PATRICK MULCAHY Divine Measure of Time and Space

The Divine Measure of Time & Space January 27, 2013
[This is a work in progress … ]
AstroQab Publishing 2010

First Electronic Edition 2010
Date of Original Publication: 13 th October, 2010
Author: Patrick Mulcahy
Cover Art by Patrick Mulcahy
Email: mallukh_ahi@yahoo.com
Website: http://www.astroqab.co.nr/
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Table of Contents
The Divine Measure of

o f Time & Space .....................................
.............................................
........ 5
Introduction ...................................
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....................................
.................. 5
The Creation Unit ....................................
.....................................................
.............................
............ 7
The Sevenfold Circle of Creation ....................................
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........... 9
The Mundane Cycle .....................................
......................................................
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....... 11
The Universal Esoteric Cycle ...................................
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............... 15
The Radian ........................
...........................................
.....................................
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....... 17
The Sevenfold Spiral ....................................
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.... 20
The Radian Comma ...................................................
..........................................................
....... 24
Characteristics
Characteristics of the UE Cycle ...................................
...............................................
............ 30
The Four Cardinal Points .....................................
....................................................
............... 30
The Mirroring Effect .......................................
........................................................
................... 33
The Divine Measure of Time....................................
...................................................
............... 38
The Year of Brahma ....................................
.....................................................
.....................
.... 38
Pi in the Hindu System ..........................
............................................
............................
.......... 42
The Precession Cycle .........................................
........................................................
............... 45
Some Further Considerations...................................
...............................................
............ 46
The ‗Seven Times‘ Prophecy ...................................
...............................................
............ 53
The Number 5040 ...............................................
............................................................
............. 56
The Divine Measure of Space...................................
..................................................
............... 58
Squaring the Circle .....................................
......................................................
.....................
.... 58
Squared Circle Methodology
Methodology ....................................
................................................
............ 60
Key Relationships Within the Squared Circle.............................
Circle............................. 63
The Great Pyramid of Giza ........................................
.................................................
......... 75
Designing the Great Pyramid ...................................
...............................................
............ 79
The 4/Pi Spiral ....................................
.....................................................
...........................
.......... 85
Esoteric Services....................................
......................................................
...............................
............. 94
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Table of Illustrations
Figure 1 - The Creation Unit ......................

..................................
.......................
......................
.....................
.......... 7
Figure 2 - The Sevenfold Circle of o f Creation
C reation ............................
........................................
..................
...... 10
Figure 3 - The Hexagonal Division of the Circle ......................
..................................
....................
........ 11
Figure 4 - Circle Radius = Hexagon Side .....................
.................................
........................
..................
...... 14
Figure 5 - The Radian Wheel .......................
...................................
.......................
......................
..................
....... 16
Figure 6 - Radian vs. Hexagon......................
..................................
.......................
......................
..................
....... 17
Figure 7 - The Universal Esoteric Cycle .....................
.................................
........................
..................
...... 21
Figure 8 - The Circle of Fifths ......................
..................................
.......................
......................
..................
....... 24
Figure 9 - The Clockwise
Clo ckwise Cardinal
C ardinal Point Cycle
C ycle .......................
...................................
....................
........ 32
Figure 10 - The Mirroring Effect .......................
..................................
.......................
.......................
..............
... 35
Figure 11 - Image from
f rom 'The Solar System' ......................
..................................
.......................
..............
... 39
Figure 12 - Day One of a Year of Brahma .....................
.................................
.......................
................
..... 40
Figure 13 – Night One of a Year of Brahma .............................
.........................................
..................
...... 41
Figure 14 – Basis of Hindu System? .....................
.................................
.......................
.......................
.............. 43
Figure 15 - The Mundane Division of the t he Circle
Ci rcle ......................
..................................
....................
........ 46
Figure 16 - Squaring the Circle (4 x 7920) ............................
........................................
....................
........ 59
Figure 17 – Major Squared Circle Relationships ...........................
.......................................
...............
... 61
Figure 18 - Calculating the Earth+Moon
E arth+Moon Circle .......................
...................................
....................
........ 64
Figure 19 - The Square (4) Evolves to a Pentagram (5) ......................
.................................
........... 65
Figure 20 - The Phi Connection ........................
...................................
.......................
.......................
..............
... 67
Figure 21 - Perimeter = Circumference .....................
.................................
........................
..................
...... 71
Figure 22 - Hexagon Side vs. Radian ...................
...............................
.......................
.......................
.............. 72
Figure 23 – The Two Expressions of the 4/Pi Ratio .....................
.................................
.................
..... 74
Figure 24 - The Great Pyramid - Phi ...........................
.......................................
.......................
................
..... 76
Figure 25 - The Great Pyramid - 4/Pi ......................
..................................
.......................
....................
......... 76
Figure 26 - The Ancient
Anci ent Egyptian
Egypti an 4/Pi Ratio....................
Ratio...............................
.......................
...............
... 77
Figure 27 - The Mendelssohn Methodology ......................
..................................
.......................
..............
... 79
Figure 28 - The Ancient
Anci ent Egyptian
E gyptian Pyramid Triangle ......................
...................................
...............
.. 80
Figure 29 - Dimensions ofo f the Great Pyramid in Cubits C ubits......................
.................................
........... 82
Figure 30 - Pyramid Pi Circle ...............
...........................
.......................
.......................
.......................
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... 83
Figure 31 - Organism and Environment .............
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................
..... 85
Figure 32 - The Process of Emergence .....................
.................................
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........ 87
Figure 33 - The 4/Pi Principle & Emergence .................................
............................................
............. 88
Figure 34 - The 4/Pi Spiral ......................
..................................
.......................
......................
......................
........... 89
Figure 35 - The 4/Pi Spiral Triangle ......................
..................................
.......................
......................
........... 90
Figure 36 - The Fibonacci Spiral ..........................
......................................
.......................
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........... 91
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The Divine Measure of Time & Space
Introduction
Cycles are a fundamental part of human life in the physical universe. Our
living planet and its biosphere are controlled and organized by cycles.
Cycles are built into the cosmic and microcosmic structure of the
universe. For example, on our planet we experience the cycle of night and
day, the cycle of the four seasons, the cycle of the 365+ day year (etc.).
Repetitive cycles allow us to predict future outcomes based on past

experience of the cycle. The Earth‘s annual cycle, for example, is
composed of four seasons that we expect will repeat each year in a
relatively consistent manner. The rational mind is naturally attuned to the
th e
logical linear repetition of cycles and our basic human perception (i.e. our
basic interpretation of the world) is founded on this particular
p articular paradigm of
logic. Our linear-based perception of the physical world is useful as we
interact with our local environment during our day-to-day lives, but the
mental body‘s rational perception is typically blind to other extraneous
subtle, but powerful forces that are also influencing our world from
behind the veil of material appearances. These forces originate and
operate below the surface of mundane reality, but their influence
penetrates into and subtly affects the world in which we live. These forces
tend to surprise the rational mind by upsetting the expected consistency
of the regular cycles of life.
Modern humanity understands that our planet‘s day/night cycle is due to

the rotation of the Earth. We are also aware that the four seasons are
caused by the tilt of the Earth on its axis in relation to the Sun around
which it orbits (i.e. in a 365+ day cycle).
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Theoretically, and based on this reasoned perception, the rational mind

might assume that our experience of the four seasons will be consistent
from year to year. This assumption is based on the fact that the course of
the seasons repeats in an endless cycle that basically follows the cycle of
planet Earth as it orbits around the Sun—a cycle that isn‘t likely to change
anytime soon. But there is another more subtle
s ubtle influence that ensures that
each cycle of the Earth around the Sun is different from the last. This
influence is due to the movement of the Sun itself in a large circular
sweep within the local region of our Milky Way galaxy. Of course, planet
Earth and all of the Sun‘s satellites are carried with the Sun on its
mysterious path through the heavens. Thus, even though every year winter
occurs at the same point in Earth‘s orbit around the Sun, i t is occurring in
a slightly different region of the galaxy (i.e. due to the Sun‘s own orbital
trajectory). Just how this changing cosmic influence effects the mundane
cycles of planet Earth is unknown, and is not generally considered by the
rational mind.
I have used the Earth‘s natural seasonal cycle as an example to

demonstrate the characteristic nature of the subtle forces that transform
an otherwise mundane cycle into an esoteric cycle. A mundane cycle is
one that is relatively obvious to the rational mind. We call a cycle
‗esoteric‘ when we also take into consideration the less obvious influences
that are effecting the cycle ‗externally‘, or from another ‗dimension‘.
Symbolically speaking, the orbit of the Earth around the Sun can be
termed a mundane cycle when it is interpreted as following a repetitive
circular (or elliptical) path. On the other hand, t he ‗esoteric‘ cycle of the
Earth‘s orbit around the Sun also acknowledges the Sun‘s own movement
through space and therefore is perceived not as a circle, but as a spiral.
This example demonstrates how an extra (usually hidden or obscure)
‗lateral‘ influence is appli ed to a ‗mundane‘ cycle to convert it into an
‗esoteric‘ cycle.
This book uses sacred geometry and mathematics to determine and

describe the universal nature of esoteric cycles. It explains the basic form
Page6
and structure of the prototypical universal esoteric cycle, which as you

will see, is very similar to the esoteric cycles described in the teachings of
Theosophy.
The Creation Unit
Our quest to determine the structural nature of the prototypical universal

esoteric cycle begins with what I call a ‗creation unit‘.
I define a creation unit as a circle with a diameter of one unit. This is the
simplest geometric form and therefore it is ideally qualified to represent
the fundamental unit of creation.1
Figure 1 - The Creation Unit
A creation unit has a diameter of 1 and the length of its circumference is


Pi ( ).
1
A ‗creation unit‘ is a form that abstractly represents an original cosmic building -
block of the Universe. There are a total of seven primordial ‗creation units‘.
Page7
Pi is a mathematical constant. It is classed as an irrational number —

meaning that its decimal never ends or repeats. It is also classed as a
transcendental number because it also has other unusual mathematical
properties.2
The nature and value of Pi is an important key that will help us determine
the prototype of the universal esoteric cycle.
cycle.
The length of the circumference of a circle is almost exactly equal to

three times the length of its diameter. The reason why the circumference
is not exactly equal to three times the diameter is a key to the mysteries
(as you will discover as you read this book).
The discrepancy is represented by the irrational transcendental decimal

portion of Pi.
The Pi constant is equal to:

This is the measure of the circumference of a creation unit. The integer
‗3‘ represents three diameter lengths of one unit. The decimal portion
continues indefinitely and symbolizes the mysterious transcendental and
irrational influence that permeates the universe. It effectively prevents
the circumference of a creation unit circle from ever completely closing. 3
2
http://en.wikipedia.org/wiki/Pi
3
That is, because it is a never-ending decimal.
Page8
The fact that a creation unit circle never closes is symbolic of the notion
that there is no such thing as a closed cycle in the universe. The rational
mind thinks it sees a closed cycle (i.e. a closed circumference) when it
looks at a circle, but this is an illusion that is based on the m ind‘s limited
perception of the mundane material world.world.
It is remarkable that mathematicians have decided to use the words

