Sacred Geometry Discovery

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"For the earth shall be filled with the knowledge of the glory of the Lord, as the waters cover the sea." HABAKKUK 2:14

Introduction to the Tree of Life as a memory castle...

Sacred Geometry is an art of memory which employs geometric forms as symbols, for the memory and organization of theosophy; where theosophy means divine wisdom. Illustrated on this page the sacramental geometry of a 'fractal memory castle' will reveal deeper understanding of the Tree of Life, and the Decalogue will be shown as integrated into its structure. Memory Castles were developed during the 1600's at European universities, and this fractal memory castle borrows from understanding that was gained during that time. The memory castles purpose is to memorize a body of knowledge and make its recollection rapid and organized. The original development of memory castles provided a means for speakers to deliver coherent and lengthy presentations solely from memory. Here, however, the memory castle is intended as an assistance for contemplation,  mapping the inner Kabbalistic worlds which are traditional guides for self understanding. 

The graphics above depict the "Tree of Life" of Kabbalah with natural geometric forms which make the memory castle rooms. Each gray square contains a pictograph of a three dimensional form, and each form is a Sephirah symbol. The 'Sephirote', the plural of Sephirah, are ten divine attributes which are said to inhabit spheres, spheres would be the traditional rooms. Gray squares in the diagram above mark the Sephirote positions. The traditional Tree is drawn with ten circles in these positions where there are gray squares. The red symbol not shown in a square is not a Sephirah, it is, however, a three dimensional object, and it is the symbol for Knowledge(Daath). The  ten Sephirote are divided into two groups of five, masculine and feminine, shown above in blue and violet respectively.

The complete fractal memory castle has been derived from the discovery that the nested relationships available among pairs of  Platonic Solids can be ordered so as to allow for a simple mnemonic of the Decalogue. The Decalogue mnemonic expresses the commandments by way of impressing the Platonic Solids with the names of the feminine Sephirote. The nested relationships between pairs of Solids demonstrate relationships between the feminine Sephirote that can exist. These ten paired relations through association with the Sephirote become memory places for each of the ten commandments. Each association represents a simplistic, generative, essence implied through understanding each commandment as a creative word.

Furthermore, in accord with this model the feminine Sephirote are shown to be modular elements of the masculine Sephirote, and these Sephirote can be taken as the five parts of the soul. The five parts are derived from the commandment to Love the Lord our God with all Heart, Soul, and Strength. Which was elaborated with the addition of Mind interposed between Soul and Strength. It seems that this was a Greek influence that separated Mind from Heart. However, it allows one to understand that Mind effects ones life in two directions, physically in the direction of ones Strength and spiritually in the direction of the Heart through the spiritual Soul.

If we can allow these four parts of the soul ( Heart, Soul, Mind Strength ) then we should allow that they exist within our sense of existence, within our mundane Senses or within occasions of higher sensory experience. This last addition is mine, completing the soul with Senses that partition our inner experience from the world in which we find ourselves. This summary should allow for an hypothesis of some kind however it may be best to regard these remarks as simply fundamental assumptions. What follows are descriptions of the geometric forms that accommodate the assumptions that I have made. The overall meanings among these relationships remain open for interpretation and further exploration. 

The feminine Sephirote, the five Elementals

The feminine Sephirote are shown in violet lines on the cover page. There are five shown, and they correspond to the Platonic Solids as follows:

Understanding(Beinah), is the Platonic Solid that has four triangular faces, the Tetrahedron. In that the Tetrahedron is the simplest of these forms, and is itself regarded as the simplest particle of these structures, it should be taken to symbolize particles in every sense. In language it would imply words. It should be remembered from Plato's dialogue "Timaeus" that the four faced figure was to symbolize fire which is a phenomenon of atomic and photon particles. As to the subatomic, it should be noted that the figure has six edges as there are six kinds of subatomic quarks(up, down, top, bottom, charmed, strange). The four faces are a reminder of the four basic fields in nature. Returning again to the word as particle, there is a counting coincidence or Gematria correspondence with the Tetragrammaton. The Tetrahedron is composed of four different features: Angles, Faces, Edges and Vertices(corners). The counts of these four are 12+4+6+4, a sum of 26 as is the sum of the Tetragrammaton (10+5+6+5).

Beinah is referred to as the Mother of form, and the four Vertices of the Tetrahedron concurrently establish a spherical surface, as it is said, the root emanation of form was spherical shells(kelippoth). Even though this form is the simplest, you can see that many associations can be made which makes the Tetrahedron a very suitable mnemonic for Understanding.

