Natural Geometry and Holy nouns

Natural Geometry and Holy nouns

AYIN-SOPH

__________Shin_________

___________|____ Aleph _

_ Mem _____|___________

"For the earth shall be filled with the knowledge of the glory of the Lord, as the waters cover the sea."
HABAKKUK Chapter Two verse 14
(click on any image to see model viewer of symbol, click and drag viewer to rotate model)

Introduction to a memory castle Tree of Life ...

The five regular geometrical solids. Forms alluded to in Plato's dialog Timeaus are employed in this paper as a means of accounting for the sacred words of the Etz Chaim or Tree of Life. Sacred means that which has been set apart, and the Platonic solids in the vacuous or wireframe form are set apart here. This is not as fortunate as it might at first seem. The solids must be limited to some specifics in order that the presentation have real meaning. Further they are mundane which is problematical in that the highest aspect of understanding tradition is heavenly and not only earthly. This is simply a metaphorical accounting which may provide another "feeling for god in everything" as someone had said of my early efforts. The religious seek to bring heaven to earth by "flying up and bringing it down" so to speak. This is healthy optimism, if we understand that, it could happen in a moment, it may require many years, a life time or not be given in one life to be understood after a resurrection of some kind. And, name is a much more important concept than number, popular numerology be damned, since personal divination quenches ones divine fire. Also life is not simply an exersize in reading, life is for us to really live; feed them, wash them, dress them up, visit the sick, and create a holy imagination in everyone through doing well for others as simply as you can figure.

Illustrated on this page the geometrical forms of a 'fractal memory castle' will reveal numerical and spatial relations to the Tree of Life, and the Decalogue will be shown as part of its structural memory; as remembering a mnemonic. Memory architectures were developed openly during the 1600's at European universities, and this fractal memory castle borrows from understanding that was gained during that time. Although, the regular geometrical solids have been with us forever it seems, and there is reason to believe that they have always been somewhat inspiring for most peoples of the earth. (In this fractal tree model there is branching potentially everywhere and the only variation in the model attractor is scalar. However, theoretically there could be a fractal rule or rules applied to branching which would make it musical instead of chaotically uniform. Both would still be projectably infinite however.) The purpose of memory architecture is to memorize a body of knowledge and make its recollection rapid and organized. The original development of such architectures provided 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 Kabbalistic worlds.

The graphics above depict the "Tree of Life" of Kabbalah with natural geometrical 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 'sephirot', the plural of sephirah's*, are ten holy attributes (sephirot are: the ten that count; wholesome countings i.e. Deborah is bee and debarot would be bees, I suppose) which are said to inhabit spheres of the first emanation of creation, spheres would be the traditional rooms. Gray squares in the diagram above mark the Sephirah positions. The traditional Tree is drawn with ten plain 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 structure, and it is a symbol for Knowledge(Daath). The ten Sephirot are divided into two groups of five, masculine and feminine, shown above in blue and violet lines respectively.

*a sephirah is a word with feminine tense and singular like a feminine name

NOTE: Most of what is written about Kabbalah in the English language should be taken with a grain of salt, and with this I would recommend that those completely unfamiliar with Hebrew and Kabbalistic concepts take time to read a rather dated summary from the Catholic Encyclopedia. Many particulars of the summary are confused, although, I believe that the general outline may be quite useful for the beginner.

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 Sephirot. The nested relationships between pairs of Solids demonstrate relationships between the feminine Sephirot that can exist. These ten paired relations through association with the Sephirot 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 Sephirot are shown to be modular elements of the masculine Sephirot, and these masculine Sephirot can be taken as the five parts of the soul. The five parts are derived from the Shemma to Love the Lord our God with all Heart, Soul, and Might. The word "might" was elaborated as 'Mind 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 follow 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 Sephirot, the five Elementals

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

Stable Understanding(Binah), would be placed upon the Platonic Solid that has four triangular faces, the Tetrahedron. 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. Remembered from Plato's dialogue "Timaeus" the four faced figure was to symbolize fire which is an energetic 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). These six edges which terminate in twelve angles seem to make a neat mnemonic place for the six names on a tetrahedral frame of any kind, and there are many in structural steel constructions. (The tetrahedron is the basic building block of strength in structural steel constructs.) The four faces can remind us of four basic fields in nature (gravitational, electromagnetic, weak and strong nuclear) gravitational might be assigned to a or the stable triangular base of the tetrahedron. This essay will consider in the main only the wire frame forms of the regular solids, athough the solid form of a tetrahedron is most interesting in not rolling well as a solid model; a tetrahedron shape with uniform mass weighted by low center of gravity.