‗irrational‘ and ‗transcendental‘ to describe the Pi decimal because these
words also perfectly describe the esoteric influence that the Pi decimal
div ine Creation.4
represents within the divine
The Sevenfold Circle of Creation
A ‗creation unit‘ symbolizes the basic building-block that the Creator uses
in the formation of a universe. As you will see below, the Creator must
adjacently combine seven creation units in order to complete the larger
circle that symbolizes the universe. In other words, the process involves
positioning seven small circles adjacent to each other to create one large
all-encompassing circle.
Like its component creation units, a universe is represented by a circle

because a circle‘s circumference delineates in the simplest way possible a
singular, self-contained region of space. The process of creation is
symbolized as unfolding within the space (or area) inside the bounding
perimeter of the circle‘s circumference.
circumference.
The various characteristics of the geometry of a circle conceal with a veil

of symbolism the deepest esoteric secrets of the universe. We are
attempting to unveil some of those mysteries in this book.
b ook.
4
But perhaps instead of ‗irrational‘ the term ‗supra-rational‘ would better
describe its esoteric influence.
Page9
The Circle of Creation is produced by positioning seven creation unit

circles adjacent to each other as follows…
Figure 2 - The Sevenfold Circle of Creation
It is not until the seventh circle is finally added that the larger Circle of
Creation (shaded pink above) is completed.
The Hebrew bible reveals this mystery in the Book of Genesis wherein the
Creation is said to occur over a period of six days —that is, six days of
form-building, and then a seventh day of ‗rest‘. In other words , the
seventh day allows the Creator to establish Himself at the hidden centre
of the physical
p hysical universe.
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The Mundane Cycle
Our normal ‗rational‘ perception of the mundane world is represented by

the six adjacent creation unit circles that touch the circumference of the
larger Circle of Creation. The arrangement of these six circles reflects (in
a symbolic sense) our characteristic linear spatial awareness, and also our
linear perception of time. This is explained below.
below.
It was the ancient Babylonians who divided the circle into six equal
portions and then each of those six portions into sixty smaller measures
that we now call degrees.
Figure 3 - The Hexagonal Division of the Circle
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Since then, the 360 degree circle has formed the basis for our sense of
direction as well as our basic measure of the angular relationship between
points in space.
This hexagonal division of the circle is also the basis for our measurement
of time. The 24 hours of a day are measured by a clock-face that is
divided into twelve equal portions, or in other words, two interlaced
hexagrams. Similarly,
Similarly, a year is divided into twelve months.
The illusory nature of the hexagonal division of space/time is represented

in the Circle of Creation by the apparent omission of the seventh (i.e.
central) creation unit circle in the process of dividing
d ividing the circumference.
The rational mind sees a sixfold division of the circumference and

therefore naturally assumes that only six elements (i.e. six creation unit
circles) are involved in the division process, but in reality a seventh hidden
central influence is also fundamentally implicated in the calculation. The
influence of the seventh circle is present in the sixfold division, but it is
not obviously included in the resulting numerical configuration.
Symbolically speaking, this reflects the rational mind ‘s tendency to be
‗deceived‘ by a superficial ‗six-fold‘ impression of the mundane world —it
fails to notice the illusive seventh component.
The seventh (i.e. central) creation unit circle symbolizes the irrational
transcendental decimal portion of Pi. It was the ancient Egyptian‘s
knowledge of this esoteric mystery that led them to use the fraction
 to

represent their sacred Pi approximation.5
The earliest evidenced conscious use of an accurate approximation for the

length of a circumference with respect to its radius is of 
in the
5
The ancient Egyptians incorporated this esoteric proportion 22:7 into the design
of the Great Pyramid of Giza.
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designs of the Old Kingdom pyramids in Egypt. The Great Pyramid at Giza,
constructed c.2550-2500 BC, was built with a perimeter of 1760 cubits
and a height of 280 cubits; the ratio 1760/280 ≈ 2π. Egyptologists such as
Professors
Professo rs Flinders Petrie] and I.E.S Edwards have shown that these
circular proportions were deliberately chosen for symbolic reasons by the
Old Kingdom scribes and architects. The same apotropaic proportions
were used earlier at the Pyramid of Meidum c.2600 BC. This application is
archaeologically evidenced, whereas textual evidence does not survive
from this early period.6
The fraction  is equal to 3  where the irrational decimal portion of Pi

is represented by .

  
   
 
  
   
As you can see, the Pi constant and  approximate each other quite
closely.
Because the seventh creation unit circle is

 of the total number of

creation unit circles that together compose the Circle of Creation it can

be said to represent the  portion of Pi7—or in other words, the decimal
0.142857 …
6
http://en.wikipedia.org/wiki/Pi
7
I.e. that portion of the Egyptian Sacred Pi estimate .
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The rational mind does not see the

 decimal portion of Pi. Instead it sees

(or thinks it sees) an exact threefold division of the circle ‘s circumference

(i.e. the ‗3‘ of 3 ), or two interlaced triangles producing the illusion of a

hexagonal division.
Figure 4 - Circle Radius = Hexagon Side
Another way of understanding the illusory perception that the hexagonal

division of the circumference represents is by realizing that each side of a
hexagon is exactly equal to the radius of the circle in which it is
inscribed. This means that the perimeter of the hexagon is equal to
exactly three times the diameter of the circle . But, we know that the
length of the circumference of a circle is equal to a little bit more than
three times its diameter. The discrepancy is represented by the irrational
decimal portion of Pi. So in other words, the hexagon is apparently failing
to include the Pi decimal in its division of the Circle of Creation ‘s
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circumference. The reason that it fails to do so is because the six sides of

the hexagon are straight lines, whereas the circumference of the circle
containing it is curved.
c urved.
The diagram above demonstrates that the radius of a circle (a) is equal to
the length of the side (a) of the inscribed hexagon. The illusion it
symbolically portrays is that the circumference of the circle is equal to
‗  ‘ whereas in fact it is equal to ‗  ‘ plus a small amount reflecting
the decimal portion of Pi. If   then the perimeter of the hexagram
 , but the circumference of the circle    (approximately).
In other words, space and time are not linear (as is usually perceived by
the rational mind) but rather, space and time are curved. Our standard
measure of space/time, because it is based on the hexagonal division of a
circle, reflects the limited illusory perspective of the rational mind and
reinforces the false perception that space/time is linear.
The Universal Esoteric Cycle
In order to determine the structure of the universal esoteric cycle we

must take into account the curvature of space and time as represented by
the curvature of the circumference of the Circle of Creation. So we must
divide the circumference not with a hexagon, but with some other form
that incorporates as part of its essential nature the irrational decimal
portion of Pi. In other words, a form that takes into account the curvature
of the circle‘s circumference.
circumference.
As it turns out, science has already discovered the esoteric measure of a

circle‘s circumference.
circumference. It is called a ‗radian‘.
Page15
Figure 5 - The Radian Wheel
[Base image courtesy of Wikipedia.]
A radian is the esoteric (or ‘non-

‘non-linear’) equivalent of one side of a
hexagon . The hypothetical figure that is the radian-based esoteric
equivalent of a hexagon has 44 curved
c urved sides, each of which is the length of
one radian, and its existence spans seven adjacent dimensions. 8 This
8
In actual fact this figure has over 15840 sides and spans more than 2520
dimensions, but its full extent is subdivided into tiers of seven progressively
linked sub-cycles. In other words, it represents an open system subdivided into
Page16
esoteric 44-sided figure does not have a name that is known in the modern
world, but we could tentatively call it a ‗radiagon ‘.
The Radian
An easy way to understand the difference between a radian and the side
of a hexagon is by considering the following diagrams
diagrams …
Figure 6 - Radian vs. Hexagon
linked groups of seven. In this book we are mainly examining the first tier of
seven sub-cycles.
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A radian represents the length of the radius of a circle as measured along

its circumference. Its measure produces an angle of 57.2958°. The
hexagonal division of the circumference, on the other hand, produces an
angle of exactly 60° (i.e. a ‗sextile‘).
In the diagram above, all three lines labelled ‗a‘ are the same length. The
red line labelled ‗a‘ is equal to one radian. The slight difference in the
angles formed by the red and blue lines is due to the curvature of the
circle‘s circumference— i.e. symbolizing the curvature of space/time.
The side of a hexagon, like a radian, is equal to the length of the circle‘s
radius, but unlike a radian, the hexagon applies a straight-line
measurement to the circumference which is inappropriate because the
circumference of a circle is curved. Thus, a hexagon represents a linear
model of reality applied to an essentially curved structure. It symbolizes
the mode of perception and thinking of the rational mind.
If, on the other hand, we divide the circumference of the circle using a
radiagon we are appropriately
appropriately applying a non- linear (or ‗lateral‘) model of
reality to a non-linear figure. This will lead to a more accurate perception
(or symbolic representation) of the curved reality of time
t ime and space.
A radian incorporates in its division of the circumference of a circle the

complete Pi constant whereas the side of a hexagon only considers the
integer ‗3‘—i.e. it ignores the irrational decimal portion of Pi.
Here‘s a mathematical demonstration of this comparison…
A creation unit circle has a diameter of one and a circumference equal to


Pi ( ).
Page18
Because the radian measure is based on a circle with a radius of one unit
(rather than a diameter of one) we need to double the standard radian
measure for this demonstration.
    

Now we divide that number into the 360 degrees of a circle.
  
   
 
Applying the same procedure to a measure based on the side of a hexagon
gives...
    
 
  

As you can see, the hexagon-based measure is exactly three. It omits the
decimal portion of Pi from its linear division of the circle‘s circumference.
Page19
The Sevenfold Spiral
The division of a circle‘s circumference by a hexagon resul ts in six equal

sectors beginning from 0° and ending at 360° (i.e. returning to 0°). In
other words, it closes the circle in a cycle that eternally repeats itself.
The change that occurs during the cycle is regular and predictable because
it represents a closed system —i.e. there are no obvious external variables
impacting on, and affecting, the course of the cycle.
In reality, however, the universe is unpredictable, and regular cycles are

always influenced by external forces that disrupt their regularity.
The radian-based (or radiagon-based) model we are developing and

exploring during the course of this book incorporates as part of its natural
structure the influence of these kinds of non-linear (or extra-dimensional)
forces. These forces are concealed behind the veil of ‗mundane
appearances‘. They emanate through that veil and encroach upon our
world from the ‗chaotic‘ realm of the supra-rational Unknown. The
intuitive mind perceives and acknowledges these forces because it is not
constrained by the rational mind ‘s natural focus on surface appearances.
A radiagon is a hypothetical abstract geometrical figure with 44 curved

sides spanning seven dimensions (or having seven tiers). The fact that it‘s
composed of seven adjacent tiers reflects the inclusion of the seventh
(i.e. central) creation unit circle in its division of the circumference of the
Circle of Creation. As you know, the hexagonal (i.e. sixfold) division of the
circumference represents a mundane perception of the world. A
radiagonal (i.e. non-linear sevenfold)
sevenfold) division of the circumference, on the
other hand, symbolizes an esoteric (or ‗intuitive‘) perception of the world .
Page20
Figure 7 - The Universal Esoteric Cycle
Page21
When a radiagon is mapped to a circle it divides its circumference into


sections of 57.2958° (i.e. one radian). This mapping results in an uneven
division of the circle and therefore a process of overlapping is required in
order to complete the cycle it represents. The overlapping process
produces a spiral form with multiple arcs (or levels) rather than one closed
circle (as is produced by a hexagonal division).
The mapping of the radiagon to a circle produces a spiral of seven arcs.

