may have belonged to or been completed by
Sneferu his son, since there are no identifying inscriptions. Sneferu built two more pyramids, the Bent Pyramid and
the Red Pyramid, the latter achieving a true pyramid shape. If it was a strictly a funerary objective one wonders why
he needed two or perhaps three pyramids. He was the father of Khufu (Cheops) who built the Great Pyramid.
The Pyramid Complex at Giza and the Great Pyramid:
There are indications that whole pyramid complex at Giza had a prior overall plan to it. An Idea gained momentum
with Sneferu that took three succeeding generations of kings to complete. The first to be built and the most
impressive was the Great Pyramid of Khufu, although the pyramid of his son Khafra is almost as large. That of
Khafra's son Menkaura, although smaller is also impressive.
There are six so-called Queen’s pyramids that although smaller are still massive, with burial chambers inside. For
example the largest near Menkaura’s pyramid was originally 28.4 meters high, (93 feet) the other two being
somewhat smaller. Three of them are aligned in a row immediately south of Menkaura’s pyramid, and three are
aligned in a row immediately east of Khufu’s Pyramid. The three beside Khufu’s pyramid are larger. So there are nine
pyramids in all. Or should that be ten, because there is a small single pyramid immediately south of Khafra’s pyramid.
Is there some significance to all this? We will come back to this question later.
Sir Flinders Petrie conducted the first very precise survey of the pyramids between 1880 and 1882. His work is still
used as a standard reference. His book “The Pyramids and Temples of Gizeh” was published in 1883 and can be
accessed freely on the net. He also evaluates various theories that had arisen, discrediting most, confirming others.
Petrie determined that the accuracy of the Great Pyramids workmanship is such that the four sides have a mean
error of only 58 mm in length and 1 minute in angle from a perfect square. The sides are closely aligned to the four
cardinal compass points to within 3 minutes of arc based not on magnetic north but on true north. Continental drift or
the precession of the poles might account for this small discrepancy. The original design dimensions, as confirmed by
Petrie’s survey and all that followed, were 280 royal cubits in height by 4x440 royal cubits around. (One royal cubit
was 0.524 meters.) As these proportions equate to 2xPi to an accuracy of better than 0.05% this was and is
considered to have been the deliberate design proportions. The height of the pyramid was the radius of a circle
whose perimeter was equivalent to the perimeter of the base.
There was an empty sarcophagus found in the king’s chamber.
The Great Pyramid was the world’s tallest building for 4,000 years. It was originally 146.6 meters or 480.9 feet tall.
Each base side was 440x0.524= 230.56 meters or 755.8 feet. There were over 2.4 million stones averaging about
two tons. To build it in 20 years meant quarrying, shaping, hauling and placing a stone every two minutes on
average, with great accuracy. The organization required to accomplish such a feat within Khufu’s lifetime was itself
Khufu’s son, Khafra’s pyramid was also built to very exacting standards. The pyramid has a base length of 215.25 m
(706 ft) and originally rose to a height of 143.5 m (471 ft) The slope of the pyramid rises at an 53° 10' angle, steeper
than its neighbor Khufu’s pyramid which has an angle of 51°50'40". Petrie confirms that this pyramid was built on the
plan of a 3, 4, 5 triangle, the only right angled triangle with all three sides successive whole numbers. This too has
significance which will be pointed out later.
The SW to NE diagonal axis of Khafra’s pyramid is offset slightly from that of Khufu’s pyramid although the diagonals
and sides remain accurately parallel. This contrasts with the larger offset of Menkaura’s pyramid which is also rotated
slightly in its relative orientation. There is an interesting theory on this at www.legon.demon.co.uk that relates to the
squaring of the circle.
There was a hiatus in building the Giza pyramids. Khufu had several sons and his immediate successor was his son
Djedefre (Radjedef) who for some reason chose to build his pyramid at Abu Roash north of Giza. There is some
speculation among scholars that there was a religious schism between elder and younger members of Khufu’s
successors. In any case Khafra again took up building at Giza eight years later when he became king after his half
brother’s death. There was an empty sarcophagus found in the main burial chamber hewn into bedrock beneath
Menkaura’s pyramid was less accurate in its construction, although its original slope was estimated by Petrie to be
51°20'25", close to that of Khufu’s pyramid at 51°50'40". The bottom courses were capped with granite which
remained rough cut on its outer edges, so that final surfacing was never completed. The upper courses were
capped with limestone and apparently finished to present a smooth slope.