Since tetrahedrons are not radially symmetrical the geometric center of a single regular tetrahedron lies above the triangle base and below the midpoint of the tetrahedron heigth. If a form is large such as a mountain, then sometimes it becomes convienent to grasp an understanding of the three physical centers that large structures have since it is unlikely that they coincide exactly with each other. These are: the geometric center, the center of mass, and of course the commonly misunderstood center of gravity which is probably best understood in comparing travellers in spacecraft with travellers on earth. 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 twelve+four+six+four*, a sum of twenty+six as is the sum of the Tetragrammaton (ten+five+six+five) also twenty+six.

*plus sign "+" read as "and"

Binah is referred to as the Mother of form, and the four Vertices of the Tetrahedron concurrently establish a spherical surface. The Tetrahedron is a three-dimensional triangle. It has heigth, length, width and a certain kind of breadth which should not be underestimated! 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.

Principled Strength(Gevurah) would be 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 a cube strong one must construct a diagonal or diagon meaning straight line segment (or polygon 2) across each face; converting the surface into a system of triangles, triangles being the only polygon to which 'rigidity' is attributable. The cube has eight corners, which are identicle vertexes since the angular displacements bearing on each edge, are in simmilar correspondence, with each edge joined at the other vertices.

The observation of four+four vertices, four above and four below, makes a suggestion for a type of regressive method in analysis. When the diagon is the second polygon, the single point becomes the first polygon, which of course could be a specific instantaneous view of a diagon in rotation; seen at one end along the perpendicular to the plane of sight. Here the cube can proceed into the real through speculation given that anyone might be thus first aquainted with the cube as affore mentioned. Credit here goes to Buckminister Fuller for an excellent term "diagon" which names the second polygon. None the less, the Cube edges(diagons) are apt as a symbol of Gevurah which also has a connotation of severity(Din). 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'. A convienient working system of meter is of course binary which has been carried by Brittan for many centuries. With a hail for Brittania the cube rotates into speculation by the powers of two from the first power, a single point(vertex).

Does it ever seem at a certain time and season that the clock is two hours off? It might be the case if you are alternating between time standards like daylight savings time (DST) and eastern standard time (EST). Why one might seem as two was solved with the adoption of the Hindu Zero symbol. Mathematics did work very well before that, and after the adoption of zero technology, so to speak, it gradually picked up speed becomming faster than before. 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. Many Paralellepiped forms are convienent in modeling standard geometrical approachs to electromagnetic field study within electrodynamic physics. In addition the surface features (Edges, Vertices, Faces, Angles) of the cube (twelve+eight+six+twentyfour) have the sum of fifty, which is associated with the five letter name of God.

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

Very 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.) Daath comes together with two kinds of complimentary fire to illuminate the splendor enduring for only a moment. Anyone should remain cautious about what is or should be truly known. There can be hazard in some things known that would be known, were they not want of hazard.

*in this older case of english
"of" would mean "layed upon with" invertedly modifying
the word "want" only; a contracted usage of a small word
as my father spoke, now from my middle aged recollection...; nevermind.

(There is a third set of connections through the center of the Cube and these contribute a suggested formation of a twelfth symbol as a possibility, 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 DUODECALOGUE which in part forms a mnemonic of the Ten Commandments. Twenty-eight symbols are employed to express the commandments as relations among the Sephirot. The cube when completely interconnected has a total of 28 edges. These twenty-eight symbols from the DUODECALOGUE compilation can be symmetrically distributed upon these twenty-eight 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).