The spiral is divided into 44 sections (i.e. radians). Each arc of the spiral
has seven sections—five sections in its main sweep, one section joining it
to the previous arc, and one section joining it to the next arc. During the
first sub-cycle, the first and fourth arcs contain six main sections plus the
two adjoining sections.9
The diagram above is a two-dimensional representation of the first seven

tiers of a radiagon. Its 44 curved sides form a spiral of seven arcs. I have
used adjoining straight lines in order to clearly identify the 44 radian
divisions of the spiral (or radiagon).
radiagon).
The sevenfold spiral is naturally divided into two mirrored halves. In the
diagram above I have represented the UE Cycle (i.e. Universal Esoteric
Cycle) as a descending spiral that reaches its lowest point halfway through
the fourth arc (i.e. at the 180° point 10, or astrologically at 0° Libra). The
cycle then rises through another three and a half arcs and returns to the
0° point (or astrologically, 0° Aries).
Note however, that the cycle returns not to exactly 0°, but rather to
 1°01‘. I call this mathematical constant (i.e. 1.014°)
1.014°) the Radian Comma.
(See below.) All these measures are approximate because the decimals
involved in the calculations
calculations are irrational.
irrational.
9
But these exceptions only occur in the first sevenfold tier of spiral arcs.
10

To be more precise, its lowest point is 180°30‘.
Page22
This means that the sevenfold spiral does not end at 0°, but rather it
flows into another sevenfold spiral that begins at 1°01‘ (and that ends at
2°02‘). In other words, this repeating spiral-cycle is eternal and ever-
changing. We can however, determine its rhythm, or harmonic pattern.
The UE Cycle is divided into tiers of seven linked sub-cycles, or a spiral of

seven arcs that folds back upon itself. The turning point of the spiral
occurs after 22 radian divisions. The spiral returns to nearly its starting
point after 44 radian divisions (or 44 sides of the radiagon). The spiral
returns to a point very close to one degree (i.e. 1°01‘) from its original
starting position. Therefore we can extrapolate and state that the UE
Cycle will experience another major return after 360 spirals —i.e. because
each new sevenfold spiral begins almost exactly one degree after the
previous spiral, and there are 360 degrees in a circle.
This means that the ‗higher order‘ return of the UE Cycle occurs after
   
11
(i.e. 2520 spiral arcs) or after    
(i.e. 15840
radians ).
But the 15840 radian point doesn‘t mark the end of the UE Cycle because
the cycle is also changing by approximately one minute of arc after each
sevenfold spiral, and because these calculations are approximations of
irrational numbers it means that the UE Cycle never actually completes (or
closes)—it is eternal.
That is one effect that the irrational transcendental decimal portion of Pi

has on the structure of our reality. Without it our universe would be
represented by the closed regularity of a circle, and an extremely limited
repetitive (relatively static) cycle of existence.
11
Or radiagon sides.
Page23
The Radian Comma
There‘s a close geometric parallel between the radian spiral and the
‗spiral of fifths‘ (of music theory). The similarity is due to the fact that
both spirals produce a very small residuum (or ‗remainder‘) when they‘ re
overlaid onto a circular template. When the ‗spiral of fifths‘ is forced to
conform to the circle of seven musical octaves its natural form must be
modified in order to remove the residuum.
The ‗spiral/circle of fifths‘ is based on the ratio 3:2 (or the fraction 3/2)
which is the most consonant non-octave interval (according to music
theory/geometry).
Wikipedia describes the circle as follows:
Figure 8 - The Circle of Fifths
Page24
“...the circle of fifths is a circle of closely related pitches or key

tonalities. Musicians and composers use the circle of fifths to understand
and describe those relationships. The circle's design is helpful in
composing and harmonizing melodies, building chords, and moving to
different keys within a composition.” 12
Our Western musical system is challenged, however, by the ‗circle of

fifths‘ because in reality, it‘s not a circle, but rather, a spiral.
“This interval [i.e. 3/2] has serious implications for the various tuning
schemes of the chromatic scale, because in Western music, 12 perfect
fifths and seven octaves are treated as the same interval. Equal
temperament, today the most common tuning system used in the West,
reconciled this by flattening each fifth by a twelfth of a Pythagorean
comma (approximately 2 cents), thus producing perfect octaves.13
“When an instrument is tuned with the equal temperament system, the

width of the fifths is such that the circle “closes”. This means that
ascending by twelve fifths from any pitch, one returns to a tune exactly
in the same pitch class as the initial tune, and exactly seven octaves
above it. To obtain such a perfect circle closure, the fifth is slightly
flattened with
with respect to its just
just intonation (3:2 interval
interval ratio).
“Thus, ascending by justly intonated fifths fails to close the circle by an

excess of approximately 23.46 cents, roughly a quarter of a semitone, an
interval known as the Pythagorean comma. In Pythagorean tuning, this
problem is solved by markedly shortening the width of one of the twelve
fifths, which makes
makes it severely dissonant.” 14
severely dissonant.”
The artificial closing of the circle of fifths is what our popular Western
music system is based on. The procedure is implemented for ‗ease of use‘
12
http://en.wikipedia.org/wiki/Circle_of_fifths
13
http://en.wikipedia.org/wiki/Pythagorean_comma
14
http://en.wikipedia.org/wiki/Circle_of_fifths
Page25
and is another example of how the linear perception of the rational mind
seeks to impose itself on our non-linear reality. If the Creator of our
Universe took the same linear-based approach to His/Her work then S/He
might remove the irrational decimal portion of Pi and round it to 
(instead of  ). Then a circle‘s circumference
would be equal to exactly three times its diameter. But then our world
would be a place of stasis and stagnation.
The Pythagorean Comma & the Earth Year Differential
The spiral of fifths is prevented from naturally becoming a closed circle by

a geometric ratio called the Pythagorean Comma. This very small measure
is considered an inconvenient discrepancy by Western music standards.
Therefore, it‘s been removed from our Western music system in order to
close the circle of fifths at precisely the same end-point as the circle of
seven music octaves.
So, the Pythagorean Comma discrepancy arises when twelve perfect fifths
are superimposed upon a circle of seven musical octaves. A slight
‗overhang‘ means that the circle of fifths doesn‘t close in perfect
alignment with the octave circle.
Here‘s how the Pythagorean Comma is calculated:

calculated:
 
         
      







   







Interestingly, the same issue occurs when the spiral of 44 radians is
mapped to a sequence of seven 360° circles (or arcs). And this suggests
that a similar geometric principle
principle is at work.
Page26
The radian spiral winds its way around seven arcs and slightly overhangs
the end-point of the sevenfold circular pathway. As explained above, this
is exactly what occurs in relation to twelve perfect fifths and seven
octaves.
What is most fascinating is that the respective ‗overhangs‘ are almost

exactly equal.
Here‘s how the Radian Comma is calcula ted:
(  
 )      







   




The Radian Comma arises because of the difference between 44 radians
and seven 360 degree
d egree spiral arcs.
     
    

   
 

 
    
  


  


  
  


   
Compare this value to the Pythagorean Comma (previously calculated):
calculated):
  


   





Page27
The difference between the Pythagorean and Radian commas is:  .
When each figure is rounded to three decimal places they‘re both equal to
.
Interestingly, the radian comma is also reflected by the ratio of the
average length of the Earth‘s tropical year to the ‗ideal‘ length of an
Earth year. The ideal length of an Earth year is figured as 360 days —
corresponding to the number of degrees in a circle. 15 The actual length of
a tropical year is approximately
approximately 
days. Their ratio is:
  

  
 


  





Compare this to the radian comma…
   

 





  

  






 
The difference is only 

This comparison between the Tropical Earth Year and the Ideal Year
provides us with another example of the contrast that exists between a
‗perfect‘ circular system and a natural non-linear system. And once again
we see the difference between the two systems very closely
approximating
approximating the radian comma value: . 
15
Visit: http://www.ancient-world-mysteries.com/360-days-earth-year.html
Page28
The Tau Constant:



In the field of mathematics there's a relatively new constant called Tau ( ) 

that‘s been promoted as a practical alternative
alternative to Pi (especiall y in radian -
(esp ecially
based mathematics).
The value of the Tau constant is   , or 6.2831...
The ancient esoteric value of the Pi constant is

.

We can therefore designate the esoteric value of Tau to be:
   , or  (6.2857...)
Notice how the esoteric Tau (44/7) reflects the underlying principle of the
radian spiral (and the UE cycle)—that is, 44 radians measured along a
winding pathway of seven spiral arcs.
Page29
Characteristics of the UE Cycle
The universal esoteric cycle is not only divided into seven spiral arcs (or
levels) but those seven arcs are also divided into two halves, and four
quarters.
The two halves of the UE Cycle mirror each other and represent the two
halves of a multi-dimensional near- circle. I say ‗near -circle‘ because th e
circle doesn‘t actually close. It is, in fact, merely the first arc of an even
bigger 360 arc mega-dimensional
mega-dimensional spiral (as explained above).
The UE Cycle reaches its halfway point after  spiral arcs (or 22 radians)
and then folds back upon itself.
The boundaries of the four quarters of the UE Cycle occur after spiral
arcs (or eleven radians) and are sign-posted by the four cardinal points (or
degrees) of the zodiac.
The Four Cardinal Points
The UE Cycle begins at the first degree of Aries (or 0° of the 360° circle).
This special degree corresponds with the Spring Equinox in the Northern
Hemisphere of the Earth. Astrologically, this point represents ‗new
beginnings‘. It coincides with the beginning of new life in the natural
world. It‘s a time of germination, the hatching of eggs, and the birth of
wild-life.
Page30
The spiral cycle of the UE Cycle proceeds from 0° Aries and winds its way
to another milestone (i.e. another cardinal point) after eleven radians
when it reaches 0° Capricorn (or the 270 th degree of the circle). It then
continues on and at 22 radians it reaches the lowest point of the cycle at
0° Libra (or the 180° point of the circle). The UE Cycle then begins its
return via an ‗upward‘ spiral , and at 33 radians it reaches 0° Cancer (i.e.
the 90° point of the circle). Finally, after a total of 44 radians it reaches
1° Aries from where it begins a new seven-tiered spiral-cycle.
This progression through the four cardinal points ( in reverse order)

suggests a symbolic correspondence with the daily path of the Sun through
the twelve astrological Houses (also in reverse order). This ‗mundane‘
cycle occurs as the Earth rotates on its own axis during the course of 24
hours. The reversed (or clockwise) movement of the Sun through the
Houses from dawn to noon to dusk is an illusion that is based on a
perspective that is ‗Earth-centric‘— i.e. limited (and distorted) by the
physical conditions of planet Earth. But all human-beings must experience
life on Earth, tread the path of material existence, and eventually
overcome the inherent limitations of physicality by becoming ‗soul-
conscious‘. It is then that our perspective of the world becomes less
subjectively oriented and we are able to move with the Sun on its true
anti-clockwise path through the Heavens —from Aries to Pisces. To do this
requires a shift of focus (symbolically and actually) from the Earth (i.e.
the subjective personality) to the Sun (i.e. the soul). In other words, being
‗Earth-centric‘ is like spinning yourself around and then declaring that the
world is, in fact, revolving around you.
Even though the UE Cycle touches the four cardinal points in the reverse
(i.e. clockwise) order, the cycle itself actually flows in the anti-clockwise
direction around the circle of the zodiac. In other words, the UE Cycle
traces the path of the Sun as it follows the true ‗esoteric‘ order of the
signs of the zodiac (i.e. from Aries to Pisces).
Page31
Figure 9 - The Clockwise Cardinal Point Cycle
Thus, as you can see, the UE Cycle embodies two cycles that are moving in
opposite directions. The 44 radian cycle that spirals its way around the
zodiac in an anti-clockwise direction, and a converse four-fold cycle that
moves (in an illusory manner) through the four cardinal points in a
clockwise direction. This interesting characteristic of the UE Cycle is
something worth pondering.
The descending half of the UE Cycle, even though it is associated with the
first six signs of the zodiac, is punctuated by the cardinal degree of
Capricorn. Similarly, the ascending half of the cycle is punctuated by the
cardinal degree of Cancer.
Page32
It may be that these four boundary markers represent ‗turning points‘ or