Strength(Gevurah) is symbolized by the six square faced figure, the Cube, sometimes called the Hexahedron. (Each one of these symbols is specified by its outer edges only and all edges are of the same length.) The outer edges of the Cube were written about by Buckminster Fuller in a work entitled "Tetrascroll", and he made famous the observation that a Cube constructed of only outer edges has no strength. In order to make it strong one must construct diagonals across each face which converts the surface into a system of triangles, triangles being the only forms to which 'rigidity' is attributable. None the less the Cube (its edges) are apt as a symbol of Gevurah which also has a connotation of severity. True strength is not rigidity and the aspect of severity opens ones understanding to the idea that the deep meaning of this Sephirah may be 'works'. Unlike Plato's notion attaching stability to the six faced figure it more rightly denotes the unstable constantly changing forces of energy. The changing shape of the element Earth, work, and all the difficulties thereof. In addition the surface features(Edges, Vertices, Faces)of the cube 12+8+6 have the sum of 26.

The Cube is the first of the forms that is compound. Interconnecting the corners of the Cube forms not only the outer edges, but also twelve diagonals across the square faces which form two symmetrically intersecting Tetrahedrons which are together the symbol for

Knowledge(Daath). This symbol, in addition to eight outer Vertices, has six inner ones that form the place for the construction of an Octahedron. This makes the symbol suitable for Daath which is referred to as a 'hole' and sometimes an abyss. (Again, this symbol does not represent a Sephirah.)

(There is a third set of connections through the center of the Cube and these contribute to the formation of a twelfth symbol, also not a Sephirah; that of The Infinite(Ein-Sof) which will be left for a later description.)

Later a compilation of symbols will be introduced entitled DODECALOGUE which in part forms a mnemonic of the Ten Commandments. Twenty eight symbols are employed to express the commandments as relations among the Sephirote. The cube when completely interconnected has a total of 28 edges. These 28 symbols from the "DODECALOGUE" compilation can be symmetrically distributed upon these 28 edges while maintaining the integrity of the compilation's ordering. So, the Cube symbol for Gevurah can in this way be a sign for the essence of the Law(Torah).

The Crown(Kether) is a Stellated Dodecahedron. As a part of the soul it represents the supernatural senses, a transcendental extension of Yesod the mundane Senses.

Beauty(Tipareth) is a Stellated Icosahedron. Its part of the soul is the Heart. This form has thirty-two Vertices, and the Hebrew word for Heart has a Gematria value of the same value. Lebb (heart) is the sum of 30+2, and the Vertices of the form also are the sum of two values; 20+12. The twenty outer points of the form connect very neatly inside of the Stellated Dodecahedrons inner Vertices, and in like manner another Stellated Dodecahedron can be constructed from the inner Vertices of the Stellated Icosahedron, and another Stellated Icosahedron within the Stellated Dodecahedron, etc.. So these forms reproduce each other infinitely; one within the other. This is the essence of the fractal. The fractal replicates at alternating nodes of 12 and 20 Vertices. At nodes of 20 two other forms are also constructed.

Wisdom(Hohkmah) is a special subset of the Dodecahedron; it is ten Tetrahedrons connected to the 20 Vertices. Hohkmah has 170 Vertices of its own, and 30 of those, the innermost vertices, form an embrace for five Octahedrons (to be described following the next form). Hohkmah represents the intellectual soul, the mind.

As has been shown in other of these symbols they can provide memory places for specific counts which have Kabbalistic attributions. This symbol for wisdom composed of ten Tetrahedrons which are symbols of the element Fire can be seen as the coming together of five Tetrahedron pairs which symbolize Knowledge(Daath).

The total vertices of each Tetrahedron pair(Daath) is counted as 14, so these five structures provide 70 vertices as a group. Seventy is the number value of the Hebrew letter Ayin. Ayin is thought to symbolize the "light of the mind" and was represented in ancient times as a circle which was a symbol for the eye. To form the symbol of wisdom from these five Tetrahedron pairs they must be merged together where additional vertices occur, as the edges of the Tetrahedrons intersect. This raises the total vertices to 170. Each Tetrahedral edge in the final construction is punctuated by seven vertices, dividing it into six line segments. Since Tetrahedrons have six edges, there are ten Tetrahedrons within the form, and each edge is divided into six segments, there are 360 line segments forming the structure of the symbol for Wisdom(Hohkmah).

So we see that the elements from which the symbol for wisdom is formed provide a place for the count of 70 which is Ayin the circle and light of the mind, and the completed form of wisdom has within its structure place for the count of 360 which has an ancient association with the measurement of circles. Dividing circles into degrees is an early method by which the circle was first intellected.

Victory(Netzaack) is formed from the edges of five Cubes and is simultaneously a composite of twelve five pointed stars one inscribed within each of the twelve pentagons that make up the faces of the Dodecahedron. This form represents the muscles, and the vital soul.

This five Cube symbol for Netzaack(Victory) has a beautiful property in its structure. Regular triangles, squares and also pentagons are repeated throughout its construction and these are the only regular polygons involved in the structure. The Platonic Solids also, only have faces which are triangles, squares or pentagons; only these three, having 3, 4, and 5 sides respectively. Mnemonically we can use these counts to form the greater count of 345 which is the Gematria value of the name Moses(Moshe). These three fundamental polygons occur together in no one other of the symbols with the exception of the Dodecahedron symbol for Yesod when it is completely interconnected.

Mercy(Hessed) is a composite of five Octahedrons that embrace the twelve inner Vertices of the Stellated Icosahedron(Heart). This form represents the Spiritual Soul, and has special properties because of its Vertices. It has 62 Vertices, the 30 outermost connect it to Hohkmah(Mind), the 12 innermost connect it to Tipareth(Heart), and there are 20 in between these that form the place for the construction of a second Dodecahedron. The second Dodecahedron implies a second continuum of fractal construction, this time in a different place within the original fractal. This process can be compounded, and this is the 'chaotic' characteristic of the fractal. What this means is that by way of continued construction within the form representing Mercy(Hessed) the fractal could be said to emanate throughout the entire soul; extended constructions reproducing the entire fractal infinitely, so that each of the fractal edges become planes; 45 intersecting planes all told.

Restating in another way, the fractal is generated from interacting constructions between two continuums. There is an unending repetition of heart structures within divine child structures; Stellated Icosahedrons within stellated dodecahedrons. As these forms are repeatedly constructed, each event of reproduced form is about four times larger than the last one constructed within it. This process of construction may be thought of as partitioning space like a set of concentric circles being drawn on paper with each new circle larger than the last. Although, a closer analogy would be the repeated construction of five pointed stars by drawing straight lines connecting the outer points of the star to form larger and larger stars. This then brings us to the second continuum of construction of the fractal.

Within the dodecahedrons within the first continuum there are constructed tetrahedrons, and within their intersection octahedrons. Again, this is soul within mind or Hessed within Hohkmah as described above. These octahedrons within tetrahedrons form the second continuum directly within the first, in other words, concentrically. The nodes of reproduction in this continuum are separated by a proportion of about two. Each new replica is about twice as large as the last within it. The simple analogy for this would be the construction of four sided diamonds within four sided squares etc.. The squares would represent mind and the diamonds soul. Again, this can be looked upon as a partitioning of space, but importantly since both continuums are established within each other and "grow" at two different rates they establish infinite numbers of copies of themselves within each other at an infinite number of different scales, which has the effect of filling the space that they are partitioning.

This all happens through the form Hessed(Mercy) which is said to be where love is expressed. There probably is not any other symbolic mnemonic that is such an apt symbol of love other than "reproduction filling space" as we have in this symbol of the spiritual soul.

A symbol for the Infinite(Ein-Sof) is the intersection of thirty-one lines at the single central point of the fractal, and as a symbol it should be thought of as a symbol of a spark of the Infinite Light(Ein-Sof-Orh).

It is not a form, and not a Sephirah, it is in space but not composed of space. It implies the beginning and end of limit; the finite unapproachable center, and the infinite reach of outer construction. The primary Vertices of all forms within the fractal are constructed along these lines. These lines fall into three groups, they are primarily diameters of three forms: the six diameters of the Icosahedron(Kingdom), the ten diameters of the Dodecahedron(the Righteous One), and the remaining fifteen are diameters of the five Octahedrons forming Mercy(Hessed). This structure can also be said to be composed of diameters of two other elements; those of single Octahedrons and single Cubes. So, we may look upon the Ein-Sof(spark) as composed of five basic elements; representative of the five final letters of the Hebrew alphabet.

A Summary of the three Pillars

The central pillar contains the forms that exhibit simple five part symmetries, and the stellations of these. The central pillar contains all symbols which constitute the means by which the elemental forms are reproduced in the fractal. The symbol for Daath is also in the central pillar, it is the vehicle by which the Octahedron is produced. It has no five part symmetry, although, it would represent a stellated Octahedron if an Octahedron were constructed within it. Daath is a compound symbol composed of two Tetrahedrons so it bears some resemblance to the compound forms which make up the three Sephirote of the right pillar.