There are many kinds of regulation which we might call Law. There are sacred laws and civil laws. Our religious bodies preserve sacred law and our civil bodies regulate civil law. These are only two large categories of law and there can be others that we might name such as scientific and mathematical laws or even cultural laws which prescribe cultural value centered in the people of a culture. Our ideas are not so much within our books as they are instead within our selves. So, we must learn to honor the distinctions between kinds of law which pertain to ourselves. i.e. ones personal culture and local civil group and religious group, etc.. There are currently in the category of medicine subcategories such as laws of fertility. It must be said that this presentation of symbols is simply a presentation of symbols which might be applied to anything, and these symbols are not a special kind of law. A nice application of the symbols might be as quick pointers to general categories, which could be made more specific by forming a structured presentation in accord with some method or rule. And, here again, the symbols are like cosmological ruled patterns(rules; meaning drawn lines) in compliment to the countings(sephirot) of Kabbalah.

Splendorous honor(Hod), the form with eight triangular faces, the Octahedron. This form is always constructed within Knowledge(Daath), it is the spiritual form, the root substance of the spiritual soul. Symbolizing Air, it is the first form to have an axis to spin on. Spinning or the whirling of vortex rings specifies the motion of fluids of which air is one. As with the Tetrahedron and Cube the surface features (E,V,F,A) (twelve+six+eight+twentyfour) provide a sum of fifty, which is associated with the five letter name of God.

The edges and vertices of the Octahedron(Hod) can be impressed with the twenty-two letters of the Hebrew Alphabet as a foreshadowing of the Alphabet's manifestation in Yesod. The Octahedron has twelve outer edges which can be impressed with the twelve elemental letters. The Octahedron when completely interconnected has three diameters which can be places for the three Mother letters. Finally these three and twelve 'edges' are bound together by seven vertices which are 'memory places' for the seven Double letters.

Greater significance will be seen in this form when exploring its compound configuration (Hesed). For now let it be understood that this form is purely secondary in relation to the construction of the other forms, and for this reason has a special place within the understanding of the 'fractal' nature of the completed geometry.

The Adult Senses or sensiblities(Yesod), the twelve pentagon faced Platonic Solid, the Dodecahedron. This form is, among the Elementals, the most complex. Yesod is said to contain the Alefbeis(Hebrew alphabet) of twenty-two letters. Like the Cube, the Dodecahedron is compound, and it is composed of twenty-two Platonic 'solids'; when five secondary Octahedrons are also included in the construction. Understanding this, the Dodecahedron becomes the focus of the entire Tree, the Tsaddik(the righteous one), the Microcosm, etc..

Here I would like to posit a bit more correspondence to be found in the dodecahedron structure. It must be remembered that this form is the most complex among the feminine forms, and that finding countable features within it which provide suitable 'memory places' for Kabbalistic concepts or the concepts of other systems is to be expected from the magnitude of the dodecahedrons natural complexity. However, as more attributes of a form are found to be suitable 'memory places' for the Kabbalistic attributes of a Sephirah it becomes more 'necessary' that the form be employed as the Sephirah symbol.

Yesod is the Sephirah of righteousness and is associated with the righteous King, and foundation. Yesod implies the entire Hebrew alphabet of twenty-two letters. One of those letters, Tsaddi, is the symbol of righteousness. Tsaddi is one among those five letters of the Alefbeis which have an alternate 'final' form when it is written at the end of a Hebrew word. Each of the twenty-two letters also serves as a numeral representing a number value. The number value of the letter Tsaddi is ninety, and when the edges of all twenty-two Platonic Solids are used to establish planes that pass through the center of the form each pentagon face of the dodecahedron is divided by the planar edges into exactly ninety irregular polygons. Among the ninety on each face are nine different polygons eight irregular triangles and one irregular quadrilateral, repeated ten times through rotations and reflections about the center of each face. (see: forty-five fractal planes)

Probably of greater interest is that the number value of the final form of Tsaddi is also expressed in the Dodecahedron symbol for Yesod. The Dodecahedron has thirty outer edges and each of the remaining twenty-one Platonic Solids express edges as linear connections within the Dodecahedron. Since there are many intersections among the connections within, all internal connections are established as multiple line segments. The line segment sum from all twenty-two Platonic Solids within the Dodecahedron(Yesod) is nine-hundred which is the number value of the final form of Tsaddi.

Nine-Hundred, ninety, and nine are all assigned to the sefirah Yesod.