‗points of crisis‘ on the path of cosmic evolution. The clockwise order of
their appearance may be associated with the need for evolving humanity
to free itself from the ‗wheel of illusion‘ that turns clockwise as the Earth
rotates on its own axis.
The Mirroring Effect
The fact that the UE Cycle is split into two mirrored halves is of great
significance.
The fourfold cardinal division of the UE Cycle, and the way that it
naturally folds back upon itself in a mirror image suggests a descent
through the first half of the zodiac and then a paralleling ascent through
the second half of the cycle. The esoteric reason why the fourfold cardinal
cycle touches the four cardinal points in clockwise order as the UE Cycle
spirals anti-clockwise through the zodiac is a profound mystery.
The mirrored movement of the UE Cycle is of the nature of a palindrome,

symbolically speaking. I touched on this subject in my earlier book The
Pi —because the structure of a palindrome
Metaphysical Significance of Pi—
symbolizes nicely the Creation concept of ‗the
‗ the end is in the beginning and
the beginning is in the end ‘.
‘.
For example, consider the foll owing palindrome…
Are we not drawn onward to ‘New

‘ New Era’?
Era’?
Page33
This is a very apt example because the UE Cycle appears to reflect the
structure of cosmic time cycles. (See The Year of Brahma below.)
The UE Cycle reflects back upon itself in the same way as a palindrome
does, and therefore we can use a palindrome to illustrate the basic form
of the UE Cycle.
A a
r r
e E
W w
e e
N N
o o
t T
D d
r r
a a
w w
n n
O
The above arrangement of the letters of the palindrome illustrates my

point.
The descending half of the UE Cycle is represented by the words ‗ Are We

Not Drawn‘. This symbolizes the downward pull of gravity that draws each
human soul into physical incarnation. The lowest point of the cycle is
symbolized by the ‗O‘ of ‗Onward‘ . The ascending half of the cycle is
represented by the words ‗Onward To New Era‘. This second half of the
palindrome symbolizes the evolutionary force that pushes every incarnate
human soul towards initiation and the transcendence
transcendence of matter.
Page34
Figure 10 - The Mirroring

M irroring Effect
Page35
The mirroring of the UE Cycle occurs across the Aries-Libra axis of the
zodiacal circle. In my diagrams I have placed Aries at the top of the circle
as was sometimes the custom in ancient times. 16 This has the effect of
associating the ordered sequence of six signs from Aries to Libra (located
on the left-hand side of the circle) with a process of deepening descent
into manifestation.
The returning half of the cycle corresponds with an ascent through the six
signs from Libra to Pisces (i.e. located on the right-hand side of the
circle). It corresponds with a process of rising out of the depths of
materiality and into higher planes of consciousness. This makes sense
because these latter six signs are associated with progressive stages of
initiation.17
The diagram above demonstrates the mirroring effect as it occurs within

the circle of the zodiac. Only the first and last portions of the UE Cycle
are shown. (The complete UE Cycle would consist of a series of seven
vertically aligned circles.)
The figure demonstrates how the first six radian divisions mirror the last
six radian divisions of the UE Cycle. If you are wondering why the zodiacal
arrangement of the lower circle mirrors (or is in opposition to) the
arrangement of the higher zodiacal circle it is because the figure is
translating a spiral path into a vertical straight line.
The following table lists all the zodiacal degrees that constitute the
complete UE Cycle. The left-hand column is mirrored by the right-hand
column (and vice versa). The rows that are highlighted red show the
position of the four cardinal points as they divide the UE Cycle into its four
quarters.
16
And still is in some Eastern traditions.
17
See the Theosophical writings of Alice Bailey for more information.
Page36
The Universal Esoteric Cycle

0. 0°Aries00’ 44. 1°Aries01’
1°Aries01’
1. 27°Taurus18‘ 43. 3°Aquarius 43‘
2. 24°Cancer35‘ 42. 6°Sagittarius 25‘
3. 21°Virgo 53‘ 41. 9°Libra08‘
4. 19°Scorpio 11‘ 40. 11°Leo50‘
5. 16°Capricorn 29‘ 39. 14°Gemini32‘
6. 13°Pisces46‘ 38. 17°Aries 14‘
7. 11°Taurus04‘ 37. 19°Aquarius 57‘
8. 8°Cancer22‘ 36. 22°Sagittarius 39‘
9. 5°Virgo40‘ 35. 25°Libra21‘
10. 2°Scorpio 57‘ 34. 28°Leo03‘
11. 00°Capricorn15’ 33. 00°Cancer46’
00°Cancer46’
12. 27°Aquarius 32‘ 32. 3°Taurus28‘
13. 24°Aries51‘ 31. 6°Pisces 10‘
14. 22°Gemini08‘ 30. 8°Capricorn 52‘
15. 19°Leo26‘ 29. 11°Scorpio 35‘
16. 16°Libra44‘ 28. 14°Virgo 17‘
17. 14°Sagittarius 02‘ 27. 16°Cancer59‘
18. 11°Aquarius 19‘ 26. 19°Taurus 41‘
19. 8°Aries 37‘ 25. 22°Pisces 24‘
20. 5°Gemini01‘ 24. 25°Capricorn 06‘
21. 3°Leo13‘ 23. 27°Scorpio 48‘
22. 00°Libra30’
00°Libra30’
Page37
The Divine Measure of Time
The Year of Brahma
The spiral-cyclic pattern of the UE Cycle is identical to the structure that

underlies the Hindu theory of world cycles. This ancient Indian system of
sacred time-cycles includes periods that define a Day, Year, and Age of
Brahma.18
Each ‗day‘ (also called a planetary ‗manvantara‘) in a year of Brahma

covers the period of the life of a planet. In Theosophical terms, it
corresponds to a planetary ‗chain‘ consisting of a series of seven ‗globes‘
(and seven ‗rounds‘).
Here is how A. E. Powell describes the cycle of seven globes…
Although the 7 globes of a chain all exist

exist simultaneously,
simultaneously, yet they are not
all, at any given time, equally active in supporting life. Broadly speaking,
at any given time, 1 globe only is active and fully functioning, the other 6
being in a dormant
do rmant condition.
condition.
The globes come into full activity, i.e., they become fully inhabited by
beings—with whom we shall deal in later chapters—
various classes of beings— chapters —in
succession. First, globe A becomes active: after a vast period of time the
life on it begins gradually to lessen and almost to disappear, passing to
18
Brahma (Sanskrit) [from the verbal root brih to expand, grow, fructify] The first
god of the Hindu Trimurti or triad, consisting of Brahma, the emanator, evolver,
and creator; Vishnu, the sustainer or preserver; and Siva, the regenerator or
destroyer. Brahma is the vivifying expansive force of nature in its eternally
periodic manvantaras. He stands for the spiritual evolving or developing energy-
consciousness of a solar system which is also called the Egg of Brahma
(brahmanda). [From: The Rakefet Theosophical Dictionary.]
Page38
globe —B. Globe A then becomes dormant, whilst B begins to

the next globe—
awaken.
After another vast period of time, globe B in turn “goes to sleep”, the
greater part of the
the life passing on to
to the next globe in order, globe
globe C.
This process continues until each globe in turn has awakened from its
sleeping condition, has supported the main stream of life for an eon, and
again become dormant. The period during which a globe is fully active,
supporting the main stream of life, is called a globe-period.
Figure 11 - Image from 'The Solar System'
The passage of the spiral-cycle of life through all 7 globes is known as a

round. A round thus consists of 7 globe-periods, or world-periods (as they
are sometimes called).19
19
The Solar System . A. E. Powell. Pp. 14-15.
Page39
Figure 12 - Day One of a Year of Brahma
The second major period of the UE Cycle is termed a ‗Night of Brahma‘. It

begins at 1 degree (i.e. 1° Aries) and ends at 2 degrees (i.e. 2° Aries). It
involves another seven-fold cycle that spirals around the circumference
circumference of
the main circle. It is essentially a period of rest during which the monads
of life who inhabited the world during its ‗day -time‘ are withdrawn into a
planetary ‗laya centre‘20 until the next day dawns and they are able to
reincarnate once again in physical form.
20
The seven Laya centres are the seven Zero points, using the term Zero in the
same sense that Chemists do, to indicate a point at which, in Esotericism, the
scale of reckoning of differentiation begins. From the Centres … begins the
differentiation of the elements which enter into the constitution of our Solar
System. [Secret Doctrine , HPB. 1:138. See also: SD, 1:145-8.]
Page40
The first Night of Brahma ends at 2 degrees

d egrees (i.e. 2° Aries). The full course
of a ‗Year of Brahma‘ thus involves a double cycle of     
,
or
    

 
 . (We will return to the very important numbers
2520 and 5040 presently.)
    
 

    

Figure 13 – Night One of a Year of Brahma
From the above we can deduce that a Year of Brahma is naturally divided
into two halves at every level, and in every aspect
aspect of its structure.
Page41
Pi in the Hindu System
A full day/night cycle of Brahma is equivalent to:


    

 
 

And a full year of Brahma is calculated as:
      

This number is interesting because, if we use a floating decimal point, it
somewhat resembles the value of Pi.
   

    
In fact, the famous Hindu yogi Paramahansa Yogananda, in his best-selling
book Autobiography
Autobiography of a Yogi,
Yogi, stated: ―The
―The life span for a whole universe,
according to the ancient seers, is 314,159,000,000,000 solar years, or
Brahma‟‖.21
„One Age of Brahma‟‖.
He was apparently suggesting that the ancient Hindu system of world

cycles is based essentially on the Pi constant. This is interesting because
the fundamental ‗creation unit‘ circle has a diameter of one and the
21
Autobiography Yogi. Paramahansa Yogananda. Footnote.
Autobiography of a Yogi.
Page42
length of its circumference is Pi —suggesting the existence of a major

universal cycle that reflects the Pi figure.
The diameter unit measure in the diagram below would need to equal 10 12
(i.e. 1000000000000)
1000000000000) in order to produce a circumference (or cycle length)
of 3,141,592,653,589 (or one Year of Brahma). Symbolically speaking, the
exponent ‗12‘ in 1012 could be associated with the twelve ‗months‘ of the
Year of Brahma. To produce a cycle equivalent to one Age of Brahma the
diameter value would need to be 10 14.
Figure 14 – Basis of Hindu System?
The ancient Hindu system is, in fact, based on multiples of the number
10822, and especially of the number 432 (i.e.   
).
  
   
  
 

     
  
 
22
In ancient India, Vedic seers had calculated the following distances which
modern scientific measurements have reconfirmed: * The distance between the
Earth and Moon is 108 times the diameter of the Moon (true) * The distance
between the Earth and Sun is 108 times the diameter of the Sun (true) * The
diameter of the Sun is 108 times the diameter of the Earth (close - the number is
closer to 109.2 ) Wikipedia
Wikipedia..
Page43
  
   
  
 

     
   

The traditional figures for an Age and Year of Brahma resemble the Pi
constant, but unlike Pi, they are rational numbers (being the result of the
multiplication of rational numbers). Pi is an irrational number (as
explained previously). Thus, it appears that the Hindu system might
represent an effort to approximate Pi using rational numbers —just as the
ancient Egyptians did with their sacred
s acred Pi fraction
.