The three symbols of the right pillar are compound and importantly they are compounds of the three elemental symbols of the left pillar. Hohkmah is a compound of ten Beinah, Hessed is a compound of five Hod, and Netzaack is formed from five Gevurah. These forms fit within each other and they are modeled on Yesod(Dodecahedron), and can be said to be within Yesod and Yesod can be said to be within Kether. Hohkmah holds within itself Hessed, and the outer form of Hohkmah is Netzaack. The form outermost to Netzaack is Yesod or Kether, and finally the innermost form, Tipareth, is embraced by Hessed. In ordinary terms this is; the Heart within the Soul within the Mind within the Strength within the Senses. These are the five parts of the Soul alluded to by the right pillar.

The left pillar geometries are a harmonious group. We might refer to them as Earth, Wind, and Fire, since these correspondences can be made from Plato's dialog "Timaeus". Each one contains a feature of six part symmetry. The Tetrahedron has six edges, the Cube six faces, and the Octahedron six outer Vertices. When the three forms are constructed with all connections between their primary vertices secondary Vertices are formed, and we can form a sum of the total Vertices among them as (4+15+7), here again 26 appears among the forms. A completely interconnected symmetrical hexagram is a proper shadow of the Octahedron , the Cube, and two Tetrahedron shadows are subsets of that diagram. All three forms have a symmetrical shadow which is a square with the corners interconnected by diagonals. Each of these forms is also found in nature as mineral crystals. It should be noted that the other two feminine forms are also represented by crystals. The Dodecahedron forms as a quasicrystal and the Icosahedron is found in nature as twelve atom crystals of boron.

The three Mothers

Ahmsh, Air, Water, Fire, written as AMS(Alef Mem Shin) represent primal forces in creation. The three symbols above are the forms that represent Water, Air, and Fire respectively corresponding to the order in which the letters Mem, Alef, and Shin are found in the Tree....

(the cover page diagram shows these letters S-A-M above three horizontal red lines in the diagram, the traditional diagrams of the tree show all 22 interconnecting lines which represent 'channels of interconnection' between pairs of the Sephirote, also shown in the cover diagram are seven blue vertical strokes which are places for the seven Hebrew letters referred to as the seven doubles, the remaining twelve diagonal interconnections representing the 'elemental' letters have been omitted)

The polyhedrons represented by the three images above have a unique relationship in that the Icosahedron(Malkute) will fit inside the Octahedron(Hod) with its corners touching the twelve edges of the Octahedron(Hod), and further, the Octahedron(Hod) can be packed into the Tetrahedron(Beinah) with the six Octahedron corners touching the midpoints of the six Tetrahedron edges. Not only do they all neatly pack together, but each of them is composed of triangular faces, and the sum(12+6+4) of their vertices is 22; which reflects what is said about the three mother letters conveying all potency of language. This is simply another example of how well these forms accommodate the expression of the Tree Of Life, even though it is a bit at variance to traditional interpretation of the Tree. Geometry is one of the Handmaidens of Kabbalah, and very beautiful she is.

The Four Worlds

These four polyhedrons mark the establishment of the four Kabbalistic worlds within the 'fractal tree of life'.

The basic module of the "fractal tree of life" is the composite assemblage of the five masculine Sephirote constructed one within the other as:

Tipareth within Hessed within Hohkmah within Netzaack within Kether

which is to say:

Heart within Soul within Mind within Might within the Supernal(senses)

The stellated and compound forms symbolizing these masculine Sephirote contain within them 22 feminine Sephirote. Twenty-one of these feminine Sephirote are ordered within Kether in accord with the Gematria structure of the divine name assigned to Kether itself and the structure of this name is also found in the ordering of the vertices in the above pictograph of the symbol for Kether; the stellated Dodecahedron.

When we examine this basic module of the "fractal tree of life" we find that the outer vertices of all geometric forms within the 'fractal tree' are contained on just four concentric spherical surfaces. Three of these concentric spherical surfaces are established by the construction of the geometric forms, one within the other, of the three forms representing the Three Mothers. The vertices of each of the three establish separate spheres, one within the other. As was stated the 'fractal tree' contains 22 platonic solids which are geometric symbols of feminine Sephirote. These three spheres of the Three Mothers establish all of the 22 forms. The outermost of the three mother spheres is established by the Tetrahedron(symbol of Beinah and the letter Shin), the Dodecahedron(symbol of Yesod) is also established with all Tetrahedral vertices on that sphere. The fourth and final sphere is established by the stellations of the Dodecahedron(Yesod) which is to extend the edges of the form to form the symbol for Kether having 12 outer vertices, which reside in that outermost sphere.

The four concentric spherical surfaces correspond to the four worlds of the Kabbalistic paradigm. Without elaboration I would like to simply conclude that when the basic module becomes infinitely reproduced by the filling of Hessed, Kether is then grasped within Malkute in reflection, the vessel receives the supernal, and the supernal the kingdom; making of the four worlds only three distinctly different worlds, although, now becoming infinitely reflected and filled with the light of the supernal world experienced in Kether.

Symbolizing the Decalogue with Geometry

If one considers the productive relationships among the elemental forms, there can be found an ordering of the forms, as one is produced from the other, which imitates the form of the Decalogue to a high degree. There are just ten productive relationships available which present a mnemonic of the Ten Commandments, and there are in addition two special reproductive relationships to consider that mnemonically represent the two Great Commandments.

The productive relationships correspond to the construction of one form(Sephirah) within another where the ordering of these constructions is three tiered. The constructions are ordered firstly by the greatest number of constructions required to create the product, then by the greatest number of vertices of the host, and finally by the number of vertices of the product. These relationships are shown among the five Platonic Solids. It is of interest to note that the sum of the faces of the Platonic Solids is 50, as there were 50 days from the Exodus to the delivery of the commandments, also the sum of the angles of these faces is exactly 40 complete rotations, and Moses was on the mountain for forty days. What follows is a list of the Ten Commandments each accompanied by a short mnemonic phrase given through the Geometry:

The Ten Commandments

(I) "I am the Lord your God who brought you out of the land of Egypt, out of the house of bondage. You shall have no other gods before me."

Glory(Hod) within Knowledge(Daath) within the Crown(Kether) within the Kingdom(Malkute)

(II) "You shall not make for yourself a graven image, or any likeness of anything that is in heaven above, or that is in the earth beneath, or that is in the water under the earth; you shall not bow down to them or serve them; for I the Lord your God am a jealous God, visiting the iniquity of the fathers upon the children to the third and fourth generation of those who hate me, but showing steadfast love to thousands of those who love me and keep my commandments."

Glory(Hod) within Knowledge(Daath) within the Senses(Yesod)

(III) "You shall not take the name of the Lord in vain; for the Lord will not hold him guiltless who takes his name in vain."

Strength(Gevurah) within the Crown(Kether) within the Kingdom(Malkute)

(IV) "Remember the Sabbath day, to keep it holy. Six days you shall labor, and do all your work; but the seventh day is a Sabbath to the Lord your God; in it you shall not do any work, you, or your son, or your daughter, your manservant, or your maidservant, or your cattle, or the sojourner who is within your gates; for in six days the Lord made heaven and earth, the sea, and all that is in them, and rested the seventh day; therefore the Lord blessed the Sabbath day and hallowed it."

Understanding(Beinah) within the Crown(Kether) within The Kingdom(Malkute)

(V) "Honor your father and your mother, that your days may be long in the land which the Lord your God gives you."

Glory(Hod) within Knowledge(Daath) within Strength(Gevurah)

(VI) "You shall not kill."

The Kingdom(Malkute) within Beauty(Tipareth) within the Senses(Yesod)

(VII) "You shall not commit adultery."

The Senses(Yesod) within the Crown(Kether) within the Kingdom(Malkute)

(VIII) "You shall not steal."

Strength(Gevurah) within the Senses(Yesod)

(IX) "You shall not bear false witness against your neighbor."

Understanding(Beinah) within the Senses(Yesod)

(X) "You shall not covet your neighbor's house; you shall not covet you neighbor's wife, or his manservant, or his maidservant, or his ox, or his ass, or anything else that is your neighbor's."

Understanding(Beinah) within Strength(Gevurah)

The Royal Commandments

"Hear, O Israel: The Lord is one; and you shall love the Lord your God with all your heart, and with all your soul, and with all your mind, and with all your strength."

The Kingdom(Malkute) within Beauty(Tipareth) within the Crown(Kether) within the Kingdom(Malkute)

"You shall love your neighbor as yourself."

The Senses(Yesod) within the Crown(Kether) within Beauty(Tipareth) within the Senses(Yesod)

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(The foregoing symbolic relationships have been compiled in a document entitled "DODECALOGUE", to view this symbolic document click on its name.)