(This observation is very important because in it we are counting three things that have some relations. This means that there are reasons, very obvious ones, which would lead us to relate or tell of them at the same time, i.e. they are ordinarily related about together. These are engineeringly different kinds of things among features of the dodecahedron that may be told by observation about that form. When doing engineering these data cannot be simply presented as 999. Although, the tripple relation can be noted as 900:90:9 indicating that each is somehow related and pointing to different attributes of one thing.)

There are twenty-two words which name the Hebrew letters, and ten words which name the sefirot, thirty-two paths of wisdom by those words. Numbers are words which name counts in counting things. We also need to name each kind of thing we count in our engineering effort. Engineering unit names, (like watt/hours, yards and degrees, pints and pounds and many others), a set of them is engineering, and an art of engineering to resolve a set well into workable specific terms which must be written terms in documentation.

Still, the primary meaning of the Dodecahedron is the implication of the outer edges, as a symbol of the skin or importantly the Senses. This would seem to imply that our Senses, and recollection of sense experience are what is 'true' about us; a symbol for the real boundaries of sensibility, our foundation. Finally, the completely interconnected dodecahedron contains only three regular polygons: triangles, squares, and pentagons having 3:4:5 sides respectively, which can be used to represent 345, the gematria value of the name, God Almighty.

(A or The, (echad or ha)) Kingdom(Malkuth), the twenty triangular faced form, the Icosahedron. As was stated, the Dodecahedron is composed of twenty-two Platonic 'solids'. The one that is the innermost is the Icosahedron(Kingdom). It is the central chamber of the 'Heart'; the Star Icosahedron. The Icosahedron(Kingdom) is also compound, although this innermost form constructs only one other form: the Star Dodecahedron within itself, symbol of the Divine Being, the Crown(Kether). So, in other words, Kether is to be found within Malkuth, and Malkuth within Kether.

This symbol is the ancient symbol for Water, so importantly the Kingdom is associated with Water, and 'Blood' instead of Earth. When completely interconnected the icosahedron defines 66 lines among its twelve vertices, which corresponds to the name, Adonai.

The masculine Sephirot, the five parts of the Soul

The cover page shows the masculine Sephirot in blue lines. These five forms correspond to the five parts of the soul, and to the five parts of an "infinitely repeating self similar three dimensional pattern of form"; a fractal.

Child Intuition (a or the) Heavenly Crown(Kether) is a Star Dodecahedron. As a part of the soul it represents the symbol only; the supernatural sense of experiences. The highest reach as a metaphor of reaching upward, a transcendental extension of Yesod the mundane Senses.

Supernally there is a crown above the head which is beyond sensations. The heavenly crown is some distance above the head. Perhaps the same small distance above the head as the distance of the feet above the ground, when descending onto the earth, delivered here upon the earth from eternity. amen the halleluyah

(There can be no idols of that which is never apparent. The symbol, put in the place of the heavenly crown, is simply implied by natural forms which have been already discovered in creation such as the forms of salt cystals. This mundane star dodecahedron when internally elaborated with the edges of twenty-one additional mundane forms, twenty of which are the edges of three common crystaline shapes, becomes a system of one-thousand line segments (containing a completly interconnected dodecahedron with five octahedrons). Line segments when considered in comparison one to another can imply string lengths in music theory. Among the one-thousand segments there are only about 15 compared lengths which would provide music, where very few tones could be played together, and the 15 tones would span about three octaves. The music perhaps, is never played upon instruments, always sung by the audience, and always sung as vowels like the native American tremelo.) When completely internally constructed the star dodechaedron contains: (1)icosahedron, (5)octahedrons, (10)tetrahedrons, (5)cubes; this as gematria expresses the name, I will be.

Ageless Wisdom(Hokmah) is a special subset of the Dodecahedron; it is ten Tetrahedrons connected to and estabished by the primary 20 dodecahedron Vertices. Hokmah has 170 Vertices, and 30 of those, the innermost vertices, form an embrace for five Octahedrons (to be described following the next form). Hokmah 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 resulting in additional vertices, 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. 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(Hokmah).