Here is a quote on the subject

s ubject from Sepharial…
―The value of π, which expresses the relations of the circumference to

the diameter of a circle, was concealed in the great cycle known as the
Age of Brahma, 311,040,0
311,040,000,000,000
00,000,000 years.
ye ars. This age is one hundred
years of Brahma, and a Brahmic year is therefore 3,110,400,
3,110,400,000,000
000,000
solar years. This number is again divided by 360, which gives the value of
the Brahmanic Day or 8,640,000,000 years. Then by adding together the
value for the age, year and the day of Brahma, we have 314159, etc.,
which is the familiar value of π or very nearly, 355 divided by 113. ‖23
   


   

   

 
 

    
 


23
The Kabala of Numbers , by Sepharial, p.12.
Page44
As you can see, the resulting figure is quite an accurate (floating decimal)
approximation of Pi.
The Precession Cycle
Interestingly, when we use a floating decimal point, the traditional figure

given for a year of Brahma is the same as the sum of twelve precession
cycles, or     . This suggests that the number 25920 is
somehow associated with the twelve months ofo f the year of Brahma.
  
  




        
    
    
The figure 25920 years for a precession cycle is an approximation of the
currently accepted scientific estimate (i.e.   ) but it appears
to have been an important symbolic figure in ancient times. Plato referred
this number to the Great Year, and it is also implied in the ancient Hindu
world cycle calculations (as shown above).
The number 25920 is derived by designating the rate of the Earth‘s

precession as 1° every 72 years (which is an approximation). Thus, a full
360° cycle of precession is calculated as:
    

 
 

Page45
Some Further Considerations
As previously explained, the mundane measure of time and space on

planet Earth is based on the ancient Babylonian division of a circle into
360 degrees. Our most common instruments for measuring space and
time—i.e. the compass and the clock—are both configured according to the
Babylonian hexagonal system of measurement.
But the hexagonal division of the circle (on its own) fails to incorporate
the subliminal influence of the mysterious esoteric principle of
time/space. The esoteric principle is represented by the original creation
unit located at the centre of the seven-fold Circle of Creation. The centre
circle does not touch the outermost circumference, and therefore, is not
directly used in the mundane division
division of space and time.
Figure 15 - The Mundane Division of the Circle
Page46
The mundane division of the circle is achieved using hexagrams (or

hexagons) that are drawn using the six creation units that touch the
circumference of the Circle of Creation. These six circles represent the
outer (or surface) appearance of things and emphasize the ‗form aspect‘
of the world, whereas the central seventh circle symbolizes the concealed
inner esoteric dimension of life in the universe.
The omission of the seventh circle in the hexagonal (i.e. 360 degree)
division of the Circle of Creation is symbolized by the fact that the number
360 has for its factors all the natural numbers from 1 to 10 except for the
number 7.
The smallest factors of 360 are:
         
As you can see, the number 7 is missing in the series.
The divine measure of time/space is represented by the number 2520.

This number is the least common multiple (LCM) of the first ten digits. So,
in other words, unlike the number 360, it does include seven in its list of
factors.
The smallest factors of 2520 are:

Page47
Thus, the number 360 can be said to represent the calendar of the
mundane year (as recorded in various ancient calendar systems 24) while
the number 2520 is associated with the divine25 esoteric (or cosmic) year.
The mundane hexagonal division of space and time is useful for measuring
the material qualities, quantities, and attributes of the physical world, but
is not (by itself) ideally suited to measuring the esoteric influences and
dimensions of life. In order to esoterically measure time we need to factor
in its spiral-cyclic nature. The mundane cycle (or circle) of time is
transformed into an esoteric spiral by the influence of a force that
impinges upon it laterally and pushes it in a particular direction —i.e.
towards the fulfilment of the divine Will and Purpose of the Creator of the
universe.
I have been referring to the central creation unit as the seventh circle in
order to differentiate it from the other six creation units that touch the
circumference of the Creation Circle. But the centre circle is, in fact, the
first (or original) creation unit to appear during the process of Creation.
The other six creation units arrange themselves around the central circle
only after the first creation unit has manifested (and established itself) as
the ‗genesis point‘ of the universe.
So, the original creation unit represents the 1st Principle of Creation,
which in Theosophical terms is called the divine Will-Purpose aspect.
Microcosmically, it ‘s associated with the plane of Atma (Divine
24
…And in its adjusted form in our modern 365+ day calendar.
25
The divine year is termed the Year of Brahma in the ancient Hindu system. As
previously noted, the Year of Brahma corresponds to the 2520 x 2 spiral-arcs of a
double circuit of the UE Cycle. As explained above, the figure 2520 does not
include the 360 Nights of Brahma —i.e. the intermediate cycles of ‗pralaya‘. (See
below for further clarification.)
Page48
Intelligence) and on the kabbalistic Tree of Life diagram its microcosmic

expression is represented by the sefirah Da‘at.
The original creation unit impresses the intention of the divine Will upon
the fabric of Creation. The 1st Principle initiates the creative process from
high levels of Being and then becomes covered (or hidden) by the six
creation units that cluster around it to form the larger Circle of Creation.
In other words, after the initial creative impulse the source of that
impulse becomes hidden below the surface of the ‗material‘ universe by
six layers of consciousness/substance. As already explained, these six
represent how/what we perceive with our normal (or mundane)
perception, and hence their use in configuring our standard systems of
time/space measurement.
measurement.
The centre creation unit thus represents an original impacting lateral

force that is the starting point of the universe, and that moves the circle
of time and space on a spiral path towards the fulfilment of the divine Will
and Purpose. The completion (or relative completion)
completion) of that spiral path is
symbolized by the number 2520 (i.e. one UE Cycle). 26
Even though the original creation unit circle represents the influence of
the 1st Principle it also expresses the esoteric (or ‗numerological‘)
qualities of the number seven. As you know, the influence of the number
seven is missing (or rather, ‗concealed‘) in the 360 degree model of
reality. This is symbolized for us by the conspicuous absence of the
number seven in the factors of 360. The esoteric model of reality,
however, as represented by the UE Cycle, is seen to remedy this
deficiency because its exemplary number 2520 includes the number seven
in its series of factors.
Numerologically, the number seven expresses the influence of Heaven

upon Earth. It represents the grounding of subtle cosmic forces, and the
26
And 5040 representing a double UE Cycle.
Page49
manifestations that arise in our lives as a result of those forces. Typically,

these kinds of influences go unrecognized, or are dismissed as nonsense by
the concrete rational mind. A good example of the type of expression (and
psychological reaction) engendered by the number seven is the crop circle
phenomenon (and the whole issue of ‗extraterrestrial visitors‘ in gen eral).
Interestingly, the geometric structure of the UE Cycle implies that our

physical universe (as a coherent whole) is very possibly moving along a
path (or trajectory) within an even larger unknown, undefined
Space/Substance.
When we think about the universe we might imagine it as an expanding

sphere of matter (in its various gradations and forms) that originally
emanated (during the ‗Big Bang‘) from a singular motionless point. But
the nature of the UE Cycle suggests that the ‗genesis point‘ of universal
emanation (i.e. the starting point of Creation) was (and still is) in a state
of motion, and therefore, so is the universe that was spawned by it. We
cannot perceive the outer macrocosmic movement of our universe because
our senses are constrained by (and within) the substance of its body.
This concept can be best understood using Theosophical principles.
ev ery entity in the universe is basically spherical 27

Theosophy teaches that every
and in motion. For example, every star is spherical and moving on its own
unique trajectory. We can therefore assume (based on the principle of ‗ as
below, so above‘) that the universe as a whole is also basically spherical
and in motion.
27
Even those beings who do not physically appear to be spherical are surrounded
by a relatively spherical aura (or etheric envelope).
Page50
Theosophy also teaches that every entity embodies a host of ‗lesser‘ (or
‗microcosmic‘) entities who together compose its vehicle (s) of expression.
But the lesser entities are necessarily unconscious of the ‗lateral‘
movement of the greater entity of whom they are a small part. In other
words, the extent of the body of the greater entity constitutes a
completely self-contained ‗universe of being ‘ for its constituent lesser
entities. The Self-created, Self-defined limit of the body of the greater
entity (i.e. its outer perimeter) forms a ‗ring-pass-not‘ that naturally
restricts the awareness and movement of the lesser entities who exist
within it. That is because the lesser entities are all formed of the same
substance as the body of their host, and they are fixed according to their
specialized functions within that body. They are physically unable to
extend themselves beyond the scope of their natural functions, or outside
of the substance of which they and their macrocosmic host are formed.
Theosophy teaches that the universe is the body of expression of a great

and living Entity (called Brahma), and because we are constituent parts of
its body (and our senses and instruments are formed of the substance of
its body) we are unable to perceive its ‗lateral‘ movement through the
surrounding super-ethereal substance in which IT exists. That rarefied
substance is of a higher order of density than the substance from which
the universe is itself made. The full extent of our physical senses (and
scientific instruments) have evolved to function within (and detect) the
substance of the universe itself and can never sense anything that exists
outside the substance of the universe.
Similarly, a water molecule in a raindrop has no awareness of, or means of

detecting, the atmosphere through which it is falling.
falling.
Page51
The number 2520 is programmed into the UE Cycle in two different ways.
It is present in the structure of one complete UE Cycle, and it also figures
importantly in the smaller, seven-fold sub-cycle.
The sub-cycle consists of a seven-arc spiral that folds back upon itself at
the  arc point (as previously described). The two halves (descending
and ascending) mirror each other across the circle of the zodiac.
The seven arcs of the sub-cycle each consist of 360 degrees, so the seven
spiral arcs represent a descending and returning passage through a total of
2520 degrees.
Sub-cycle:
  
   


When the seven-fold sub-cycle returns, its starting point shifts 1° anti-
clockwise around a larger 360 degree circle. Because each degree of the
circle comprehends a sub-cycle of seven spiral arcs so the total number of
spiral arcs in one UE Cycle is 2520.
UE Cycle:
      


 
Another interesting mathematical characteristic of the number 2520 is

that it‘s an EPORN ( Equal Product Of Reversible Numbers). An EPORN is a
Page52
number that can be expressed as the product of two reversible numbers in

two different ways.
In fact, of all numbers, 2520 is the first and smallest EPORN. This unusual
numerical characteristic
characteristic of 2520 is illustrated as follows…
    


    