The main burial chamber was hewn into bedrock beneath the pyramid base. The sarchophagus was found empty
with the lid missing, although pieces of the lid were found. Unlike those of Khufu and Khafra it was of ornate design,
but it was unfortunately lost when the ship carrying it to Britain sank. Apart from this sarcophagus the Giza pyramids
generally lacked any decor or inscriptions of any kind. This is in sharp contrast to later works and tombs.
The technology certainly existed to attain great accuracy in Menkaura’s smaller pyramid, suggesting another factor
in the overall plan. We will come back to this later.
The Divergence of Later Pyramids:
Pyramid building continued off and on for about 2000 years but later pyramids were smaller and often shoddily built,
lacking the accuracy and significance of the Giza complex. In general tomb building hewn out of bedrock took
precedence over pyramid building and became characterized by extravagance as at the Valley of the Kings near
Luxor. The tombs doubled as a storehouse of treasures to accompany the pharaoh in the afterlife. Elaborate décor
portrayed the pharaohs as gods, a feature completely absent from the Great Pyramids at Giza.
Whether or not the Giza pyramids were intended as tombs, they certainly have obvious cosmological significance
implicit in the stark simplicity of their geometrical design, lacking in hieroglyphics or internal décor. Indeed the only
likeness of Khufu that has ever turned up is a 3” ivory carving bearing his cartouche. Petrie found it at the ancient
site of Abydos, between Giza and Luxor (Thebes). Almost nothing is known of Khufu. There is none of the self
aggrandizement that characterized the tombs of later periods.
An Overall Plan Spanning Four Generations:
There must have been some incredible incentive, some driving impulse or insight that instigated such massive
undertakings that strained the whole population of Egypt. The undertaking enlisted the cooperation and support of
the whole populace. The Weighing of the Heart theme is obviously inconsistent with forced labor on such a scale.
It may or may not have begun with Imhotep, the builder of Djoser’s Step Pyramid, which seems to have been an ad
hoc affair that evolved during construction. It nevertheless could be indicative of insights that came as a staged
development. Djoser’s successor Sekhemkhet left an unfinished step pyramid complex nearby. There were no
inscriptions apart from his name inscribed on the stoppers of clay jars found in the pyramid. When his still sealed
sarcophagus was opened in the mid 20th century, it was found empty.
Sneferu (2613 to 2589 BC), the first king of the Fourth Dynasty, was certainly possessed of a compulsion to build
pyramids, probably three in all. He passed his compulsion on to his son Khufu, who reached extraordinary heights.
Then another two generations followed at Giza in what has the earmarks of an overall plan spanning four
generations. There are specific relationships between the pyramids, their geometry, and their history, to clearly
The circle suggests a time-like dynamic principle whereas a straight line of fixed length suggests a static linear
extension in space. Hence the tools of the sacred geometry were the ruler and compass. And one way to reconcile
the two is to draw the square whose perimeter is the same as the perimeter of a circle. It is called squaring the circle.
There was an interesting book published by John Michell, City of Revelation, 1972. In it he explores many
correspondences between esoteric subjects, from St. John’s revelation of the New Jerusalem, to the ground plan of
Glastonbury Abbey, to Stonehenge, to the Great Pyramid, to gematria, to ancient systems of measure, and so on.
Without going into the sense or reliability of all of them, a few salient features stand out as undeniable, and that
A central theme of Michell’s book is that there was a sacred geometry that was passed down through works in stone
as a secret tradition of masonry. It may be connected to the establishment of the Masonic Lodge but the connection
is lost in the mists of history somewhere in the Middle Ages along with the sacred geometry associated with it.
Nevertheless the emblem of the Masonic Lodge is the ruler and compass.
There is a common feature of the creative process that does not make rational sense. If we draw a circle on a piece
of paper the diameter of the circle can not be measured completely accurately in the same units used to measure the
circumference, and vice versa, and yet the circle is clearly there on the page. For this reason the ratio between
diameter and circumference, known as Pi, is called an irrational number. It has no completely discrete value. It goes
on for ever and ever with no discernable pattern to the endless sequence of numbers. The diameter and
circumference are said to be incommensurable. It is like the indeterminacy principle of quantum mechanics where the
static position and dynamic momentum of a moving particle can never be known accurately together.
