the hexagonal tree of life

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

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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: [email protected]

Website: http://www.astroqab.co.nr/

mailto:[email protected]

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.)

http://members.optusnet.com.au/~astroqab/formation_tree_life.htm


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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:


37

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.


38

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.


39

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.


40

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.


41

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:


42



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.


43

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


44

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.


45

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...


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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.


47

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.


49

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


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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.


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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.

\


52

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.


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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.


54

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.

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

Figure 57 : Tree of Life Polyhexagon


59

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.

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

Figure 58: The Genesis & Tree of Life Polyhexes


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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.

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

Figure 59 : The Merging of the Spirit & Form Polyhexes


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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.

The three highest sefirot represent the sacred trinity of divine Powers. The seven Double

letter pathways symbolize the seven Heavens/Palaces2, 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 (שמים), and Hekhalot (היכלות).

Figure 60 : The Tree of Life Polyhexagon


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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 (ך), Maym final (ם), Nun final (ן), Peh final (ף), and Tzaddie

final (ץ). The implication of this geometric symbolism is fascinating.

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

Figure 61 : The Soul Principle (Green)


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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

as a vehicle of consciousness at the 4th

Initiation (called the Great Renunciation). It is then transcended and replaced with ‘monadic’ consciousness.


64

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 (יהוה) basically 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.

Figure 62 : Polyhexagon Tree of Life — Hebrew


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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 (ם) = Mother letter

Kaf Final (ך) and Peh Final (ף) = Double letters

Nun Final (ן) and Tzaddie Final (ץ) = Simple 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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Most kabbalistic commentators assign the three Mother letters to the three horizontal

pathways of the Tree diagram6, 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 (ב) is 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.

Figure 63 : Tree of Life — Traditional & Polyhex Designs


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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.

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.

Figure 64 : The Hierarchical Arrangement of the Hebrew Letters


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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

It is true that twelve of the Tree of Life pathways are diagonal (or sloping), but the 8.(גבולי

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, ש) of creative inspiration flows from Da’at to

Geburah (i.e. representing the Abstract Mind) along the Sameck (Sagittarius) pathway. The

Water (Maym, מ) of 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


71

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 (ש) and Maym (מ) flowing 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’.

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).

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


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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).

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 book14) 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.

Figure 66: The Polyhex Tree of Life (Revised)


74

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.

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the hexagonal tree of life

Spiritual

pentahex triangular

triangular

simple pentahexes

life snowflake

hebrew letter

life triple

shin pentahex

pentahex hexagonal