It is quite extraordinary that the mathematical structure of the number
2520 exhibits this mirroring quality because, as you know, in terms of the
UE Cycle, it represents a series of sub-cycles that mirror each other in
much the same way.
0 0
1 1  
2 2
x
The ‘Seven Times’ Prophecy
The number 2520 is associated with a very well-known biblical prophecy

called the ‗Seven Times‘ prophecy . If you Google ‗2520 prophecy ‘, for
example, you will see countless pages on the WWW devoted to this
mysterious prophecy.
Page53
The 2520 year period (of the ‗Seven Times‘ prophecy) is especially
associated with the following eleven verses of the Bible.28
Leviticus 26:28. Then I will walk contrary unto you also in fury; and I,
even I, will chastise you seven times for your sins.
The name ‗Seven Times prophecy‘ comes from the above verse (and
surrounding verses) of Leviticus .
Interpreters of the Bible have theorized that when the scripture refers to
a ‗time‘ it is referring to a ‗prophetic year‘ of 360 days. Thus, ‗seven
times‘ refers to a period of      
, or .
It has also been generally agreed by Biblical interpreters that a day of

divine time is equivalent to a year of Earth time. Th erefore, the ‗Seven
Times‘ prophecy is usually said to refer to a period of 2520 Earth
E arth years.
The following biblical verses describe a half ‗Seven Times‘ prophetic

period of 1260 days, or 
cycles. This is significant because the UE sub-
cycle is naturally divided into two mirrored
mirrored halves of  spiral-arcs each.
Daniel 7:25. And he shall speak great words against the most High, and
shall wear out the saints of the most High, and think to change times and
laws: and they shall be given into his hand until a time and times and the
dividing of time.
Daniel 12:7 . And I heard the man clothed in linen, which was upon the
waters of the river, when he held up his right hand and his left hand unto
heaven, and sware by him that liveth for ever that it shall be for a time,
28
All quotes are from the King James Version.
Page54
times, and an half; and when he shall have accomplished to scatter the
power of the holy people, all these
these things shall be
be finished.
Revelation 11:2. But the court which is without the temple leave out,
and measure it not; for it is given unto the Gentiles: and the holy city
shall they tread under foot forty and two months. 29
Revelation 11:3. And I will give power unto my two witnesses, and they
shall prophesy a thousand two hundred and threescore days, clothed in
sackcloth.
Revelation 11:9. And they of the people and kindreds and tongues and
nations shall see their dead bodies three days and an half, and shall not
suffer their dead bodies to be put in graves.
Revelation 11:11. And after three days and an half the Spirit of life from
God entered into them, and they stood upon their feet; and great fear
fell upon them which
which saw them.
Revelation 12:6. And the woman fled into the wilderness, where she
hath a place prepared of God, that they should feed her there a thousand
two hundred and threescore days.
Revelation 12:14. And to the woman were given two wings of a great
eagle, that she might fly into the wilderness, into her place, where she is
nourished for a time, and times, and half a time, from the face of the
serpent.
Revelation 13:5. And there was given unto him a mouth speaking great
things and blasphemies; and power was given unto him to continue forty
and two months.
A common designation of the ‗half cycle‘ is ―a

― a time, and times, and half a
time‖
time‖ which is interpreted as follows…
29
Forty-two months of 30 days is equivalent to 1260 days.
Page55
 
  
  

  
  
      
The ‗half cycle‘ (i.e. of 3½ spiral arcs) is important because the first half
cycle carries us to the point of deepest and densest manifestation, while
the second half cycle takes us to a new high point of elevation, and a new
beginning.
The Number 5040
I mentioned earlier that every day of Brahma is followed by an equal

period of night, and that therefore the Year of Brahma actually involves a
double UE Cycle, or two revolutions of the Circle of Creation.
Therefore a full Year of Brahma covers 720 degrees and is actually

comprised of      . This is mathematically significant
because it derives a very important number in the field of Esotericism …

    

The number 5040 is equivalent to 7! (‗seven factorial‘) and therefore, in
an esoteric sense, represents a very potent expression of the number
seven…
Page56
                


And it also has another interesting attribute…
        


The number 5040 is particularly significant because it embodies a special

geometric relationship between the Earth and the Moon.
   

   

      
 
 
  
     

Symbolically, we could say of the Earth+Moon that the Earth represents
the 360 ‗Days of Brahma‘, and the Moon represents the 360 ‗Nights of
Brahma‘ because, as previously explained, the number 5040 represents
the double UE Day/Night cycle of the Year of Brahma.
Amazingly, these figures also provide a solution to the ancient geometric

puzzle known as ‗squaring the circle‘.30
30
The following solution to the problem is based on equalizing the circumference
of a circle with the perimeter of a square.
Page57
The Divine Measure of Space
Squaring the Circle
The phrase ‗squaring the circle‘ is traditionally associated with the

challenge of drawing a circle and square of equal areas using only a ruler
and compass. Historically, this has been proven to be an impossible task
due to the irrationality of Pi.
A variation on the original challenge involves equating the length of the

circumference of a circle with the length of the perimeter of a square. As
far as I know, this is also impossible to achieve with a straight-edge and
compass, but an approximate figure is sometimes drawn using the
established dimensions of the Earth and Moon. The procedure revealed
below however, involves the use of functions and relationships that exist
within the natural geometry of a circle inscribed
inscribed within a square.
s quare.
Before we begin constructing a squared circle diagram from scratch using

abstract geometric functions let‘s look at the Earth-Moon ‗geophysical‘
model of the design we are aiming to replicate. (See the diagram below.)
The diagram below shows a green square precisely enclosing the Earth,
and a red concentric circle surrounding it at a distance. The radius of the
red circle is proportionally equivalent to the combined radii of the Earth
and the Moon (i.e. 5040 miles). Note the number 5040.
The linear distance between the green dots (i.e. positioned at the corners
of the green square) is 7920 miles —i.e. corresponding to the mean
diameter of the Earth. The radial distance between the red dots (i.e.
Page58
placed at each quarter arc of the red circle‘s circumference) is also 7920
miles (i.e. using  for ) .31 32
Figure 16 - Squaring the Circle (4 x 7920)
31
If scientific Pi is used the result is approximate.
32
It is interesting to note that 
        . We already know that
        
 .
Page59
In other words, the perimeter of the green square, and the circumference
of the red circle are the same length —i.e.  
  

 .
The number 5040 (as represented in the diagram by the radius of the red
circle) is thus seen to be a major key to transforming linear measurement
(i.e. the squared Earth) into non-linear measurement (i.e. the red circle
of the Earth+Moon). It does so also in the double UE Cycle of the Year of
Brahma (as previously explained).
Squared Circle Methodology
The following model for squaring the circle is not based on the known
geophysical dimensions of the Earth and Moon. Nor is it based on a
convoluted application of the Phi constant. The construction of the
diagrams (below) is based on the simple internal geometry of the
squared circle figure itself . The included
included diagrams are designed to reveal
the natural geometry of the squared circle, and in the process, also to
lend weight to the historical ‗Pi Theory‘ of design of the Great Pyramid of
Giza.
The following simple exercise demonstrates

demonstrates the ratios involved.
1. First, draw a circle 20cm in diameter. This circle represents the

Earth. (I suggest using 20cm because it‘s easy to convert. See
below.)
2. Next, draw a square that encloses the circle and adjacently
touches its circumference. Of course, the length of each side of
the square will be 20cm (i.e. because the diameter of the circle is
20cm).
3. Now, draw a second concentric circle of radius length 12.73cm —
i.e. its circumference extends 2.73cm beyond the circumference of
the 20cm circle. This circle is concentrically positioned around the
first circle. This second circle‘s diameter is 25.46cm.
Page60
4. Finally, draw a small circle whose circumference adjacently

touches the first circle. This circle represents the Moon. The centre
of this third small circle can be located anywhere on the
circumference of the large outer
o uter concentric circle.
Figure 17 – Major Squared Circle Relationships
The measurements used in the procedure described above are all

proportionally based on the radius of the first circle (which can be any
length). The larger concentric circle (i.e. the second circle drawn) always
has a radius that is
 () times the radius of the first circle. The

 is the ratio of the area of the square to the area of the inscribed
value

circle. This ratio  is the key to the entire process of successfully
squaring the circle.
Page61
By using the key value , the Earth and Moon circles are always
precisely in proportion to their cosmic counter-parts. Why this is so is a
great mystery. It suggests that the 
ratio has special physical and
symbolic significance for our Earth/Moon system. (See below.)
When the  ratio is used to square the circle, the length of the
circumference of the outer concentric circle is always exactly the same
length as the perimeter of the square (whose corners it overlaps). And,
the ratio of the area of the outer circle to the area of the square is also
equal to  
(or ). (See below.)
Many people use the standard geophysical approximation 1.273 (or 0.273)
when they are drawing the squared circle diagram. 33 They use it because
they don‘t realize that the true number/ratio (of which 1.273 is merely an
approximation) is built into the very foundation of the geometry of the
circle in the square. This oversight also leads people to mistakenly attach

the Phi ( ) constant to various dimensions of the squared circle.
Fortunately, because the  ratio is implicit in the geometry of the
squared circle, these erroneous correspondences can beb e discarded.
   

   

  
 
The standard 0.273 figure is determined scientifically using the calculation
above, and is then added to a circle radius of 1 unit to draw the second
concentric circle. This represents the conventional geophysical approach
to squaring the circle.
33
This figure is sourced from the actual geophysical dimensions of the Earth and
Moon (i.e. as recorded by Science).
Page62
The method employed herein, however, replaces the 1.273 approximation

with the true and perfect value 
. No prior knowledge of the
geophysical measurements of the planets are required in order to
construct the diagram because this value exists fundamentally as part of
the geometry of the squared circle. This means that ancient geometers 34
had no need to physically measure the dimensions of the Earth and Moon
with scientific instruments in order to weave those dimensions into their
architecture.
The natural geometry employed in this method also demonstrates that the

apparent presence of Phi ( ) in the squared circle figure is basically an
illusion. What looks like Phi in the geometry of the figure is, in fact, the
expression √ 
  .
     

   
This value35 is extremely close to Phi (   



)—the difference
being only 
—but it‘s derivation is very different (as you can see).36
Key Relationships Within the Squared Circle
The total area of the square in the squared circle diagram is equal to
    . Esoterically, the number 4 symbolizes the prison of the Form
principle.
34
For example: The ancient Egyptians.
35
Note: I have used sacred Pi ( 
) in the calculation. When scientific Pi is used

the value is 1.6189931...
36
This is not to say, however, that the expression‘s close proximity to Phi is
insignificant.
Page63
In the diagram (below) the blue ‗Earth‘ circle is wholly contained within
the boundaries of the four sides of the square. The area of the blue circle

is equal to the Pi ( ) constant. (See below.)
The red circle in this sacred geometric scheme produces numbers that are

close to the value of Phi ( ). I don‘t know the reason for this coincidence ,
but it makes esoteric sense because, in the diagram, the red circle
symbolizes ‗non-linear‘ time and space . 37 Note, however, my previous
remarks explaining that the numbers, although close to Phi, are not
actually Phi. (Also, see below for further elucidation.)
Figure 18 - Calculating the Earth+Moon Circle
37
The Phi constant is associated with many spiral forms that occur in nature.
Page64
The radius of the red circle corresponds to the Earth+Moon radius (i.e.
5040 miles) and in the more abstract version of the squared circle diagram
this radius is close to the value of the square root of Phi ( √ 
). My belief,
however, is that the correct representation of the red circle‘s radius is
 .
Similarly, I believe that the true area of the red circle is 
, which is
very close in value to      
( ). Both of these expressions are
almost exactly equal to 5 square units. I maintain however, that 
is
the correct expression and produces the correct
co rrect value.

The area of the red circle ( 5 sq. units) can be understood to symbolically
transcend the bounded area of the square (4 sq. units). This symbolism is
based on the very different esoteric meanings of the numbers 5 and 4, and
their associated pentagram and square.
Figure 19 - The Square (4) Evolves to a Pentagram (5)
Page65
The various area ratios present in the figure are crucial to determining
(and verifying) many of the diagram‘s other dimensions. You will discover
as we proceed that they are all involved in an amazing process that
symbolizes transcendence beyond the physical plane. The key to the
whole scheme is the ratio 
.
This  ratio is critically important because it‘s our starting point.
Symbolically, it describes the initial conditions of the Earth circle, and
mathematically, it gives us the value of the radius of the larger red circle.
Knowing this ratio, we don‘t need to rely on the scientifically measured
dimensions of the Earth and Moon in our calculations, or on an artificially
introduced Phi constant.
      