The Great Pyramid is thus a monumental example of squaring the circle, since its height is related to the perimeter of
its base by the ratio 2 Pi. In this sense it is a monument that reconciles subjective spiritual dynamics to objective
Squaring the circle is often dismissed by academics as simply impossible since Pi is an irrational number, but they
miss the point. The point is that the linear regularities of space are reconciled with the cyclic dynamics of the heavens
through living processes. And the heavens are dominated by circular motions. The universe is a throbbing living
reality. That was the theme of the sacred geometry.
The Relationship of the Earth and Moon to the Khufu and Khafra Pyramids:
As Michell points out, if we draw a scale diagram of the earth and moon so that they are touching, and enclose each
in a square, a surprising relationship becomes obvious. It is illustrated in the diagram (borrowed from my book
The triangle with apex at the center of the moon and base across the diameter of the earth has the same proportions
as the Great Pyramid of Khufu. The circle scribed through the center of the moon concentric with the earth has a
circumference equal to the square that contains the earth to an accuracy of 0.0588%. The earth radius of 6,378 km
and moon radius of 1,738 km are taken from standard reference tables. They could be out by that much given the
irregularities of the earth and moon surfaces. So the dimensions of Earth and Moon themselves represent squaring
the circle. Also the triangle that joins the square of the moon to the square of the earth has sides in the ratio of 3, 4,
and 5. This same ratio corresponds to the slope of Khafra’s pyramid as confirmed by Petrie according to his
meticulous survey. Michell also points out that the two concentric circles are in the same proportion as the bluestone
and sarsen circles of Stonehenge.
It is a remarkable coincidence that the actual dimensions of Earth and Moon correspond to squaring the circle. There
are other remarkable coincidences that will be explored later.
The Nine Giza Pyramids and Fractals of Unity:
Try a little test. Sketch four separate circles on a piece of paper. Now try and draw all of the ways that four circles can
be arranged with respect to their inside and outside. For example one can be drawn inside one of the others but
separate from the remaining two. Or the remaining two can also be drawn one inside the other but separate from the
other two. Or the circles can be drawn as four concentric circles. And so on. How many ways are there? You will find
that there are only nine possible ways to do it.
These nine ways that derive from four circles correspond to the nine Terms of System 4 at www.cosmic-mindreach.
com. The circles represent active interfaces. They share a common active center inside and a common passive
periphery outside. Because they share a common active center we can call them centers. How they relate to one
another in each Term and between Terms outlines the basis of meaning. This System is introduced on the website.
We will come back to this later.
All nine ways relate to the same four circles that are employed to explore a unifying principle of subjective to objective
relationships. This is analogous to squaring the circle, since the circle represents the subjective animating principle
inside living creatures as they relate to the linear regularities of objective experience outside. This self-similar
analogy is apparent in the way the whole of experience is integrated. It is representative of the cosmic order.
Number systems thus also exhibit self-similar analogous patterns that are hierarchically subsumed. This means that
arithmetic and mathematics in general derive from the way the cosmic order works, not vice versa.
The Egyptians assigned no symbol for zero although their number system was designed on the base ten. They had
separate symbols for 1, 10, 100, 1,000, 10,000 100,000, and 1,000,000 and so they generally had no need for a
symbol for zero. Their number system was based on progressive whole number powers of ten. In other words it was
logarithmic to the base ten but only in terms of the whole number digits from 1 to 9. There are only nine repeated
ones between 1 and 10. There are only nine repeated tens between 10 and 100. There are only nine repeated 100’s
between 100 and 1,000 and so on. The subsumed repetition of nine holistic elements of counting is essentially a
fractal representation of Unity.
Egyptian Fractions as Fractals related to the Giza Pyramids:
The fractal representation of unity extends to fractions. Egyptians allowed only unit fractions. They saw fractions as
reciprocals of whole numbers. Their understanding of numbers, mathematics and measurements was implicitly
holistic in nature and related to the cosmic order.
