As described in depth in ‘What’s the Point’ the second article, or lesson, in this series on the meaning of Sacred Geometry, it was explained that the entire process of Sacred Geometry begins with a mere point. From that initial point either a straight line or a circle may be generated. This initial generating point was likened to a seed. It was proposed that this was the basis for the New Testament parable of the Grain of Mustard Seed, an analog for the mass singularity in which the potential of the entire universe of Space and Time was contained at the first moment of Creation.

It was also proposed that corresponding to this ultimate infinitesimal point of infinite potential which preceded all of manifest reality, there exists scale invariant analogs in the form of mass singularities at the core of ‘black holes’ scattered throughout the universe, serving as nuclei for the formation of galactic systems. And, it was also proposed that corresponding to these external singularities there exists an inner, spiritual nucleus of infinite potential, in the form of the Bindu, the gateway to Superconsciousness. The idea was presented that the operations of Sacred Geometry recapitulate and symbolize the primordial process of cosmic manifestation, both internal and external.

As discussed in that article a brief examination of various ancient or sacred traditions reveals that there exist surprising parallels between archaic and contemporary models of Creation.The early 20th century author, kabbalist and ceremonial magician,

Charles Stansfield Jones, writing under his magical *nom de plume* Frater Achad, gave a succinct and effective description of the occult model of Primordial Creation that echoes the varied traditions briefly surveyed in Lesson Two.

“To begin at the beginning . . . the Qabalists postulated the AIN or NO-THING as the Zero From which, in a mysterious manner, the Universe arose. Next, they say, the AIN SUPH, or Limitless Space, became the Nature of the AIN, and this conception was followed by that of AIN SUPH AUR or the Limitless Light of Chaos. It was not until this Limitless Lighthad concentrated Itself to a Centre that the First Positive Idea arose, and this was called

Kether and attributed to the Number One . . .”

“Let us accept the term AIN as representing That of which Nothing is known, nor can be known, except through the positive manifestations which arise from it. When we attempt to imagine AIN SUPH—Limitless Space—our minds tend to rush on and on, only to fall back before the Profundity of the Great Deep; yet we have to admit the possibility of Infinite extension in space . . . When the AIN SUPH AUR became concentrated upon a Single Centre, it compressed the Light into Substance of Light, which is Life. Or, in other words, the Concentrated Light became an inconceivably powerful Force or Energy in the centre of Kether. This Pure Being, or Living Substance, owing to its reaction from the Invisible Centre, tends to expand towards Infinity. This gives us the idea of theSubstance of the Universe ever expanding, ever occupying more and more of the Limitless Space of AIN SUPH, while the Primal Centralizing Urge still continues to contract upon the Infinitely Small, or the AIN . . . Kether is then the junction of these Two Infinites, but particularly represents the concentration of the Light to a Point on its way to the Infinitely Small . . .”1

The study of Sacred Geometry teaches us that the ** “Living Substance, ever expanding into Limitless Space,”** proceeds according to the principles of Geometry and Proportion. Johannes Kepler articulated this understanding when he said

**Kether is the highest emanation of the Kabbalistic Tree of Life, which itself is a manifestation of the geometry of Creation through its structure of 10 emanations and 22 paths of elemental force linking them into a pattern of Time, Space and Energy.**

*“Geometry existed before the creation. It is co-eternal with the mind of God…Geometry provided God with a model for the Creation…”*The tree represents a map of Creation, unfolding from the primordial Unity to the infinite diversity of manifested reality, expressed according to a certain mathematical progression based upon the square root of three, the holy Trinity.

In the previous lesson I posed the question: * “If the primordial point, the singularity, is the creative seed, what, might we ask, constitutes the womb of the ‘supernal mother’? Geometry provides the answer as we shall see.”* In this article I am going to walk you through an exercise that demonstrates the opening moves in the emergence of the entire corpus of geometric symbolism and is itself an excellent example of simple geometrical reasoning. It is the construction of the Vesica Piscis, or the Vessel of the Fish, actually Fishes Bladder. The figure is sometimes referred to as a

**mandorla**, which in Italian means ‘almond’, and hence, is sometimes also called the ‘Mystic Almond’ due to its shape.

The geometric construction of the kabbalistic Tree of Life begins with the Vesica Piscis. If you have not read the two articles preceding this one I would strongly urge you to do so before proceeding. Let us establish some preliminaries. As with all areas of human activity that evolve and grow through time, the practice of geometry has its own vocabulary.

Many of the terms used in Geometry are also used in everyday language but without the precision of meaning that is necessary for the effective practice of the geometric art. If you are a sincere student of the Mysteries, but you are not already familiar with the fundamental ideas of geometry please make the effort to begin learning the basic terminology now. If you are mathematically learned please bear with us as we cover some preliminaries.

So let us define a few terms.

A **circle** can be thought of as the set (or locus) of points equidistant from a fixed point called the center. The set of points so designated is called the **circumference** of the circle. Any line connecting the center of the circle to its circumference is called a **radius** of the circle (plural: radii). By the definition of a circle, all radii of any circle must be equal in length. A circle can have an infinite number of radii. Any line passing through the center of a circle that is terminated on both ends by the circumference is termed a **diameter** of the circle. A diameter of a circle must necessarily be twice the length of its radius. A circle can have an infinite number of diameters.

A **triangle** is a figure composed of three straight lines. If all three lines are the same length the triangle is said to be **equilateral**. If two sides are the same length and the third side different the triangle is called **isosceles**. If all three sides are of different length the triangle is said to be **scalene**. A triangle with one right angle is called a **right** triangle. A right triangle can be either isosceles or scalene.