So we see that, the elements from which a or the (echad? or ha?) 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. The ten tetrahedrons of this symbol forming five pairs, corresponds to the gematria of the name, Yah.

there is no word or thought only brilliant light. halleluyah for I am and understand god is

Steadfast Mercy(Hesed) would be impressed upon a composite of five Octahedrons that embrace the twelve inner icosahedral Vertices of the Star 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 within Hokmah(Mind), the 12 innermost connect it around the icosahedron of Tifaret(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(Hesed) 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; forty-five fractal 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; Star Icosahedrons within Star 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 Hesed within Hokmah 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 Hesed(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. The thirty+one vertex pairs of this symbol correspond to the gematria for the name, El.

god in steadfast mercy brings principle strength

Luminous Beauty(Tifaret) is a Star Icosahedron. Its part of the soul is the Heart. This form has thirty-two Vertices, and the Hebrew word for Heart the same Gematria value(a spelling check sum used in handwritten transcriptions mostly). Lev(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 Star Dodecahedrons inner Vertices, and in like manner another SD can be constructed from the inner Vertices of the SI, and another SI within the SD, 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. When the edges of this form are shown transparently it has 91 apparent vertices, and 31 of its thirty-two real vertices can be seen, one of two vertices can be seen at the center surounded by three concentric sets of ten. The star-dodecahedron has the same properties as the star icosahderon when illustrated by the same means. Although, the star-dodecahedron differs by providing 111 apparent dodecahedron vertices when shown transparently. The image above shows only 21 real star-icosahedron vertices as does the former star-dodecahdron image make a similar demonstration of 21 real vertexes.

god in halleluyah, it is well with my soul

Enduring Victory(Netzak) 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.

formation, the warmth of the day, my feet... and amen the halleluyah

This five Cube symbol for Netzach(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 3:4:5 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.

AYIN-SOF

Keter

A symbol for an Infinite(Ayin-Sof) could be* the intersection of thirty-one lines at the single central point of the fractal, and this very symbol should be thought of as a symbol of a spark of an Infinite Unapproachable (Ein-Sof-Orh)Light. This symbol suffers as highly qualified in that Ayin-sof has no symbol.

( *this is not a writting... who read it first?... )

No thing and Not a form, and not a Sephirah, Ayin-Soph is in space although not composed of space*. This is an unending hollow: the hollow which hallowed our universe, a hollow earth very fearful, a hollow stone jetting steam beyond boiling from the volcanic ground, Chefas Peter the large hallowed apostle of good natured health, an infertile dove's egg shared bye the first woman. The hollow of purely human heart. This place implies the beginning and end of limit; the unfathomable center with undefineably brilliant sparks; and also the infinite reach of outer construction outgoing everywhere, thirteen billion light-years in only ninety-six steps.

(*Primary Vertices of all forms within the fractal constructed along thirty+one lines.)

The 31 lines, which would be a spark of infinite light, fall into three groups. The lines span 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(Hesed). 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.

(*infinite lines which in part span the diametric paths within a regular polyhedron)

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 Star 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 Sephirot 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. Hokmah is a compound of ten Binah, Hesed is a compound of five Hod, and Netzach 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. Hokmah holds within itself Hesed, and the outer form of Hokmah is Netzach. The form outermost to Netzach is Yesod or Kether, and finally the innermost form, Tipareth, is embraced by Hesed. 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-sulfide.

The three Mothers

AMSh, 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 Sephirot, 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(Malkuth) 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(Binah) 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 thought 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 Sephirot constructed one within the other as:

Tifaret within Hesed within Hokmah within Netzach within Kether

which is to say:

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

The Star and compound forms symbolizing these masculine Sephirot contain within them 22 feminine Sephirot. Twenty-one of these feminine Sephirot 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 Star 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 Sephirot. These three spheres of the Three Mothers establish all of the 22 forms. The outer sphere of the three mother spheres is established by the Tetrahedron(symbol of Binah 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, forming the symbol for Kether: having twelve vertices in outermost reach, residing in the fourth and 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 Hesed, Kether is then grasped within Malkuth 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 faces of the host, and finally by the number of faces 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."

Splendor(Hod) within Knowledge(Daath) within the Crown(Kether) within the Kingdom(Malkuth)

(II) "You shall have no other gods before me. 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."

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

(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."

Understanding(Binah) within the Crown(Kether) within The Kingdom(Malkuth)

(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."

Splendor(Hod) within Knowledge(Daath) within the Foundation(Yesod)

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

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

(VI) "You shall not kill."

The Foundation(Yesod) within the Crown(Kether) within the Kingdom(Malkuth)

(VII) "You shall not commit adultery."

The Kingdom(Malkuth) within Beauty(Tipareth) within the Foundation(Yesod)

(IIX) "You shall not steal."

Strength(Gevurah) within the Foundation(Yesod)

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

Understanding(Binah) within the Foundation(Yesod)

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

Understanding(Binah) within Strength(Gevurah)

The Royal Commandments

"Hear, O Israel: The Lord our God is one Lord: and thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy might (mindful strength)."

The Kingdom(Malkuth) within Beauty(Tipareth) within the Crown(Kether) within the Kingdom(M)

"Thou shalt not avenge, nor bear any grudge against the children of thy people, but thou shalt love thy neighbor as thyself: I am the Lord."

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

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

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Outline of the Discovered Correspondences

The theme of my original study of geometrical metaphysics was to make illustrations of geometric circumstances where each of the first twelve integers were implied. The motivating thought was that the ancients probably were very much concerned with representing their theosophical ideas with integer representations, magical numbers, instead of the more modern fascination with computation and irrational or transcendental values. The golden section and various irrational square roots are repeatedly demonstrated in ancient temples, etc., but the ancients did not need to be aware of the irrational nature of these proportions to build ancient structures. All proportions in ancient works are simply constructed with compass and straight edge. So I was thinking of monuments like Stonehenge where the presentation of countable objects seems to be thematic. My hope was to discover the ancient key to magical numbering. What I discovered, however, took me beyond any interest in magical uses of number.

The ordered set of productive relationships illustrated in the DUODECALOGUE document were discovered first, and were first drawn as twelve pairs of Platonic Solids in a blank book that I was using to make drawings for my study of Sacred Geometry which began in 1982. The first ten pairs were identified as a unique group describing the 'simple products' of the Platonic Solids, and were placed on a single page in two columns of five pairs. The remaining two pairs were placed on the following page representing the group of 'reproducing' forms.

Among the Platonic solids there are only three 'solids' that have the property of producing Platonic Solids; these are the Icosahedron, Dodecahedron, and Cube. So the left element of each pair was selected from this group of 'host' forms with each product form as the rightmost element. The product form is constructed inside of the host either by the complete interconnection of the primary vertices of the host or by the interconnection of secondary vertices that have been formed from the complete interconnection of the host. The complete interconnection of the host form is considered to be a single construction, and the pairs of Platonic Solids were ordered firstly by the number of constructions required to produce the product from each host.

The ordered pairs have been rendered in the DUODECALOGUE document with the intermediate constructions between them. You will note that the first group of four symbols at the top of the left tablet are rendered in three colors. Each color represents a level of construction, as three separate constructions are required to produce an Octahedron within an Icosahedron. Each of the remaining productions on the left tablet are in two colors, and represent two level constructions. All productions on the right tablet are in one color, and are single constructions given by complete interconnection of the host. It is here that we have the first reflection of the Decalogue within the compilation.

The Lord makes reference to Himself within each of the first five commandments and no reference in the remaining five, as the first five productive relationships are compound and the other five are singular. The reflection of the Decalogue becomes more acute when we consider the placement of the products on the tablets.

The first product the Octahedron symbolizing Splendor is shown in correspondence to the first commandment, the fourth, and the fifth, these are in essence: to Honor God, Honor the Sabbath day, and Honor Mother and Father.

The second product the Cube symbolizing Strength appears in correspondence to the second, and eighth commandments: the prohibition of making images, and prohibition of stealing; each use of Strength.

The third product the Tetrahedron symbolizing Understanding is found in the third, ninth, and tenth commandments which make reference to: the use of the Lords name, the telling of the truth, and thoughts of another's house; all examples of the use of mind or Understanding.

The remaining two commandments, the sixth and seventh, are complimentary, the prohibition of murder is the Senses(Dodeca') within the Kingdom(Icosa') , and the prohibition of adultery is the Kingdom within the Senses.

There are two more productions that can be made, and they are extensions of the last two, but they are in a special group because they represent 'reproduction' of two host Platonic Solids. These are known as the Greatest or Royal Commandments.

The first, love of God, places the "Child of God"(Kether:SD) within the Kingdom(Icosahedron), and an image of the Kingdom(Icosa') within his Heart(Tipareth:SI). [Kingdom within the Kingdom]

The second, love of neighbor , places the "Child of God"(Kether:SD) within the Heart(SI) of the "Righteous One"(Yesod:D). [Senses within the Senses]

Finally these two reproductive relationships are each based on the continuum formed from the two Star forms of the Dodecahedron and Icosahedron, so like the actual commandments themselves which are considered to be two forms of the same expression, these reproducing relationships are two aspects of the same continuum.

without the bees the flowers would not come in seasons, without the eggs we have no birds, so you would see,
with two we get four

Building Polyhedron Models

The key to the comprehension of this presentation of geometrical metaphysics, is the building of models. When one makes the models the mind becomes filled with new understandings of the forms. Each of the forms can be viewed in more than one way, there are some optical illusions to be explored, and actually building them gives one experience with them. It is something that we are all familiar with, play. Just as the child incorporates ideas of the world into his soul by play so must the geometer. To view some models constructed of wooden and bamboo sticks click Here.

Concluding remarks

My conclusion is that the language of Kabbalistic theosophy dictates the structure of the geometric forms presented above. You must pardon me for not making this completely clear, by way of presenting the structural correspondences of names of God with the symbols of the Sephirot. The reason for the omission is that one must be mentally prepared for the discovery of letter by letter accounting to be found within these symbols. One "must not" take any of these symbols to be names of God. The only way to understand each correspondence without suffering "grief" is to understand that the symbol under consideration has itself been formed from the Gematria structure of the traditional divine name, and not that the geometric structure is the name.

The presentation is merely a kind of numerical accounting so it is not a powerful expression. The concept of name is much more potent than the concept of number. Through names generalization is possible. Even in mathematics the abiltity to name variables provides the generation of algebras which are vastly more far reaching than arithmetic providing the ability to operate on any kind of number or functional definition. Naming can name any idea, not being limited to number ideas.

After making the initial discovery I had reflected upon asking God, when I was no more than seven years old, to help me discover the way in which His Law was written in nature. I am grateful for what I have found, even though I think the implications extend well beyond my ability to comprehend. It is only recently that the Kabbalistic structure was identified, and I am not a Kabbalist. The Kabbalah has been referred to as a file cabinet by some, so please try to overlook the way in which things seem to be misfiled. Of course my label for this is Sacred Geometry, and I might be the only geometer claiming divine inspiration for such information, but that is how I see it. Although, I at this point only have vague ideas as to how such knowledge might by used, hopefully someone reading this page will find great benefit.

Hopefully you found this presentation of Geometrical Kabbalah of some interest. If you have any questions or there are ideas that you would like to discuss, please contact me at the following e-mail address.

Peace,

J.Vincent Beall(Bell)

May 2, 1997

Post Script note in further conclusions: The five regular forms must have been imagined in part by the first men who realized early magisterium within cave safety, and also the many values of cave products. Probably mostly mineral and metal crystaline materials. Only three crystal grails were generally avilable, for grading stone with the grate of the complete grail in five stones. The other two, five part symmetrical crystals, are known as realities only lately, and both are small cryptocrystaline forms too small to see without a powerful magnification from electron microscopes. So, it may be that about the time that quasicrystal surface treatment, had been developed, for very light weight frying pans that evenly heat, science as science began in caves, simply ended. The last of the five crystal grails, had been found in the little dodecahedron shaped alloys.

Then again it may have been the boron-silicon crystals, related to the boron-sulfur icosahedrons, which were found last in studies of boron doped silicon, silicon crystals, so useful in making electricity from light. We're so sorry uncle Albert, but we haven't done a blessed thing all day...solar cooked pancakes, maybe. and solar powered roaming lamp at my feet, perhaps. This past winter was season two-hundred and twenty and two for my personal catalog of seasons and winter was my winters fifty-sixth. An R2D2 lamp at my feet, that would move with me on flashlight signals might help, when I awake from sleep during the night.

Peace, do not leave home without that, His and yours...

May 2, 2007

Sacred Geometry Discovery
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Natural Geometry and Holy nouns
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