This brings us to another peculiarity of the numbers from 1 to 9. The reciprocals of the numbers 3, 6 and 9 repeat
themselves to infinity. They yield the infinite numbers 0.3333etc., 0.16666etc., and 0.11111etc. The reciprocal of the
number 7 is also an infinite number but it is the repeating sequence of the remaining six numbers, namely
The numbers from 1 to 9, apart from 7, when each divided by 7 also result in the same repeating six digit sequence
but it starts at different places in the sequence. The number 7 divided by itself is Unity of course, so there is
something about the reciprocal of 7 that regenerates the other numbers endlessly as they relate to unity. Numbers as
fractals of unity proliferate as reciprocals of themselves and the number 7 represents the memory of the
computational proliferation. In fact any number divided by 7 has decimal positions that endlessly repeat the same six
digit sequence, unless the number is an exact multiple of 7. The Egyptians had a system of division that worked in
such a way that they must have known this. They must also have known about irrational numbers that are
This is a further indication that the Giza pyramids represent the comic order in their stark geometry and in the overall
plan. The obvious fact that there are three large pyramids and six small ones suggests a direct correspondence
without getting into a lot of spurious speculation. The reciprocals of 3, 6 and 9 relate to the three large pyramids. The
repeating six member sequence of 7 relates to the six small pyramids. The tiny tenth pyramid indicates the decimal
system their numbers are based upon.
The Pyramids, the System and the Cosmic Order:
There are direct parallels between the Giza pyramid complex and the System as introduced on the website. The
System as an expression of the cosmic order consists of a nested hierarchy of subsumed higher Systems that all
derive from requirements implicit in the nature of universal wholeness. Each higher system is a discrete elaboration
of the lower systems that precede it. The lower systems thus transcend and subsume the higher systems such that
the whole nested hierarchy is a subsumed elaboration of System 1 and universal wholeness. Each higher system is
nevertheless complete unto itself as it relates to phenomenal experience. All possible structural varieties of
experience are thus encompassed. It is to this extent a Theory of Everything. It is not possible to conceive of
anything outside it. More accurately it is a Universal Methodology that must find consistency with the living empirical
evidence much like Ancient Egypt’s living insights. It is not a Theory of Everything as an abstract construction distinct
from a living reality such as the kind that science currently seeks.
In System 2 there are only two ways that two circles can be drawn with respect to inside and outside. One of the
circles must be universal and unique and the other must be particular and many in order for System 2 to be a
consistent elaboration of System 1 and universal wholeness. The two circles can be separate facing one another, or
one circle can be inside the other. This implicitly defines both a subjective and an objective basis to experience.
In System 3 with three circles there are four possible ways to draw them with respect to inside and outside. Two of
these Terms are universal and two are particular but they combine to work in pairs as an elaboration of System 2.
The pairs alternate between an objective and a subjective orientation.
In System 4 with four circles there are nine ways to draw four circles, each called a Term, and they interact as an
elaboration of System 3. Three of the terms are universal and six are particular. The six particular terms occur in
alternating synchronous groups of three. One group has an objective orientation. The other group of three has a
reciprocal subjective orientation. So one grouping is orthogonal or at right angles to the other, like the two groups of
three small pyramids at Giza. The six particular terms also relate in a subjective to objective orientation as polar pairs
that correspond to the three universal terms and the three large pyramids at Giza.
These observations together with the stark geometry of the Khufu and Khafra pyramids that correspond so
accurately to the proportions of Earth and Moon make a powerful statement. These two largest pyramids indicate the
reconciliation of the living subjective dynamics of the circle that we experience inside with the stark objective linearity
of space and time that we experience outside. These two perspectives must find reconciliation through living
Menkaura’s pyramid did not have to be geometrically accurate to take its place as the third member of the infinite
triad. In fact it is rotated slightly as if on purpose to set it apart for this reason. Its rotation together with its three
orthogonal pyramids suggests a dynamic relationship between the subjective and objective orientations implicitly
represented in the other two pyramids.
Taken also in the context of the Weighing of the Heart as described in Part 1, it is a statement that portrays life itself
as the reconciliation of otherwise incommensurable elements between the living subjective dynamics and the static
objective regularities of the cosmic order.
Ancient Egypt's Theory of Everything - Part 2
|Ancient Egypt’s Theory of Everything
The Pyramids as Cosmic Temples
Robert Campbell 2008
Squaring the Circle: