The Hexagonal Tree of Life

The Hexagonal Geometry of the Tree of Life November 17, 2010

The Hexagonal Geometry
Of the Tree of Life
By Patrick Mulcahy

This short treatise explores (using polyhex mathematics) the special
relationship that exists between the kabbalistic Tree of Life diagram and the
geometry of the simple hexagon. It reveals how the ten sefirot and the twenty -
two pathways of the Tree of Life are each individually and uniquely linked to
the hexagonal form. The Hebrew alphabet is also shown to be esoterically
associated with the geometry of the hexagon.

This treatise is intended to be a primer for the further study and esoteric
application of polyhex mathematics.

First Edition (Expanded)
Copyright 2008 © Patrick Mulcahy
AstroQab Publishing 2008
Email: mallukh_ahi@yahoo.com
Website: http://www.astroqab.co.nr/

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Table of Contents

Preface .................................................................................................................................. 6
Note to the 2010 Edition ....................................................................................................... 7
General Introduction ............................................................................................................. 8
Introduction to the Polyhex Tree of Life .............................................................................. 11
The Polyhex Tree of Life ...................................................................................................... 15
The Three Highest Sefirot ................................................................................................ 15
The Seven Lower Sefirot .................................................................................................. 16
The Malkut Tetrahex ....................................................................................................... 18
The Holy Name ................................................................................................................ 20
The Twenty-two Pathways .............................................................................................. 21
The Shin Pentahex ........................................................................................................... 22
Towards A Pentahex Alphabet ............................................................................................ 24
The Twelve Simple Pentahexes........................................................................................ 27
Pentahex Word Forms ..................................................................................................... 31
Constructing Polyhex Tree of Life Mandalas ........................................................................ 34
Introduction .................................................................................................................... 34
A Tree of Life Hexagonal Mandala (34 Polyhexes) ........................................................... 35
Tree of Life Sefirotic Mandalas (10 Polyhexes)................................................................. 39
A Tree of Life ‘Snowflake’ Mandala (32 Polyhexes) .......................................................... 40
One Hundred and Forty-seven Hexagons ........................................................................ 42
Closer to the Tree of Life ................................................................................................. 50
A Pentahex Hexagonal Mandala (22 Pentahexes) ............................................................ 51
A Pentahex Triangular Mandala (22 Polyhexes) ............................................................... 52
A Pentahex Hexagonagram (22 Polyhexes) ...................................................................... 53
Polyhex Divination .............................................................................................................. 55
Appendix ............................................................................................................................. 57
The Polyhex Key to the Pathways .................................................................................... 57
The Geometric Model .................................................................................................. 57
Esoteric Services ................................................................................................................. 75

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List of Illustrations

FIGURE 1: THE HEXAGONAL POSITIONING OF THE TEN SEFIROT ........................................................................................ 8
FIGURE 2: THE MONOHEX & THE DIHEX ..................................................................................................................... 9
FIGURE 3: THE THREE TRIHEXES................................................................................................................................ 9
FIGURE 4: THE TWO PRIMORDIAL VEILS ................................................................................................................... 11
FIGURE 5: THE THREE TRIHEX SEFIROT ..................................................................................................................... 12
FIGURE 6: THE SEVEN TETRAHEX SEFIROT ................................................................................................................. 12
FIGURE 7: THE TWENTY-TWO PENTAHEXES ............................................................................................................... 13
FIGURE 8: THE SINGLE HEXAHEX CONTAINING A HOLE ................................................................................................. 14
FIGURE 9: THE FIRST THREE SEFIROT OF THE TREE OF LIFE ............................................................................................. 15
FIGURE 10: THE POLYHEX TREE OF LIFE .................................................................................................................... 17
FIGURE 11: TWO EXAMPLES OF TETRAHEXES TILING THE PLANE ..................................................................................... 18
FIGURE 12: THE MALKUT TETRAHEX TILING THE PLANE ................................................................................................ 18
FIGURE 13: THE BINAH & MALKUT POLYHEXES .......................................................................................................... 19
FIGURE 14: THE KETER-MALKUT HEPTAHEX .............................................................................................................. 19
FIGURE 15: THE KETER-MALKUT HEXAGRAM ............................................................................................................ 20
FIGURE 16: YHV & YHVH - EXAMPLE POLYHEXES ...................................................................................................... 21
FIGURE 17: TWO SIMPLE PENTAHEXES TILING THE PLANE ............................................................................................. 22
FIGURE 18: THE SHIN PENTAHEX TILING THE PLANE .................................................................................................... 23
FIGURE 19: THE BINAH , MALKUT, & SHIN POLYHEXES ................................................................................................. 23
FIGURE 20: THE SHIN PENTAHEX & HEBREW LETTER ................................................................................................... 24
FIGURE 21: THE TWELVE SIMPLE PENTAHEXES ........................................................................................................... 25
FIGURE 22: THE THREE MOTHER PENTAHEXES ........................................................................................................... 25
FIGURE 23: THE SEVEN DOUBLE PENTAHEXES ............................................................................................................ 26
FIGURE 24: THE TWELVE SIMPLE PENTAHEXES & HEBREW LETTERS ................................................................................ 27
FIGURE 25: THE THREE SIMPLE FIRE PENTAHEXES ....................................................................................................... 28
FIGURE 26: THE THREE SIMPLE WATER PENTAHEXES ................................................................................................... 28
FIGURE 27: THE THREE SIMPLE AIR PENTAHEXES ........................................................................................................ 29
FIGURE 28: THE THREE SIMPLE EARTH PENTAHEXES .................................................................................................... 29
FIGURE 29: THE TWO PENTAHEX TRIANGLES ............................................................................................................. 31
FIGURE 30: PENTAHEX ALPHABET YHVH .................................................................................................................. 32
FIGURE 31: PENTAHEX ALPHABET DBR .................................................................................................................... 32
FIGURE 32: THE ANGEL MICHAEL - PENTAHEX ALPHABET ............................................................................................. 33
FIGURE 33: THE FIRST TWELVE POLYHEXES FORM A HEXAGON ...................................................................................... 35
FIGURE 34: A PENTAHEX HEXAGONAL MANDALA ....................................................................................................... 36
FIGURE 35: A 34 POLYHEX TREE OF LIFE HEXAGON ..................................................................................................... 37
FIGURE 36: THE POLYHEX TREE OF LIFE HEXAGON - LABELLED ....................................................................................... 38
FIGURE 37: A TEN SEFIROT POLYHEX HEXAGRAM ....................................................................................................... 39
FIGURE 38: A TEN SEFIROT POLYHEX HEXAGON ......................................................................................................... 39
FIGURE 39: A PENTAHEX SNOWFLAKE MANDALA ....................................................................................................... 40
FIGURE 40: A TREE OF LIFE POLYHEX SNOWFLAKE ...................................................................................................... 41
FIGURE 41: A TREE OF LIFE POLYHEX TRIANGLE .......................................................................................................... 42
FIGURE 42: ANOTHER TREE OF LIFE POLYHEX TRIANGLE ............................................................................................... 43
FIGURE 43: THE APEX TRIANGLE IS MIRRORED ........................................................................................................... 44
FIGURE 44: A TRIANGULAR POLYHEX MANDALA - LABELLED .......................................................................................... 45
FIGURE 45: A TREE OF LIFE POLYHEX TRIPLE DIAMOND ................................................................................................ 46
FIGURE 46: TREE OF LIFE TRIPLE DIAMOND - LABELLED ................................................................................................ 47

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FIGURE 47: THE TREE OF LIFE TRIPLE DIAMOND BLUEPRINT .......................................................................................... 48
FIGURE 48: TREE OF LIFE TRIPLE DIAMOND MANDALA – HORIZONTAL VERSION ................................................................. 49
FIGURE 49: THE HORIZONTAL TRIPLE DIAMOND MANDALA - LABELLED ............................................................................ 49
FIGURE 50: CLOSER TO THE TREE OF LIFE .................................................................................................................. 50
FIGURE 51: A PENTAHEX HEXAGONAL MANDALA ....................................................................................................... 51
FIGURE 52: A PENTAHEX TRIANGULAR MANDALA ....................................................................................................... 52
FIGURE 53: A PENTAHEX HEXAGONAGRAM ............................................................................................................... 53
FIGURE 54: A HEXAGONAGRAM ............................................................................................................................. 54
FIGURE 55: COMBINED HEXAGON & HEXAGRAM ....................................................................................................... 54
FIGURE 56 : THE MERGING OF SPIRIT & FORM .......................................................................................................... 57
FIGURE 57 : TREE OF LIFE POLYHEXAGON ................................................................................................................. 58
FIGURE 58: THE GENESIS & TREE OF LIFE POLYHEXES .................................................................................................. 59
FIGURE 59 : THE MERGING OF THE SPIRIT & FORM POLYHEXES ...................................................................................... 60
FIGURE 60 : THE TREE OF LIFE POLYHEXAGON............................................................................................................ 61
FIGURE 61 : THE SOUL PRINCIPLE (GREEN) ............................................................................................................... 62
FIGURE 62 : POLYHEXAGON TREE OF LIFE — HEBREW ................................................................................................. 64
FIGURE 63 : TREE OF LIFE — TRADITIONAL & POLYHEX DESIGNS .................................................................................... 66
FIGURE 64 : THE HIERARCHICAL ARRANGEMENT OF THE HEBREW LETTERS ........................................................................ 68
FIGURE 65: TREE OF LIFE : 3+7+12 PATHWAYS ......................................................................................................... 71
FIGURE 66: THE POLYHEX TREE OF LIFE (REVISED) ...................................................................................................... 73

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Preface

This little book does not seek to expound upon, or try to explain the esoteric meaning of the
kabbalistic Tree of Life. My previous book Formation of the Tree of Life explored in some
depth those mysteries.

This treatise does, however, provide some thought-provoking geometric correspondences
that can deepen our understanding of the Tree of Life diagram through the process of visual
meditation and contemplation.

I have deliberately not elaborated on the possible meaning of much of this visual material.
Rather, I have preferred to let the images speak for themselves as much as possible.

This book has been written for students of the Kabbalah who already have some knowledge
of the kabbalistic Tree of Life.

I believe that the images contained herein have the potential to stimulate the intuitive mind
of the reader, and to open up pathways of understanding that perhaps exist just below the
surface of consciousness.

It is my hope that these images will stir the latent knowledge within you and bring to your
conscious awareness the divine illumination of the soul.

Patrick Mulcahy (16th June, 2008.)

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Note to the 2010 Edition

The main difference between this 2010 edition of The Hexagonal Geometry of the Tree of
Life and the original version is that I’ve added an Appendix.

The Appendix is significant because it adds another polyhex-based dimension to our study of
the kabbalistic Tree of Life.

The content of the Appendix is taken from the 2010 edition of my book The Formation of
the Tree of Life. The reason I copied it into this book is because I understand that not
everyone who is interested in the polyhex Tree of Life will want to wade through the rather
dense Kabbalah of Formation of the Tree of Life.

I hope you enjoy the new content.

Patrick Mulcahy (17th November, 2010.)

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General Introduction

As you may know, the pattern of the kabbalistic Tree of Life diagram is constructed using
the geometry of the hexagon as its basis. (See the new Appendix.)

The ten sefirot of the Tree diagram are positioned according to the principles of hexagonal
geometry.

Figure 1: The Hexagonal Positioning of the Ten Sefirot

What may be new to you is the fact that the symbolism of the hexagon extends also to the
ten individual sefirot and the twenty-two individual pathways of the Tree of Life.

The hexagonal geometry of the ten sefirot and the twenty-two pathways is based on what is
called 'polyhex mathematics'.

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A polyhex figure is one that is constructed by joining hexagons together adjacently in
various configurations.

Polyhexes are classified into groups according to the number of hexagons that are used in
the construction of each member of each group. Special names are used to identify each
group of polyhexes.

A ‘monohex’ (or ‘1-hex’) is used to describe a single hexagon. There is only one possible
monohex shape.

A ‘dihex’ (or ‘2-hex’) is a shape that incorporates two adjacent hexagons. There is only one
possible dihex shape.

Figure 2: The Monohex & the Dihex

A ‘trihex’ (or ‘3-hex’) is a shape that consists of three adjacent hexagons. There are only
three possible trihex shapes.

Figure 3: The Three Trihexes

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A ‘tetrahex’ consists of four adjacent hexagons, and a ‘pentahex’ consists of five adjacent
hexagons.

Below is a table showing the number of unique polyhexes in each of these groups.

Polyhex Number in Group
1-Hex 1 Monohex
2-Hex 1 Dihex
3-Hex 3 Trihexes
4-Hex 7 Tetrahexes
5-Hex 22 Pentahexes
Total: 34 Polyhexes

The mathematical sequence 1, 1, 3, 7, 22 is interesting because it describes the basic
elements of the structure of the kabbalistic Tree of Life.

Polyhex Number Tree of Life
0-hex 0 Ain (‘Absolute Unknown’)
1-hex 1 Monohex Ain Sof (First Veil)
2-hex 1 Dihex Aur (Second Veil)
3-hex 3 Trihexes 3 Highest Sefirot
4-hex 7 Tetrahexes 7 Lower Sefirot
5-hex 22 Pentahexes 22 Pathways

Thus, the hexagonal geometry that is used to construct the basic layout of the Tree of Life
diagram is also seen to extend to its individual component parts (i.e. to the sefirot and
pathways).

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Introduction to the Polyhex Tree of Life

Here is how I basically interpret the polyhex symbolism as it is applied to the kabbalistic
Tree of Life:

The first two groups of polyhexes symbolise the primordial veils: Ain Sof (‘Limitlessness’)
and Aur (‘Light’).

These two polyhex groups are significantly different to those that follow because they
contain only one polyhex each.

Figure 4: The Two Primordial Veils

It is not until we reach the third polyhex group (i.e. the 3-hex group) that variation becomes
possible and multiple forms are generated.

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Figure 5: The Three Trihex Sefirot

The three trihexes symbolise the highest three sefirot of the Tree of Life.

Figure 6: The Seven Tetrahex Sefirot

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Variation increases when a fourth hexagon is added to form the 4-hex group.

The seven tetrahexes of the 4-hex group symbolise the seven lower sefirot of the Tree of
Life.

When a fifth hexagon is added a total of twenty-two 5-hex forms are generated.

The twenty-two pentahexes correspond with the twenty-two pathways of the Tree of Life
(and also the 22 letters of the Hebrew alphabet).

Figure 7: The Twenty-two Pentahexes

Thus, the various components of the structure of the Tree of Life diagram can be seen to
correspond with the sequence of polyhex groups from 1-hex to 5-hex.

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The first four orders of polyhexes reflect the main body of the Tree diagram (i.e. the two
primordial veils and the ten sefirot) while the fifth order of polyhexes is associated with the
twenty-two pathways of the Tree of Life.

These first five classes of polyhexes are geometrically differentiated from the higher order
groups that follow them because none of the first five groupings contain a polyhex with a
hole in it.

It is not until the sixth order of polyhexes is reached that a hole appears, and then in only
one of the total 82 hexahexes in the 6-hex group.

Figure 8: The Single Hexahex Containing a Hole

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The Polyhex Tree of Life

The Three Highest Sefirot

The principles of hexagonal geometry upon which these polyhex forms are based allow the
figures in each polyhex group to be fitted together adjacently in the manner of a jigsaw
puzzle. But unlike a jigsaw puzzle, polyhexes can form multiple and various shapes
depending on how they are arranged.

As I mentioned previously, the single monohex symbolises the first primordial veil–called
‘Ain Sof’, and the single dihex represents the second primordial veil–called ‘Aur’.

Significantly, these two polyhexes fit snugly together to form (what I consider to be) the first
trihex–corresponding to the first sefirah of the Tree of Life, called Keter.

Figure 9: The First Three Sefirot of the Tree of Life

So the monohex of Ain Sof combines with the dihex of Aur to produce the first trihex
representing Ain Sof Aur (‘Limitless Light’) and the first sefirah Keter of the Tree of Life.

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The two remaining trihexes then naturally align with the sefirot Hockmah and Binah of the
Tree of Life.

The symbolism of Ain Sof (‘Limitlessness’) and Aur (‘Light’) combining to form the first
sefirah Keter reflects the idea of Keter being so close to the Limitless Light (Ain Sof Aur) that
their point of separation is barely definable.

I have assigned the ‘straight line’ trihex to Hockmah and the curving trihex to Binah.

Thus, the Keter trihex resembles and symbolises a ‘point’, the Hockmah trihex symbolises a
‘line’, and the Binah trihex symbolises a ‘curve’.

This basic symbolism is also reflected downward in the tetrahex sefirot directly beneath
Keter, Hockmah and Binah–i.e. in Tifaret, Hesed and Geburah. (See below.)

You will notice that the curving shape of the Binah trihex is also reflected down to the
Malkut tetrahex.

The Seven Lower Sefirot

In arranging the seven tetrahexes upon the Tree of Life diagram I have used a rationale that
links the higher three sefirot with the lower seven sefirot.

The Tifaret tetrahex is thus a natural extension of the Keter trihex.

Similarly, the Hesed tetrahex is a natural extension of the Hockmah trihex, and the Geburah
tetrahex builds naturally upon the Binah trihex.

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Figure 10: The Polyhex Tree of Life

In Geburah the Binah curve becomes an undulation–like the twisting movement of a
serpent.

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The Malkut Tetrahex

The Malkut tetrahex has been assigned because it has a unique geometric property that
separates it from the other six tetrahexes in the 4-hex group. (Note that its shape also bears
a significant resemblance to the Binah trihex.)

The first six tetrahexes are able to ‘tile the plane’ by translation in two directions. In other
words, they can completely cover an area by being stacked on top of each other.

Figure 11: Two Examples of Tetrahexes Tiling the Plane

The seventh (or Malkut) tetrahex, on the other hand, is unable to tile the plane in that
manner. It is only able to tile the plane by rotating 180° in the plane.

Figure 12: The Malkut Tetrahex Tiling the Plane

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You will notice a similarity between the shape of the Binah trihex and the Malkut tetrahex.

The Binah-Malkut polyhex geometric similarity reflects the special relationship that is
thought to exist between these two sefirot.

Figure 13: The Binah & Malkut Polyhexes

Interestingly, this assignment of the Malkut tetrahex also means that it is able to combine
symmetrically with the Keter trihex to form (what I consider to be) the first heptahex of the
7-hex order of polyhexes.

Figure 14: The Keter-Malkut Heptahex

The Keter-Malkut heptahex is similar in structure to a hexagram (or Jewish ‘Star of David’).

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Figure 15: The Keter-Malkut Hexagram

The symbolism of this Keter-Malkut polyhex relationship is (I feel) quite apposite with the
esoteric meaning of the Tree of Life diagram.

The Holy Name

The seven tetrahexes, by definition, consist of clusters of four hexagons. It is tempting
therefore, to associate their component hexagons with the four letters of the holy
Tetragrammaton YHVH.

Similarly, the three hexagons of the three trihexes can be associated with the three letters
of the holy Trigrammaton YHV.

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Figure 16: YHV & YHVH - Example Polyhexes

The Twenty-two Pathways

It is not as easy to assign the twenty-two pentahexes individually to their Tree of Life
pathways as it is to assign the other polyhexes to their associated sefirot.

One major clue that helps us however, is that twelve of the twenty-two pentahexes are
geometrically differentiated from the others and therefore appear to correspond with the
twelve Simple letters of the Hebrew alphabet (i.e. as set forth in the book: The Sefer
Yetzirah).

Like the six Hesed to Yesod tetrahexes, these twelve pentahexes are able to ‘tile the plane’
(without rotation) by being stacked on top of each other.

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Figure 17: Two Simple Pentahexes Tiling the Plane

I am unsure (at this stage) whether the remaining ten tetrahexes can be somehow
geometrically subdivided into a group of three and a group of seven–i.e. corresponding with
the subdivision of the Hebrew alphabet into three Mother letters and seven Double letters.

It is known however, that they can be differentiated geometrically into two groups of one
and nine.

The Shin Pentahex

Of these ten, a single pentahex is distinguished from all the other twenty-one pentahexes
because it can only tile the plane ‘isohedrally’.

This means that the layered symmetry of the tiling process is periodically interrupted by a
necessary rotation of the pentahex shape.

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Figure 18: The Shin Pentahex Tiling the Plane

I have chosen to assign this special pentahex to the path of Shin.

The Hebrew letter Shin is one of the three Mother letters of the Hebrew alphabet.

I have assigned the Shin pentahex primarily because it bears a geometric similarity to the
Binah trihex and the Malkut tetrahex.

Figure 19: The Binah, Malkut, & Shin Polyhexes

The kabbalistic system of the Sefer Yetzirah associates the Hebrew letter Shin with the
sefirah Binah (and therefore also indirectly with the sefirah Malkut).

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In my understanding the Hebrew Mother letter Shin also has a special esoteric association
with the twelve Hebrew Simple letters. (Similarly, the Mother letter Maym is associated
with the seven Hebrew Double letters, and the Mother letter Alef with the three Hebrew
Mother letters themselves.)

Interestingly, you will notice that the Shin pentahex bears a certain resemblance to the
actual shape of the Hebrew letter Shin.

Figure 20: The Shin Pentahex & Hebrew Letter

Towards A Pentahex Alphabet

The twenty-two pentahexes can be associated with the twenty-two letters of the Hebrew
alphabet.

Twelve pentahexes are geometrically different to the other ten and therefore can
reasonably be assigned to the twelve Simple Hebrew letters.

These twelve are in fact actually geometrically ‘simpler’ than the other ten pentahexes in
the 5-hex group, which is why they are able to tile the plane in the way that they do.

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Figure 21: The Twelve Simple Pentahexes

Interestingly, due to the unique irregularities of the shapes of the remaining ten pentahexes
they can all be assigned to the three Mother and seven Double letters of the Hebrew
alphabet by similarity of shape.

Figure 22: The Three Mother Pentahexes

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Figure 23: The Seven Double Pentahexes

If you are very familiar with the Hebrew alphabet then you may notice some quite
remarkable similarities between these ten pentahexes and the three Mother and seven
Double Hebrew letters.

The pentahex associated with the Hebrew letter Peh is probably the least like its Hebrew
letter counterpart, but it’s not too dissimilar to the ‘final form’ of the letter Peh, or the
modern English letter P.

The Tav pentahex is also quite dissimilar to the modern shape of the Hebrew letter Tav, but
it resembles quite closely the primitive form of the ancient Hebrew Tav. This ancient Tav
shape is identical to its modern English language equivalent–the letter T.

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The Twelve Simple Pentahexes

As I mentioned above, unfortunately it is difficult to assign the twelve simple pentahexes to
their Hebrew letter equivalents by shape in any obvious way.

Therefore, the approach I have taken is to identify processes of change–four in particular–
that symbolically move through the shapes of the twelve pentahexes.

Figure 24: The Twelve Simple Pentahexes & Hebrew Letters

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Following the lead of the Sefer Yetzirah I have organised the twelve Simple pentahexes into
four astrologically based groupings.

I have assigned the three Fire sign pentahexes according to a process that moves through
them–a process that resembles the raising of a tower.

Figure 25: The Three Simple Fire Pentahexes

The Heh (Cardinal Fire) pentahex symbolises the initial lifting and pushing vertically forward
and upward of the tower structure. The Sameck (Mutable Fire) pentahex represents the
second stage of the tower erection process. The Tet (Fixed Fire) pentahex represents the
completed erection and stabilization of the tower.

The three Water sign pentahexes describe a process that resembles a wave breaking on the
surface of the ocean.

Figure 26: The Three Simple Water Pentahexes

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The Het (Cardinal Water) pentahex represents the initial surging and mounting of the wave-
form. The Qof (Mutable Water) pentahex then shows the wave-form breaking. And finally,
the Nun (Fixed Water) pentahex symbolises the flattening and stabilising of the ocean (i.e.
after the wave has broken).

The three Air pentahexes simulate the movement of a pendulum.

Figure 27: The Three Simple Air Pentahexes

The Lamed (Cardinal Air) pentahex symbolises the initial pulling back of the weight of the
pendulum. The Zayn (Mutable Air) pentahex represents the swing of the pendulum to its
opposite extremity. The Tzaddie (Fixed Air) pentahex shows the weight of the pendulum
returned to a balanced state of equilibrium.

The three Earth pentahexes symbolise the process of a stone block being pushed
horizontally into position.

Figure 28: The Three Simple Earth Pentahexes

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The Ayin (Cardinal Earth) pentahex represents the initial push that starts the stone block
moving. The Yud (Mutable Earth) pentahex represents the actual movement and
momentum of the block as it slides forward towards its final position. The Vav (Fixed Earth)
pentahex symbolises the stone block stabilised and at rest after it has been pushed into
place.

Of course, I am unsure whether this method I have used to assign the twelve pentahexes to
the twelve Simple Hebrew letters is appropriate or correct, but I think it serves as a good
starting point for experimentation.

Please feel free to assign them differently if you discover a system of allocation that is more
to your liking.

The Simple letter zodiacal correspondences are as follows:

Heb. Letter Quality Astro. Sign
Heh Cardinal Fire Aries
Sameck Mutable Fire Sagittarius
Tet Fixed Fire Leo
Het Cardinal Water Cancer
Qof Mutable Water Pisces
Nun Fixed Water Scorpio
Lamed Cardinal Air Libra
Zayn Mutable Air Gemini
Tzaddie Fixed Air Aquarius
Ayin Cardinal Earth Capricorn
Yud Mutable Earth Virgo
Vav Fixed Earth Taurus

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Pentahex Word Forms

The hexagonal geometry of the twenty-two pentahexes means that they can be joined
together in various configurations and patterns.

Figure 29: The Two Pentahex Triangles

For example, the twenty-two pentahexes when combined in a certain way are able to form
two equilateral triangles of eleven pentahexes each, and there are various other interesting
geometric shapes that can also be built using the twenty-two pentahexes.

In the figure above, the three Mother pentahexes are coloured yellow, the seven Double
pentahexes are coloured blue, and the twelve Simple pentahexes are coloured red.

All this leads us to an interesting application of the twenty-two pentahexes.

If we view the pentahexes as individual letters of a geometrically based alphabet then words
constructed using this alphabet will assume unique shapes according to the way that their
component letters are fitted together.

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For example, the four letters of the Holy Name YHVH can be arranged like this:

Figure 30: Pentahex Alphabet YHVH

And there are other arrangements of the four YHVH pentahexes that are also possible and
that produce different shapes.

The following figure illustrates another example of a word constructed using the pentahex
alphabet.

Figure 31: Pentahex Alphabet DBR

The example pentahex word-form shown in the figure above is the Hebrew word DBR,
meaning ‘word’.

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Pentahex word-forms can be employed magically wherever sigils are normally used.

For example, angelic names can be formed from the pentahex letters and the resulting sigils
used in magical operations.

Figure 32: The Angel Michael - Pentahex Alphabet

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Constructing Polyhex Tree of Life Mandalas

Introduction

As you know, there are thirty-four polyhexes included in the first five orders of polyhexes,
from 1-hex to 5-hex.

These thirty-four polyhexes symbolise the two primordial veils, the ten sefirot, and the
twenty-two pathways of the Tree of Life.

The kabbalistic Tree of Life however, is traditionally ascribed only thirty-two components–
i.e. the thirty-two ‘paths of wisdom’ of the Sefer Yetzirah.

The two primordial veils, Ain Sof (1-hex) and Aur (2-hex), are generally omitted from the
Tree of Life component tally as they are thought to be beyond manifestation, or outside of
the realm of the Creation.

There are thus, two basic types of polyhex Tree of Life mandalas that can be assembled–a
34 polyhex type mandala, and a 32 polyhex type mandala.

A 32 polyhex mandala utilizes only the 3-hex, 4-hex, and 5-hex polyhexes, whereas the 34
polyhex mandala also includes the 1-hex and 2-hex polyhexes.

But, in actuality, the 32 polyhex Tree of Life mandala also discretely contains the 1-hex and
2-hex polyhexes in the structure of the first of the 3-hex polyhexes (i.e. the trihex that is
allocated to the first sefirah Keter).

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The sefirah Keter, as you know, represents the first subtle manifestation of the Ain Sof Aur
and as such, exhibits in its own polyhex form the union of the 1-hex (Ain Sof) and 2-hex
(Aur) polyhexes.

A Tree of Life Hexagonal Mandala (34 Polyhexes)

The thirty-four polyhexes (i.e. 1-hex to 5-hex) can all be fitted together to form a remarkable
composite hexagon figure.

To create this polyhex Tree of Life mandala we begin by arranging the first twelve polyhexes
in the following hexagonal pattern:

Figure 33: The First Twelve Polyhexes Form a Hexagon

These twelve polyhexes represent the main sefirotic body of the polyhex Tree of Life (plus
Ain Sof and Aur).

The various components of the Tree of Life represented in the hexagon (figured above) are
colour-coded as follows:

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o Ain Sof: White
o Aur: Grey
o Three Highest Sefirot: Yellow
o Seven Lower Sefirot: Blue

The next step in the building process requires that we arrange the twenty-two pentahexes
in a similar hexagonal pattern as follows:

Figure 34: A Pentahex Hexagonal Mandala

As you know, these twenty-two pentahexes are associated with the twenty-two pathways of
the Tree of Life.

Now, to complete this polyhex Tree of Life mandala we must place the large pentahex
hexagon around the perimeter of the smaller sefirotic hexagon as follows:

36


Figure 35: A 34 Polyhex Tree of Life Hexagon

Here is how the component polyhexes constituting the figure above are colour-coded:

o Ain Sof: White
o Aur: Grey
o Twenty-two Pathways: Red

In the figure above, we have effectively constructed a hexagonal mandala of the kabbalistic
Tree of Life, built from the complete series of polyhexes from 1-hex to 5-hex.

I believe this symbolic arrangement deserves some contemplation.

37


Figure 36: The Polyhex Tree of Life Hexagon - Labelled

There are a total of thirty-four polyhexes and 150 individual hexagons in the image above.

When we exclude the 1-hex and 2-hex polyhexes (of Ain Sof and Aur) we are left with the
traditional tally of thirty-two Tree of Life components comprising a total of 147 individual
hexagons.

We will create some 32 polyhex Tree of Life mandalas below.

38


Tree of Life Sefirotic Mandalas (10 Polyhexes)

The ten combined sefirotic polyhexes of the 3-hex and 4-hex groups can be arranged to
form several interesting shapes. Among them...

Figure 37: A Ten Sefirot Polyhex Hexagram

Figure 38: A Ten Sefirot Polyhex Hexagon

In both figures above, the three trihex sefirot (i.e. of Keter, Hockmah, and Binah) are
coloured yellow, and the seven tetrahex sefirot (i.e. the lower sefirot) are coloured blue.

39


A Tree of Life ‘Snowflake’ Mandala (32 Polyhexes)

To produce a traditional (i.e. 32 component) version of the polyhex Tree of Life we can
combine the small ten polyhex sefirotic hexagon (figured above) with the following
pentahex ‘snowflake’ design.

Figure 39: A Pentahex Snowflake Mandala

I designed the figure above to incorporate all the twenty-two pentahexes (i.e. symbolising
the twenty-two pathways of the Tree of Life).

When we combine the pentahex snowflake (above) with the smaller sefirotic hexagon we
produce the following composite figure.

40


Figure 40: A Tree of Life Polyhex Snowflake

The figure above includes all the thirty-two polyhex shapes from 3-hex to 5-hex, and
therefore represents the complete traditional version of the Tree of Life diagram.

The figure comprehends the ten sefirot and the twenty-two pathways, but excludes the two
primordial veils of Ain Sof (1-hex) and Aur (2-hex).

But note that Ain Sof and Aur are both included in the symbolism of the Keter trihex (3-hex).

Here is how the component polyhexes constituting the figure above are colour-coded:

41


o Twenty-two Pathways: Red

One Hundred and Forty-seven Hexagons

The previous figure (i.e. the Tree of Life ‘snowflake’) is comprised of exactly 147 adjacent
and adjoined hexagons, as is the following figure...

Figure 41: A Tree of Life Polyhex Triangle

At the apex of this design is the yellow Keter trihex. Below and to the right of Keter is the
yellow Hockmah trihex, and opposite it (on the left-hand side) is the yellow Binah trihex.

42


The blue tetrahexes of the seven lower sefirot occupy the middle space between the three
yellow trihexes of the highest sefirot.

One triangular hole is in the midst of the blue sefirotic polyhex triangle.

The red pentahexes (symbolising the twenty-two pathways) are spread out as the lower
foundational aspect of the polyhex Tree of Life triangle.

Three triangular holes are in the midst of the twenty-two pentahexes.

As you can see, I have sought to organise the thirty-two polyhex components of the figure
(above) in a way that is symbolically meaningful.

Figure 42: Another Tree of Life Polyhex Triangle

43


My next polyhex design (figured above) contains three roughly triangular-shaped corner
sections of 49 hexagons each–that is, 3 x 49 = 147.

In this figure the sefirotic triangle (coloured yellow and blue) at the apex is mirrored by a
triangular-shaped vacant area in the pentahex pathway section (coloured red) directly
beneath it.

Figure 43: The Apex Triangle is Mirrored

These 32 polyhex Tree of Life mandalas all contain 147 individual hexagons because the
three trihexes of the 3-hex polyhex group are constructed by joining together a total of nine
hexagons, the seven tetrahexes of the 4-hex group are made from 28 hexagons, and the
twenty-two pentahexes of the 5-hex group are made from 110 hexagons.

44


o 3-hex : 3 x 3 = 9 hexagons

Total : 9 + 28 + 110 = 147 hexagons

Conclusion : 32 Polyhexes = 147 Hexagons

Figure 44: A Triangular Polyhex Mandala - Labelled

Here follows another of my polyhex Tree of Life designs...

45


Figure 45: A Tree of Life Polyhex Triple Diamond

This pattern was a challenge to conceive and a difficult polyhex puzzle to solve.

I was looking to create a polyhex design that contained three identically shaped sections
with 49 hexagons in each section. (3 x 49 = 147.)

Thus, not only does the design represent the thirty-two components of the kabbalistic Tree
of Life, but it also reflects the three ‘jubilee’ ring cycles of the 147 year master cycle of
Jacob’s Wheel.

46


The number 147 keeps cropping up in my esoteric investigations and appears to be very
significant–particularly in relation to Jacob’s Wheel.

Figure 46: Tree of Life Triple Diamond - Labelled

Ideally, in the design, I would have liked to have positioned the three yellow trihexes of the
supernal sefirot one in the centre of each of the three diamond shapes.

I would also have liked to position the three tetrahexes of the middle pillar–that is, the
Tifaret, Yesod, and Malkut tetrahexes, in the upper diamond with the Keter trihex. The
Hesed and Netzah tetrahexes would ideally be placed in the right-hand diamond along with
the Hockmah trihex, and the Geburah and Hod tetrahexes in the left-hand diamond with the
Binah trihex.

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If you think you can find a solution, why not give it a try!

Figure 47: The Tree of Life Triple Diamond Blueprint

In the end, I resorted to rearranging the three 49 hexagon diamond sections into a new
horizontal configuration (as per the figures below) and was then able to find a solution that
nearly satisfied my specific polyhex placement requirements (as detailed previously).

In the design below, I have positioned the three supernal sefirot trihexes in a manner that
reflects their respective Tree of Life placements.

I have also aligned the seven lower sefirot tetrahexes according to their respective locations
on the three pillars of the Tree of Life–that is, left, right, or centre.

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The twenty-two pathway pentahexes have been arranged randomly on the basis of
wherever they will fit into the design.

Figure 48: Tree of Life Triple Diamond Mandala – Horizontal Version

Figure 49: The Horizontal Triple Diamond Mandala - Labelled

49


Closer to the Tree of Life

Figure 50: Closer to the Tree of Life

As you can see, the image above is closer in design and appearance to the actual kabbalistic
Tree of Life diagram.

The ten sefirotic polyhexes are arranged according to the placements of their Tree of Life
counterparts.

The twenty-two pathway pentahexes, however, are randomly arranged.

50


A Pentahex Hexagonal Mandala (22 Pentahexes)

Figure 51: A Pentahex Hexagonal Mandala

The figure above incorporates all the twenty-two pentahexes, but in a quite compact
hexagonal form.

51


A Pentahex Triangular Mandala (22 Polyhexes)

Figure 52: A Pentahex Triangular Mandala

The figure above arranges the twenty-two pentahexes in a triangular form with ten holes.

The ten holes can be considered to symbolise the ten sefirot of the Tree of Life amidst the
twenty-two pentahex pathways.

52


A Pentahex Hexagonagram (22 Polyhexes)

Figure 53: A Pentahex Hexagonagram

Again, I have designed this figure (above) so as to contain all twenty-two of the Tree of Life
pathway pentahexes.

I invented the word ‘hexagonagram’ to describe this figure because it combines both the
hexagon and hexagram shapes.

53


Figure 54: A Hexagonagram

Figure 55: Combined Hexagon & Hexagram

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Polyhex Divination

The thirty-two polyhexes of the 3-hex to 5-hex polyhex groups can be used practically as
tools for meditation and divination.

For the purpose of divination (and meditation) thirty-two polyhex tiles should be created or
obtained.

o 3-hex : 3 tiles
o 4-hex : 7 tiles
o 5-hex : 22 tiles

Total: 32 tiles

The 1-hex and 2-hex polyhexes are not employed in divination because they represent the
two primordial veils, and therefore are beyond the manifested Creation as symbolised by
the main body of the kabbalistic Tree of Life.

The three 3-hex tiles should be coloured yellow and be inscribed with the numbers 1, 2, and
3 (and possibly the Hebrew words for the supernal sefirot: Keter, Hockmah. and Binah).

The seven 4-hex tiles should be coloured blue and be inscribed with the numbers 4 to 10
(and possibly also the Hebrew words for the names of the seven lower sefirot).

The twenty-two 5-hex tiles should be coloured red and be inscribed with the twenty-two
letters of the Hebrew alphabet.

Alternatively, the thirty-two polyhexes can be coloured according to your own inspiration
and taste.

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During the process of divination the polyhex tiles are used in the same manner as Celtic
rune-stones–that is, they are placed in a cloth bag and drawn out singly, or scattered upon a
flat surface in a cluster.

There are virtually no limits to the methods of use of the thirty-two polyhexes during
divination. Use your imagination to discover different ways of using them.

For visual meditation purposes the polyhex tiles can be arranged into word-forms or shapes
(as per my example designs above) and their sefirotic and pathway correspondences
meditated upon during the construction process.

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Appendix

The Polyhex Key to the Pathways

The validity of the Tree of Life pathway formula (as expounded in this book) is also
supported by my recent (circa 2010) research into sacred polyhex geometry. My new
findings in the field of polyhex geometry demonstrate that the hierarchical 3:7:12 division of
the pathways is a natural result of the polyhex geometric model of Creation.

The Geometric Model

The manifestation of the kabbalistic Tree of Life involves the coming together of two
primary principles called Spirit and Matter (or Form). The merging of these two principles
produces a third middle principle called Consciousness (or more accurately, Soul). The
merging process can be illustrated very simply using two overlapping circles. (See the figure
below.)

Figure 56 : The Merging of Spirit & Form

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Before the beginning of Time and Space the two circles are separated, but eventually they
approach each other, intersect, and partly merge together. The area of their overlap forms
the Soul/Consciousness principle.

To build the complex form that is the Tree of Life diagram we must replace the Spirit and
Form circles with polyhexagons.

The special polyhexagonal form upon which the Tree of Life is based is a structure that
consists of nineteen individual hexagons arranged in a composite hexagonal shape. This
special polyhexagon is characterized by two sub-groups of component hexagons. Firstly, a
group of seven hexagons, six of which are located around the perimeter of the polyhex, and
a seventh located precisely at its centre. The second sub-group consists of twelve individual
hexagons that fill the main body of the polyhex construct.

Figure 57 : Tree of Life Polyhexagon

You may recognize the significance of the figures seven and twelve as coinciding with the
number of Hebrew Double and Simple letters respectively. This is a very important
correspondence that we will examine more closely below.

The Tree of Life polyhexagon is a special modification of another 19-Hex polyhex that I call
the Genesis polyhexagon. In my understanding, the Genesis polyhex is the natural organic
form upon which the Tree of Life polyhexagon is based. The Tree of Life polyhexagon

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rearranges the nineteen hexes of the Genesis polyhex so that the inner dynamics of the
Genesis polyhexagon are more easily perceived. The Tree of Life polyhex reveals the
structure of the pathways of interconnection that exists between the various hexes of the
Genesis polyhexagon.

The Genesis polyhexagon resembles an organic cell. It has a nucleus and an outer
membrane. Cosmically, we can understand the nucleus as composed of the seven
astrological planets (or planetary gods/logoi) while the outer membrane consists of the
twelve zodiacal signs (or major angelic/devic forces). The interaction between these two
groups of divine Beings produces All-That-Is. The Tree of Life polyhexagon reveals more
clearly how the two groups energetically interact with each other.

Figure 58: The Genesis & Tree of Life Polyhexes

There are two types of Genesis polyhexagons—a Spirit polyhex, and a Form polyhex. They
both have a seven-fold nucleus surrounded by a twelve-fold membrane. Correspondingly,
there are also two Tree of Life polyhexagons and it is these latter two polyhexes that we are
considering in this chapter.

The merging of the two Tree of Life polyhexagons produces interesting results. The Tree of
Life diagram is now clearly revealed in the composite hexagonal design (below right). The
three white hexagons at the top of the yellow polyhexagon represent the three highest

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sefirot of the Tree of Life—Keter, Hockmah, and Binah. They remain wholly detached from
the Form principle polyhexagon (shaded blue).

Only seven yellow hexagons remain visible in the main body of the Spirit polyhexagon.
These represent the seven Hebrew Double letter pathways of the Tree (i.e. as I have
delineated them in this book). The correspondence to my original pathway findings is
precise as I will demonstrate shortly.

Figure 59 : The Merging of the Spirit & Form Polyhexes

In numbering the component hexagons (below) I have placed all of the even numbers on
the right side of the polyhexagon and the odd numbers on the left. I have chosen to use this
convention merely to identify the right side as ‘male’ (even) and the left side as ‘female’
(odd). But it is also interesting to note that this convention shows the natural sequence of
the lower sefirotic hexagons (shaded grey) progressing from Tifaret (4) to Hod (5) which is
contrary to the traditional numbering system which has Netzah (6) following numerically
after Tifaret (4). Thus, this odd/even rule results in a more accurate presentation of the

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‘hierarchy of the planes’1 because Hod (as Mercury) represents the lower mental plane and
therefore it should come before Netzah who as Venus symbolizes the astral plane.

Figure 60 : The Tree of Life Polyhexagon

The three highest sefirot represent the sacred trinity of divine Powers. The seven Double
letter pathways symbolize the seven Heavens/Palaces 2, the seven planetary Spirits (or
Logoi), the seven angels who attend the Throne of God, etc. They represent supernal
powers guiding and administrating the lower worlds. Each of these seven Double letter
pathways is the ensouling spiritual source of one of the seven lower sefirot (or ‘planetary
spheres’).

The Form principle polyhexagon includes the seven lower sefirot (i.e. corresponding to the
seven grey shaded hexagons) and the twelve Simple letter pathways (corresponding to the
twelve blue shaded hexagons above).

1
That is, as described in the teachings of Theosophy.
2
Hebrew: Shamayim (‫ ,)שמים‬and Hekhalot (‫.)היכלות‬

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An important consequence of the polyhexagon merging and overlapping process is that five
of the Spirit principle pathways become concealed beneath the upper portion of the Form
principle polyhexagon—that is, they become hidden below the surface of the newly formed
Soul principle. My thinking is that these five concealed pathways correspond to the five
Final Hebrew letters—Kaf final (‫ ,)ך‬Maym final (‫ ,)ם‬Nun final (‫ ,)ן‬Peh final (‫ ,)ף‬and Tzaddie
final (‫ .)ץ‬The implication of this geometric symbolism is fascinating.

Figure 61 : The Soul Principle (Green)

The Soul principle is formed as a result of the intersection, overlapping, and interaction of
the Spirit and Form principles. This area of overlap is indicated in the diagram above by the
five green hexagons and their four bounding grey hexagons (i.e. numbered with white
numerals). The soul is thus seen to be an admixture of Spirit and Form.

The Soul principle is also seen to be a relatively impermanent principle (or state of being). It
manifests only when the Spirit principle merges with the Form principle. The significant
implication of this is that when it comes time for the Spirit and Form principles to once again
separate themselves the Soul principle dissolves (or disassembles). In terms of our
geometric symbolism, the area of intersection between the Spirit and Form polyhexagons

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disappears. This process of separation occurs during what Theosophists call a period of
‘pralaya’3.

Your first feelings on hearing this proposition may include some disappointment. You may
feel that the dissolution of your soul will lead to the loss of your identity (or your existence
as an individual). This is not truly the case however. The soul is merely a temporary vehicle
that allows the Spirit to involve itself in the substance of matter.4 Your spirit (i.e. monad)
remains in existence during the period of Pralaya.

Before the soul’s dissolution during Pralaya, its essence is absorbed into and stored in the
monad. I believe this process is made possible by the existence of an occult mechanism
involving the five Hebrew Final letter pathways.

The five Final letter pathways are concealed beneath the five visible Simple letter pathways
of the Soul principle. The five Finals belong to the Spirit principle polyhexagon, but they are
covered and concealed by the Simple letter ‘soul pathways’ of the overlapping Form
principle. My theory is that the esoteric function of the five Final letter pathways is to gather
in and absorb the Knowledge that has been gained during the term of the soul’s existence.
By ‘Knowledge’ is meant the combined Wisdom and Understanding (or ‘love-wisdom’) that
the soul has acquired during its term of existence. These qualities are collected and stored
within the monad during each period of Pralaya.

Thus, when the Spirit polyhexagon detaches itself from the Form polyhexagon it carries with
it the five concealed Final letter pathways because they are part of its fundamental
structure. So, nothing of value is lost during a Pralaya.

3
Pralaya (Sanskrit) [from pra away + the verbal root li to dissolve] Dissolving away, death, dissolution, as when
one pours water upon a cube of salt or sugar: the cube of salt or sugar vanishes in the water, dissolves, and
changes its form. So during a pralaya, matter crumbles or vanishes away into something else which is yet in it,
surrounds it, and interpenetrates it. Pralaya is often defined as the state of latency or rest between two
manvantaras or great life cycles. [From: The Rakefet Theosophical Dictionary.]
4
In fact, human-beings are destined to evolve beyond soul-based consciousness. The soul becomes obsolete
th
as a vehicle of consciousness at the 4 Initiation (called the Great Renunciation). It is then transcended and
replaced with ‘monadic’ consciousness.

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Figure 62 : Polyhexagon Tree of Life — Hebrew

The Hebrew letters are arranged on the Spirit and Form polyhexagons as per the diagram
above.5

The seven Double letter pathways are shaded yellow. The twelve Simple letter pathways are
shaded blue. The five Final letters (coloured blue) are positioned on five yellow hexagons
near the base of the Spirit polyhexagon (above left). They are bounded by four hexagons
containing the letters of the Tetragrammaton—YHVH (orange lettering). This small subset of
hexagons bounded by YHVH (‫ )יהוה‬basically represents the monad (i.e. the monadic triad).
The overlapping hexagons of the soul are merely a garment the monad wears when it
descends into the realm of ‘physical existence’.

The twelve Simple letters are expressions of the Form principle, or Great Mother ‘Shekinah’
principle.

5
But note that the arrangement of the twelve Simple letters is somewhat flexible.

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The seven Double letters have two variant forms—a ‘soft’ pathway form representing
supernal spiritual forces, and a ‘hard’ sefirotic form (i.e. identified by a dagesh, or dot)
representing lower, more material projections of the supernal pathways.

The five Finals are differentiated as follows:

 Maym Final (‫ = )ם‬Mother letter

 Kaf Final (‫ )ך‬and Peh Final (‫ = )ף‬Double letters

 Nun Final (‫ )ן‬and Tzaddie Final (‫ = )ץ‬Simple letters

Thus, all the three major Hebrew letter groups—Mothers, Doubles, and Simples, are
represented in the five Finals.

My current hypothesis is that Maym Final collects the essence of the two Mother letter
pathways (i.e. the vertical pathways of Maym and Shin) that infuse the Form principle
polyhexagon. Similarly, Kaf Final and Peh Final gather the essence of the seven Double letter
sefirot of the Form principle polyhexagon. Lastly, Nun Final and Tzaddie Final collect the
essence of the twelve Simple letter pathways of the Form principle polyhexagon. So when
the Spirit principle polyhexagon detaches and separates from the Form principle the
threefold essence of its experience within the substance of matter (i.e. as recorded by the
soul) is taken with it via the five Hebrew Final letter pathways.

You may notice that there are only nineteen pathway hexagons included in the natural
structure of the polyhex Tree of Life design. (See the diagram below.) Three of the twenty-
two pathways of the Tree of Life appear to be missing. I have also omitted them from the
traditional drawing of the Tree of Life (below left). The missing three are the three Mother
letter pathways—Alef, Maym, and Shin.

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Figure 63 : Tree of Life — Traditional & Polyhex Designs

Most kabbalistic commentators assign the three Mother letters to the three horizontal
pathways of the Tree diagram 6, but my findings indicate that their more metaphysically
meaningful placements are with the three vertical pathways of the Middle Pillar of the Tree
of Life.

In my view the Double letter Bayt (‫ )ב‬is assigned to the highest horizontal pathway (i.e.
joining Hockmah with Binah) because it is responsible for formulating the first of the seven
lower sefirot—Da’at. The sefirah Da’at arises as a result of the interaction between
Hockmah and Binah—an interaction facilitated by the horizontal Bayt pathway. The hard
form of the letter Bayt consequently emerges as the distinguishing archetype of the sefirah
Da’at.

The Tree of Life formation formula (as described in this book) posits the location of the
Hebrew letters on the sefirot and pathways of the Tree diagram in a descending hierarchy
that follows the sequence 3:7:12—that is, Mothers-Doubles-Simples. For this reason, the
formula cannot be reconciled with the traditional numeric method of assigning the letters to
the horizontal, vertical, and diagonal pathways. The old method places the three Mother
letters on the three horizontal pathways, the seven Double letters on the seven vertical

6
Without much explanation besides the obvious numerically based correspondence.

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pathways, and the twelve Simple letters on the twelve diagonal pathways of the Tree
diagram.

The traditional methodology appears, at first glance, to be most agreeable, but I believe it is
an erroneous correspondence. The geometric framework of the Tree diagram is built
according to the fundamental esoteric ratio 3:7:12 because it represents a blueprint of the
divine Creation. The Hebrew alphabet is also constructed according to the 3:7:12 ratio
because it represents the twenty-two divine archetypes. It does not necessarily follow
however, that the two are meant to be combined together purely on the numerical basis of
this 3:7:12 ratio. In fact, the Sefer Yetzirah states:

One on three,
three on seven,
seven on twelve,
And all are bound, one to another.
These are the twenty-two letters…7

Obviously, a 3:7:12 hierarchical arrangement of the Hebrew letters is indicated by this
statement. Clearly, it is a contradiction of the principles of the Sefer Yetzirah to widely
intermingle the three classes of Hebrew letters on the Tree diagram (i.e. as does the
orthodox numerical letter-pathway arrangement).

To put it simply, the Sefer Yetzirah is saying...

o One (All-encompassing Ain Sof) on three (supernal sefirot)

o Three (supernal sefirot) on seven (supernal pathways)

o Seven (lower sefirot) on twelve (lower pathways)

o And they are all interconnected

(See the figure below.)

7
Sefer Yetzirah, 6:5-6.

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The Zohar is similarly supportive of the hierarchical approach. In its anthropomorphosis of
the Tree of Life the Zohar associates the three highest sefirot with the head of Adam, and
the lower sefirot with his body. These correspondences combine well with the Sefer
Yetzirah’s assignment of the seven Double letters to the eyes, ears, nostrils, and mouth.

Figure 64 : The Hierarchical Arrangement of the Hebrew Letters

Logically, the seven Double letter pathways (symbolizing seven physical attributes of the
human head) need to be arranged upon the head of Adam at the top of the Tree of Life
diagram.

The Sefer Yetzirah assigns the twelve Simple letter pathways to the torso, limbs, and internal
organs of the human body, and therefore logically these should be located within the lower
portion of the anthropomorphic Tree diagram—that is, below the head.

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One of the key arguments for the old numerically-based arrangement is that the Sefer
Yetzirah describes the twelve Simple letter pathways as ‘diagonal boundaries’ (‫אלכסון‬
‫ 8.)גבולי‬It is true that twelve of the Tree of Life pathways are diagonal (or sloping), but the
diagonal pathways do not resemble ‘boundaries’ on the Tree diagram. In the hierarchical
pathway arrangement however, the twelve Simple letter pathways do resemble diagonal
boundaries (or ‘sloping sides’) because they form the twelve bounding sides of six adjacent
triangles that surround Tifaret in the lower portion of the Tree diagram. (See the diagram
above left. The twelve Simple letter pathways are coloured blue.)

The three vertical Mother letter pathways unite the four sefirot of the Middle Pillar of the
Tree of Life, but they are not included in the polyhexagonal structure of the Tree. This
reflects the mysterious nature of these three pathways. They represent the fluid essence of
Being. Their movement, interaction, and intermingling produces the flowing stream of
awareness that we call consciousness, and necessarily also the illusion of time and space.
Because of this, they move within and beneath the surface of consciousness and are not
directly perceived by it. Only their effects are perceptible. What is inferred is that
consciousness is not a fixed enduring principle, but rather, is a fluid process.

Interestingly, in the polyhexagon Tree diagram, the Shin (Fire) pathway is invisibly located
beneath the Tet (Leo) hexagon. The Fire of Shin is associated with Nachash—the ‘Kundalini
Serpent’. Tet also means serpent, and Leo (ruled by the Sun) is a fire sign that epitomizes
the fiery principle of consciousness.

The Maym (Water) pathway is invisibly located beneath the Tzaddie (Aquarius) hexagon. Of
course, Aquarius is known as the ‘Water Carrier’ 9, and is also strongly associated with the
consciousness principle. These two signs, Leo and Aquarius, form the vertical axis of the
Fixed zodiacal cross.

The Alef (Air) pathway is invisibly located beneath the Bayt (Saturn) pathway hexagon. This
is appropriate because Bayt is chief of the seven Double letters, and Alef is chief of the three
Mother letters. Also, Alef is thought to stand invisibly behind Bayt during the process of
Creation (which is why Berashit10 begins with a Bayt and not an Alef).

8
Sefer Yetzirah, 6:1. Note that it does not, however, describe the three Mother letter pathways as
‘horizontal’, or the seven Double letter pathways as ‘vertical’.
9
The Tzaddie pathway carries the Water of Maym from the sefirah Hesed to the sefirah Geburah.
10
That is, the Book of Genesis.

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You may have noticed in the diagrams that the two Simple letter pathways Qof (Pisces) and
Sameck (Sagittarius) are placed differently in this polyhex arrangement—i.e. compared to
the arrangement I described in earlier chapters. The reason for this slight deviation is that
although normally these two pathways extend from Hesed and Geburah to Yesod, they are
also capable of extending to Da’at (i.e. from Hesed and Geburah). This only occurs during
the process of ‘divine inspiration’ (i.e. conscious contact with the monad)—an experience
that rarely happens in an average person’s lifetime. In the polyhexagon Tree diagram these
two pathways are placed in their elevated positions adjacent to Da’at. Thus, this particular
pathway configuration reflects the conscious alignment strived for by Disciples (or Tzadikim)
on the Path. The state of consciousness that this configuration represents grows stronger
and more stable as a disciple continues on his/her path of inner development.

For disciples and initiates, the Fire (Shin, ‫ )ש‬of creative inspiration flows from Da’at to
Geburah (i.e. representing the Abstract Mind) along the Sameck (Sagittarius) pathway. The
Water (Maym, ‫ )מ‬of spiritual intuition flows from Da’at to Hesed (i.e. representing the
Intuitive Mind) along the Qof (Pisces) pathway.

While the Sameck and Qof pathways remain exclusively attached to the sefirah Yesod they
symbolize the presence in a person’s mind of worldly illusions (Sagittarius) and glamours
(Pisces). These illusions are fostered within, and perpetuated by the collective mindset of
mundane human society. They effectively fill the pathways of Sameck and Qof with a
darkness that blocks access to the higher sefirot Geburah and Hesed. But as a person frees
themselves from the glamours and illusions of society they simultaneously become
consciously open to the higher sefirot and thus to divine impressions entering into and
through the human soul (i.e. flowing from Da’at to Hesed and Geburah).

The divine inspiration that is received through the pathways of Sameck and Qof is expressed
and applied (in the mundane world) via the lower and complementary pathways of Zayn
(Gemini) and Yud (Virgo). Together these four pathways constitute the Mutable Cross of the
Heavens (i.e. the cross of mutable zodiacal signs). On the Tree of Life, the primary creative-
evolutionary purpose of the Mutable Cross is to positively transform the human expression
of the psychic influence of the astrological Moon (i.e. as it is felt within the sefirah Yesod).

The main evolutionary function of the four signs (and pathways) of the Cardinal Cross is to
stabilize the expression of the astrological Sun upon the Path of Righteousness. Thus, on the
Tree of Life diagram we see the four cardinal sign pathways anchoring and holding the

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sefirah Tifaret (i.e. the Sun) upon the Middle Pillar of the Tree. Their role is to hold Tifaret
steady so that it doesn’t drift too far towards the left-hand Pillar of Severity, or conversely,
too far towards the right-hand Pillar of Mercy. As long as Tifaret’s central position upon the
Middle Pillar is maintained then it retains direct access to ‘HaShem’ 11—that is, to the forces
of Shin (‫ )ש‬and Maym (‫ )מ‬flowing through the ‘invisible’ Middle Pillar pathways. When this
condition is fulfilled, a human-being becomes a Tzadik—i.e. one who is faithfully aligned
upon his/her ‘path of righteousness’.

Figure 65: Tree of Life : 3+7+12 Pathways

The four signs (and pathways) of the Fixed, or Cherubic Cross are responsible for refining the
four main faculties12 of human consciousness as represented by the four ‘directional’
sefirot—Hesed, Geburah, Hod, and Netzah. These faculties are associated with the four
elements and the four magical implements—the Cup, the Wand, the Sword, and the Disk (or
Pentacle). The four cherubic ‘animals’ represent active intelligent expressions of the four
esoteric elements. They signify powerful forces that are responsible for the diversity of life
and life experience on our planet. On the Tree diagram their four pathways form a square

11
Hebrew – The ‘Name’, spelled Shin-Maym, referring to the divine Name of God.
12
That is, the intuitive mind, the abstract mind, the rational mind, and the astral body (or desire mind).

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enclosure whose four walls intersect at the four ‘corner’ sefirot—Hesed, Geburah, Netzah,
and Hod.

The 3-7-12 hierarchical structure of the Tree of Life diagram inclines me to think that an
update is needed to the original polyhex Tree model that I described in my book The
Hexagonal Geometry of the Tree of Life. The model I depicted in that book was constructed
as follows…

 3-Hex (x3) = 3 supernal sefirot

 4-Hex (x7) = 7 lower sefirot

 5-Hex (x22) = 22 pathways

The above arrangement is useful as a basic model of the Tree of Life, but if a more
metaphysically accurate presentation is desired then a change is needed because (as has
been demonstrated) the seven supernal Double letter pathways (i.e. located in the upper
part of the Tree diagram) manifest before the seven lower sefirot and the twelve Simple
letter pathways. The seven Double letter pathways should therefore be associated with the
seven tetrahexes (i.e. 4-hexes).

The revised model looks like this…

 3-Hex (x3) = 3 supernal sefirot

 4-Hex (x7) = 7 supernal pathways (Doubles)

 5-Hex (x22) = 7 lower sefirot (Doubles) + 12 lower pathways (Simples) + 3 Mother
letter pathways

Although the original model appears simpler, the revised model represents a more accurate
depiction of the Tree of Life manifestation process. The formation of the Tree of Life begins
with the appearance of the three supernal sefirot. Then the seven supernal pathways
emerge between them. And then finally the seven lower sefirot and twelve lower pathways
manifest.

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Thus, it is more logical to have the simpler polyhexes (i.e. those with fewer component
hexagons) arranged upon the upper portion of the Tree, and the more complex polyhexes
(i.e. the 5-Hexes) assigned to the lower section (coloured red in the diagram).

Figure 66: The Polyhex Tree of Life (Revised)

In the diagram13 the three trihexes are coloured yellow, the seven tetrahexes are coloured
blue, and the twenty-two pentahexes are coloured red. The twelve pentahexes that I have
ascribed to the twelve Simple letters are (as explained in my book 14) geometrically different
from the other ten pentahexes in the 5-hex group because they are able to ‘tile the plane’ in
the most simple manner possible. As shown in the diagram above, the remaining ten
pentahexes (of the group) are assigned to the seven Double letters and the three Mother
letters.

The three Mother letter pathways are displayed on the right-hand side of the diagram. They
are concealed beneath the other pathways of the Tree. They flow into the Tree of Life in a
manner similar to the entry of the soul into the confines of its prepared physical vehicle. The

13
This arrangement of the polyhexes is only one of many possible variations.
14
See: The Hexagonal Geometry of the Tree of Life.

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three Mother letter pathways galvanize and energize the sefirot of the Middle Pillar, and
they trigger the commencement of the creative-evolutionary process that occurs within the
branches of the Tree.

Thank you for reading this book.
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Thank you!

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