    
The  ratio is rounded to  

from which is a value
derived using the sacred Pi fraction ( ). When the standard scientific
value of Pi is used, the figure is  which when rounded down
produces the same approximation:
approximation: .
It is quite remarkable that  happens to provide us with the correct

conversion ratio for the combined radii of the Moon and the Earth!
In the diagram, the radius of the blue circle represents the radius of the
Earth. It is designated as 1 unit. The radius of the red circle adds the
radius of the Moon to the Earth‘s radius—correctly asserting that the
Moon‘s radius is equivalent to  of the Earth‘s radius. The estimated
mean radius of the Earth is 3960 miles. Therefore, the mean radius of the
Moon is 
  
  
 . This is the commonly accepted
approximation.
Page66
It has been noted by past researchers that the ‗Earth+Moon‘ radius of the
red circle is very close to the value of the square root of Phi ( √  ), and
this is, of course correct. But the true derivation of the radius comes from
the  ratio—i.e. the ratio between the area of the square (4) and the

area of the inscribed blue circle ( ) representing the Earth. There is an
important symbolic reason for this, which is explained
explained below.
      

     
     √   

 
The closeness of these two values suggests a symbolic association

between the red circle and the Phi constant.
Figure 20 - The Phi Connection
This close link also carries over to the area value of the red circle. The
total area of the red Earth+Moon circle is calculated as …
Page67
       
        

   


The  fraction is derived directly from the inherent geometry of the
squared circle. The alternative Phi-based calculation      
( ) is
alien to the natural geometry of the figure. It‘s introduced artificially in
order to point out an interesting approximation (and correspondence) to
the true  result of 
.
The ratio of the area of the red circle to the area of the blue circle is…
 : 
       

      


Page68
 
This result is also quite close to the value of Phi ( )— but it‘s
definitely not Phi because it‘s based on a different geometric principle.
The principle it‘s based on is naturally present in the geometry of the

circle inscribed within a square. The numerical closeness, however, does
suggest another symbolic link to the famous mathematical
mathematical constant
co nstant Phi.
The right-angled triangle in the diagram is a fascinating geometric figure.


One side gives the radius of the blue Earth circle ( ). Another side gives
the radius of the red Earth+Moon circle ( 
), and the hypotenuse value
( √ 
 ) is remarkably close to Phi.
Page69
The position of the triangle in the diagram suggests that the hypotenuse
may symbolize a path of transition from the blue Earth circle to the larger
red Earth+Moon circle —i.e. because the hypotenuse links the
circumference of the former with the circumference
circumference of the latter.
Thus, the triangle‘s various dimensions reveal some very interesting

geometric relationships
relationships that exist between the blue and red circles.
The geometry of the squared circle diagram above provides (in principle)
the solution to the challenge of squaring the circle because the
circumference of the red circle is exactly the same length as the
perimeter of the square. Solutions derived using other methods tend to
produce results that are only approximations.
approximations.
  

   
    
In standard presentations of the solution the circumference is shown only

to closely approximate the length of the square‘s perimeter. That is
because in standard presentations the standard approximation 1.273 (or
 ) is used instead of the key geometric ratio  . When the key ratio is

used, the two instances of Pi ( ) in the calculation cancel each other out
and leave an exact figure of 8 units for the length of the circumference.
If, instead, the approximate figure 1.273 is used in the calculation (i.e. in
place of  ) the result is 7.99849 which is, of course,
course, not as satisfying.
Page70
Figure 21 - Perimeter = Circumference
As you can see, the squaring of the circle has essentially transformed
linear measurement (i.e. the sides of the square) into non-linear
measurement (i.e. the circumference of the circle). The straight-line
distance between the square‘s black dots is two units, as is the curved
distance between the circle‘s red dots. This image, therefore, is
reminiscent of the diagram used previously to show the difference
between the side of a hexagon and a radian.
The special ratio that provided the key to this transformative process is
. It corresponds with the Earth+Moon mean radius of 5040 miles —which
Page71
is the number of spiral arcs (i.e.   ) in a Year of Brahma (or double
UE Cycle).
Figure 22 - Hexagon Side vs. Radian
The final ratio we need to examine is the ratio between the area of the
red circle and the area of the square…
   

  
   
Page72
Significantly, the result is precisely the same ratio as our starting ratio —
i.e. the ratio between the area of the square and the area of the blue
Earth circle…
  
   
  

The relationship between the three figures —the red circle, the square,
and the blue circle, can thus be expressed as follows:
    

Which reduces to …
   
 
(The square is represented by the middle term of each ratio.)
Page73
This is a final verification that our methodology is correct. The 

ratio is
the ratio that, right at the start, enabled us to begin the process of
squaring the circle. It did so by providing us with the correct length of the
radius of the red Earth+Moon circle. And so we end at the beginning —with
one important difference.
Figure 23 – The Two Expressions of the 4/Pi Ratio
The area of the red circle is seen to symbolically ‗overcome‘ the

limitations of the square (which the blue Earth circle could not do). The

new ‗evolved‘ version of the ratio (above right) has the square ( ) as the
denominator (i.e. positioned below) and the red circle ( 
) as
symbolically ‗transcendent‘ (i.e. positioned as the numerator above the
square).
It appears, therefore, that the original ratio of the area of the square to
the area of the Earth circle (i.e. 
) is a symbolic reference to the
Page74
original restrictive physical conditions of the Earth —i.e. when it initially is

wholly contained by the elemental square of linear-base
l inear-based
d materialism.
The solution to Earth overcoming the limitations of the dominion of the

square is symbolized by the ratio of the area of the red Earth+Moon circle
to the area of the square. In this new expression of the 
ratio the red
circle assumes the dominant position above the elemental square,
symbolizing both a rising above, and an expansion beyond the original
limitations of the Form principle. The circumference of the red circle
exceeds the boundaries of the square symbolically suggesting the
emergence of consciousness from the ‗egg of matter‘. This is a profound
sacred geometric message that appears to be programmed into the
physical structure of our Earth-Moon system.
The red circle can therefore be understood to symbolize the Earth in an

evolved state—i.e. after it has evolved beyond the limitations of the Form
principle (as represented by the square).
Symbolically speaking, the means by which the blue Earth circle is able to
make the transition to (and become) the red ‗evolved and expanded‘
circle is via the hypotenuse expression√ 
  . Its value is extremely
close to Phi, and therefore may symbolically refer to a spiral path. Be that
as it may, it is also likely that the √ 
  expression has a deep
symbolic meaning (and esoteric application) in its own right. All we need
to do is find it.
The Great Pyramid of Giza
From the above we can deduce that the dimensions of the Great Pyramid
of Giza are based on the squared circle diagram (i.e. as presented in this
Page75
book). The standard explanation for the pyramid‘s dimensions usually


involves an artificially (and convolutedly) introduced Phi ( ) constant.
Figure 24 - The Great Pyramid - Phi
But as I have demonstrated, the most likely and most appropriate

geometric basis for the proportions of the Great Pyramid is the ratio
.
Figure 25 - The Great Pyramid - 4/Pi
Page76
The  ratio is approximately equal to 

when the Egyptian sacred Pi
fraction  is substituted for Pi. This result increases the likelihood that
the ancient Egyptians employed the geometric model described herein
because their measurement system was based on the number 28.
 

   


    


 

   


The  connection becomes easier to see when the following diagram
is considere
co nsidered
d…
Figure 26 - The Ancient Egyptian 4/Pi Ratio
Page77
The  ancient Egyptian approximation of the  ratio is actually

derived from the squared circle diagram above.
The total area of the grey square is  . The area of the blue circle
is —this being the ancient Egyptian approximation for Pi. Therefore,
the ancient Egyptian 
ratio is calculated as…
4: 
:
  
 
   
So, the expression   describes the area of a square with a side
length of two units, while a circle inscribed within that square has an area
of and a radius of one unit. Is it a coincidence that the ancient
Egyptian system of measurement is fundamentally based on all three
numbers included in this simple mathematical expression?
  

   
   
   
 
Page78
Designing the Great Pyramid
In his book Das Ratsel der Pyramiden 38 , Kurt Mendelssohn presents an

ingenious diagram showing how the ancient Egyptians could have squared
the circle using the 
principle.
His methodology is based on a stack of four adjacent circles of 1 unit

diameter—or what I have termed ‗creation units‘.
Figure 27 - The Mendelssohn Methodology
To draw the radius of the blue Earth circle (and mark the midpoint of the
perimeter of the grey square) one creation unit circle is rolled the length
38
Das Ratsel der Pyramiden , Kurt Mendelssohn. Pg. 85. Weltbild Verlag GmbH,
Augsburg 1993.
Page79
of its circumference (i.e. from a to b, as shown above). The length of the

blue circle‘s radius ( and the circumference of each creation unit circle) is

equal to Pi ( ). The Pi radius corresponds to the distance from the centre
of the base of the Great Pyramid to the midpoint of its base side. In other
words, it represents half the length of the base of the Great Pyramid.
P yramid.
The corresponding proportional height of the Great Pyramid is determined

by adding together the four ‗creation unit‘ diameters (as per the diagram
above).
So the resulting right-angled triangle resembles the famous 3-4-5

Pythagorean triangle except that the ‗3‘ is replaced with Pi (i.e.
) and the hypotenuse becomes  (i.e.). 39
Figure 28 - The Ancient Egyptian Pyramid Triangle
Significantly, the three sides of the triangles (above) mirror the area
values of our original squared circle diagram. In that diagram the area of
the grey square is equal to 4, the area of the blue circle is equal to Pi ( ), 
and the area of the red circle is equal to  
.
39
Note that this value is very close to  
−that is, the area of the red
Earth+Moon circle in our original squared circle diagram. It is also very close to
   as previously explained.
Page80
    

When the sacred Pi fraction is used in the calculation then the triangle‘s
hypotenuse becomes  , or  . The fractional value  is
what the ancient Egyptians would have used when they were employing
the principle to determine the dimensions of the Great Pyramid.
Thus, Mendelssohn‘s diagram reveals a practical ly useful method for

building the principle into the design of the Great Pyramid of Giza. In
my opinion, this methodology proves that the Great Pyramid was built in
accordance with the  ratio, and that its association with the Phi
constant (i.e.  ) is merely a coincidence —albeit an esoterically
meaningful coincidence.
The strongest evidence that the ancient Egyptians used this methodology
when they designed the Great Pyramid is revealed in the numbers that
they chose for the pyramid‘s dimensions. When Mendelssohn‘s pyramid
blueprint is used as a basis, these specially selected figures allow for
incredibly easy and precise conversions during the pyramid building
process. And this in turn, suggests that the ancient Egyptian system of
cubit measurement was originally based on the geometry of the Great
Pyramid.
The height of the Great Pyramid is estimated to have been  
.
This value is quite probable because the ancient Egyptians employed a
measure of   to one cubit. As explained previously, their
measurement system appears to have been closely associated with the
expression    .
Page81
If we convert the elements of Mendelssohn‘s basic diagram to the known

dimensions of the Great Pyramid we find that the length of the
circumference of each of the four creation unit circles must be .  
The side length of the Great Pyramid is thought to have been ,  
so one rotation of a creation unit circle needs to measure half that length
(i.e.  
) from the centre of the base to the midpoint of the base
side.
Of great significance is the fact that a circle whose circumference

measures  
has a diameter of almost exactly .40  
Figure 29 - Dimensions of the Great Pyramid in Cubits
40
The precise figure is 
  
.
Page82
The fact that four adjacently stacked circles with a diameter of

  each have a circumference of exactly  
is surely
beyond coincidence. These round numbers (of cubits) indicate the
deliberate intention of the builders to choose measurements that are
easily derived from the pyramid blueprint (as presented above). To
my mind, this proves beyond doubt that the ancient Egyptians designed
the Great Pyramid using the ratio, and that their measurement system
is also based on it.
Notice that the four ‗creation unit‘ pyramid circles reflect the ancient
Egyptian sacred Pi fraction.
Figure 30 - Pyramid Pi Circle
Strongly reinforcing the pyramid design theory is the fact that we also
see in these cubit figures another allusion to the expression    .
     
  
Page83
The above expression is relevant to the design of the Great Pyramid only
if the builders employ ed Mendelssohn‘s pyramid design methodology. As
previously explained, the ancient Egyptian system of measurement is
fundamentally based on the numbers 4, 7, and 28.
 
   
 
  

 
    

 
 
   

       
Here are some interesting WWW references pertaining to the geometry of

the Great Pyramid of Giza.
Pi and the Great Pyramid
Mystic Places – The Great Pyramid
The Golden Ratio & Squaring the Circle in the Great Pyramid
Squaring the Circle: Marriage of Heaven and Earth
Page84
The 4/Pi Spiral
From the above, it appears that a hidden message in the design of the
Great Pyramid is that Space (like Time) is also curved and spiral-shaped
(i.e. non-linear). This is suggested by the fact that encoded within the
physical dimensions of the Great Pyramid is a representation of the 4/Pi-
based ‗squared circle‘ blueprint. But how are we to understand this notion
of Space being non-linear and spiral-shaped?
I have intimated (above) that the squared circle diagram symbolizes a

basic principle of evolution in the universe. I will now elaborate further on
that theme.
Symbolically speaking, a circle enclosed within a square represents:
Square − The fixed limitations of the environment within which an

organism (or organised collective) evolves.
Circle – An organism that is evolving within the fixed limitations of

its environment (i.e. within the square).
Figure 31 - Organism and Environment
Page85
In the diagram, the blue circle

c ircle symbolizes
symbolizes a singular
s ingular autonomous
autonomous entity (or
unified collection of entities). The entity is evolving along a path of
evolution, but its development is constrained by a set of fixed laws that
characterize its natural environment (i.e. as symbolized by the square).
sq uare).
As evolution proceeds, eventually the entity reaches a certain limit (or

plateau) of growth within its environs. As this critical threshold is reached
the entity spontaneously develops a new ‗higher‘ faculty that facilitates
its entry into a new realm of experience —i.e. into a qualitatively different
environment in which it can continue to evolve its newly acquired faculty.
Modern science has called this mysterious evolutionary process the

principle of ‗emergence‘. It is a relatively new scientific theory that
recognizes and acknowledges the periodic occurrence of major
evolutionary ‗jumps‘ to new levels of potentiality . These ‗quantum jumps‘
produce organic complexities and novel qualities unseen in lower levels,
and they build upon and (in a sense) transcend the activity of the lower
levels. According to the ‗strong emergence‘ view -point the newly
emergent phenomenon also has a top-down causal influence on the lower
level phenomenon upon which it is overlaid. Therefore, we see in this
theory of Emergence the approach of science to the metaphysical
teachings of Theosophy.41
A good example of the principle of Emergence at work is the major

evolutionary shift that occurred when homo-sapiens (as a collective
entity) developed the ability to use reason, and the human mental faculty
emerged. At a microcosmic level this process also involved the emergence
of the ‗ego‘ (i.e. a strongly unifying sense of individuality) within each
member of the human family. Thus, the theory of Emergence helps to
explain scientifically why the human race (at a critical stage of evolution)
collectively developed new qualities (and complexities) that
fundamentally distinguished
distinguished them from the other animal species.
41
For a comprehensive analysis of the theory of Emergence I thoroughly
recommend the book: Mind & Emergence by Philip Clayton. Oxford University
Press, 2004.
Page86
Figure 32 - The Process of Emergence
The process of Emergence is symbolized (in the diagram) by the

lengthening of the radius of the blue circle to that of the red circle. This
allows the entity to expand beyond the confines of the dark grey square
into a new environment with new boundaries and limitations —i.e. as
represented by the larger light grey square. Mathematically, in order to
achieve this expansion to a new level of emergence the radius of the
original blue circle is multiplied by 4/Pi (see below). Note however, that
the blue circle continues to exist and function within its original domain
(i.e. the dark grey square). The red circle represents
represents the further extension
of the entity‘s awareness into a new higher realm of activity.
As previously explained, the circumference of the red circle is the same

length as the perimeter of the dark grey square —i.e. that contains its
lower expression (the blue circle). This correlation indicates (in a symbolic
manner) that the entity has achieved a ‗breakthrough‘. The entity has
successfully integrated the evolutionary requirements of its previous
environment (i.e. as represented by the dark grey square) and this has
triggered an expansion (of consciousness) to the next level of emergence.
Page87
The newly emerged evolving entity is now represented by the red circle,
or to put it more accurately, the red circle represents a new level of
awareness for the evolving entity. It still maintains its connection with
(and awareness of) the lower level of its manifestation (as represented by
the blue circle). The light grey square symbolizes the new environmental
parameters that constrain the entity‘s growth in its newly emerged form.
When the entity eventually reaches the next threshold of evolution a new
‗higher‘ level of awareness will emerge and the red ci rcle will expand
beyond the natural limitations that have been fixed by the light grey
square. In mathematical terms, the radius of the red circle (like that of
the blue circle before it) will be multiplied by 4/Pi thereby enabling it to
shift to the next ‗higher‘ level of emergence.
Figure 33 - The 4/Pi Principle & Emergence
Page88
Thus, according to the theory of Emergence, the process of evolution is

punctuated by an intermittent series of transcendent jumps to new higher
levels of manifestation (and also awareness, or being). Symbolically
speaking, the ‗quantum jump‘ to each new level of emergence is made
possible (and implemented) by the 4/Pi principle. In the diagram (below)
the radius of every concentric circle in the expanding series is multiplied
by 4/Pi in order to produce the next larger concentric circle. This has the
effect of producing a continuous spiral that moves outward from the
circumference of one circle to the next.
Figure 34 - The 4/Pi Spiral
Page89
In the diagram above, the radius of the original blue circle is  
. The
next higher level of emergence is represented by a circle with a radius of
 
  
 . The following level of emergence is represented by a
   
circle of radius . Then, the next level of emergence is
characterized by a circle whose radius is equal to     
 
.
Thus, symbolically speaking, the appearance of each consecutive level of
emergence involves a multiplication
multiplication of the 4/Pi principle.
Notice (in the diagram) an interesting geometric property of the first two
4/Pi spiral triangles …
Figure 35 - The 4/Pi Spiral Triangle
As previously explained, the length of side ‗a

‗ a‘ of the triangle above is very

close to the Phi ( ) constant, but it is derived differently to Phi. This
Page90
coincidence means that the 4/Pi spiral triangle is quite similar to the
famous Phi ‗Golden Triangle ‘.
The 4/Pi spiral also somewhat resembles the ‗Fibonacci spiral‘ , but in
fact, they differ quite considerably. The Fibonacci spiral is based on a
series of adjacent squares, whereas the 4/Pi spiral is based on a series of
adjacent triangles the sides of which represent the radii of a series of
‗squared circle
c ircles‘.
s‘.
Figure 36 - The Fibonacci Spiral
[Image courtesy of Wikipedia.]
As you can see, each section of the Fibonacci spiral is contained within a
perfect square. The equivalent sections of the 4/Pi spiral are contained
within rectangular forms.
The 4/Pi spiral is thus symbolic of an evolutionary movement through a

progressive series of distinct, but interdependent, emergent levels of
Page91
manifestation. Entry into each new level is symbolized by the squaring of

the circle—a feat that‘s achieved using the 4/Pi principle.
The Great Pyramid can therefore be said to symbolize the profoundly

significant periodic ‗quantum leaps‘ into new realms of manifestation and
experience that naturally occur during the course of evolution. Another
name for this rare and mysterious aspect of the evolutionary process is
‗Initiation‘.
Page92
Thank you for reading this book.

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like to make a donation!
Thank you!
Page93
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What are some of the main concepts discussed in the text?

What are some of the main concepts discussed in the text?

How does the text describe the relationship between time, space and evolution?

How does the text describe the relationship between time, space and evolution?

The Divine Measure of Time & Space January 27, 2013

[This is a work in progress … ]

AstroQab Publishing 2010

The Divine Measure of

Figure 1 - The Creation Unit ......................

The Divine Measure of Time & Space

Repetitive cycles allow us to predict future outcomes based on past

Modern humanity understands that our planet‘s day/night cycle is due to

Theoretically, and based on this reasoned perception, the rational mind

I have used the Earth‘s natural seasonal cycle as an example to

This book uses sacred geometry and mathematics to determine and

and structure of the prototypical universal esoteric cycle, which as you

The Creation Unit

Our quest to determine the structural nature of the prototypical universal

Figure 1 - The Creation Unit

A creation unit has a diameter of 1 and the length of its circumference is

Pi is a mathematical constant. It is classed as an irrational number —

The length of the circumference of a circle is almost exactly equal to

The discrepancy is represented by the irrational transcendental decimal

The Pi constant is equal to:

It is remarkable that mathematicians have decided to use the words

The Sevenfold Circle of Creation

Like its component creation units, a universe is represented by a circle

The various characteristics of the geometry of a circle conceal with a veil

The Circle of Creation is produced by positioning seven creation unit

Figure 2 - The Sevenfold Circle of Creation

The Mundane Cycle

Our normal ‗rational‘ perception of the mundane world is represented by

Figure 3 - The Hexagonal Division of the Circle

The illusory nature of the hexagonal division of space/time is represented

The rational mind sees a sixfold division of the circumference and

The earliest evidenced conscious use of an accurate approximation for the

The fraction  is equal to 3  where the irrational decimal portion of Pi

Because the seventh creation unit circle is

The rational mind does not see the

Figure 4 - Circle Radius = Hexagon Side

Another way of understanding the illusory perception that the hexagonal

circumference. The reason that it fails to do so is because the six sides of

The Universal Esoteric Cycle

In order to determine the structure of the universal esoteric cycle we

As it turns out, science has already discovered the esoteric measure of a

Figure 5 - The Radian Wheel

[Base image courtesy of Wikipedia.]

A radian is the esoteric (or ‘non-

Figure 6 - Radian vs. Hexagon

A radian represents the length of the radius of a circle as measured along

A radian incorporates in its division of the circumference of a circle the

Here‘s a mathematical demonstration of this comparison…

A creation unit circle has a diameter of one and a circumference equal to

The Sevenfold Spiral

The division of a circle‘s circumference by a hexagon resul ts in six equal

In reality, however, the universe is unpredictable, and regular cycles are

The radian-based (or radiagon-based) model we are developing and

A radiagon is a hypothetical abstract geometrical figure with 44 curved

Figure 7 - The Universal Esoteric Cycle