The triangle above demonstrates the standard labeling for any triangle. The vertices, or corners, which are mere points without dimension, are indicated by the use of capital letters as shown. The sides of a triangle are labeled with small case letters with the usual practice being to name any side by the same letter name of the vertex opposite. This gives us two different ways of designating a given side, either by the single lower case letter such as * Side a* above, or by the use of the two capital letters naming the end points of that side. For example

*in the above triangle could also be called*

**Side a****could also be named**

*Side BC, Side b***, and so on.**

*Side AC*It is common practice to draw a triangle with one side horizontal, as side AB in the triangle above. This side, which is below the rest of the triangle, is called its **base**. The vertex (corner) opposite the base is called the **apex** of the triangle. Also in the above triangle the dashed line CD, drawn perpendicular to the base up to the apex, is called the **altitude** of the triangle. The length of this line is frequently thought of as the height of the triangle. Of course any side of the triangle can serve as the base

. That being the case a triangle can have three possible altitudes. A general definition for altitude would then be: *An altitude of a triangle is a line drawn from any vertex perpendicular to the opposite side.*

Now let us state the very first proposition of Euclidean geometry:

**All radii of the same circle are equal.**

Pretty simple, that’s all there is to it. But that’s where geometry starts, with the most basic expression of obvious relationships, things that all of us who are reasonably sane can agree upon once we have established the meaning of our terms.

Let us now state one of the fundamental axioms of mathematical reasoning:

**Two things both equal to a third thing are equal to each other.**

This turns out to be one of the most useful of axioms and we call upon it frequently in the process of geometric reasoning. Mathematically we would say: If **a = b and b = c, then a = c**.

Both the fundamental axiom and the First Proposition are employed in understanding theVesica Piscis and are necessary to initiate the process of geometric construction. Having now disposed of some basic definitions and axioms, let us proceed.

A Vesica emerges out of the intersection of two circles of equal radius, placed so that the circumference of each lies on the center of the other. First we draw a single circle. We perceive that the first act is the placing of compass point to paper—the emergence of the primordial Singularity. We then draw, or construct, our circumference. We next draw a second circle by placing our compass point anywhere on the circumference of the first circle, making sure that the second circle is exactly the same size (radius) as the first circle, thusly:

The resultant area of overlap, or intersection, is the Vesica. In the third image above the ‘line of centers’ has been inserted, defining the width of the Vesica. The Vesica is endowed with important and unique properties. Again quoting Frater Achad regarding the occult properties of the Vesica:

**“The curious and marvelous properties of the Vesica Piscis and of the Rectangle formed on its length and breadth, have been subjects of profound speculation…”2**

In his researches Jones discovered the relevance of the Vesica Piscis to the design of the kabbalistic Tree of Life, as mentioned above, recognizing that it provided a powerful geometric key for unlocking the cryptic meaning within that most potent of occult symbols. Through simple geometric reasoning we come to understand that the length to width ratio of the Vesica is expressed as the Square Root of Three to One, or** √3 : 1**.

In other words, if the width of the Vesica is 1, then the length can be expressed as the Square Root of Three, which, rounded off to three decimal places is approximately** 1.732…** The dots mean that the decimal value of the Square Root of Three is an irrational number that continues forever, without terminating or repeating, exactly like the more well known value of Pi.

The Vesica with axes of length and width shown.

Contemporary scholar and geometrician Rachel Fletcher who regularly contributes a column to the **Nexus Network Journal** expounds upon the fundamental symbolic meaning of the Vesica:

“The vesica piscis signifies the mediation of two distinct entities; the complementariness of polar opposites, as when two extremes complete and depend upon one another to exist. One circle may signify the breath of spirit, which is eternal; the other may signify the body physical, which is forever changing and adapting. The vesica piscis itself symbolizes that which mediates spirit; or the psyche or soul.”3

Stansfield Jones mentions the Rectangle formed on the length and breadth of the Vesica as sharing in the ‘curious and marvelous properties’ that make the Vesica the subject of ‘profound speculation.’

The Vesica with superimposed rectangle

This particular rectangle has come to be known as the root 3 rectangle, for obvious reasons, in that the ratio of its long side to its short side expresses the Square Root of 3 to 1. This particular rectangle exemplifies a very special property of self-reproducibility that was called Dynamic Symmetry by artist and author Jay Hambidge, who in the early 20th century, rediscovered the proportioning system utilized by the ancient Greeks (among others) in their arts, crafts and architectural design.

This rectangle was also extensively employed by Freemasons of the Middle Ages in the erection of their architectural masterpieces. The unique and powerful properties of Dynamic Symmetry are explored extensively in our classes and seminars on Sacred Geometry. In reference to the use of the Vesica as a proportioning figure by medieval builders and to its symbolic significance, the early 20th century architect, mystic and Freemason,

William Stirling wrote:

“It is known both to freemasons and architects that the mystical figure called the Vesica Piscis, so popular in the Middle Ages, and generally placed as the first proposition of Euclid, was a symbol applied by the masons in planning their temples. Albrect Dürer, Serlio, and other architectural writers depict the Vesica in their works, but presumably because an unspeakable mystery attached to it these authors make no reference to it. Thomas Kerrich, a freemason and principal librarian of the University of Cambridge, read a paper upon this mystical figure before the Society of Antiquaries on January 20th,1820. He illustrated his remarks with many diagrams, illustrating its use by the ancient masons, and piously concludes by saying, ‘I would by no means indulge in conjectures as to the reference these figures might possibly have to the most sacred mysteries of religion.”

Dr. Oliver, speaking of the Vesica, says, ‘This mysterious figure Vesica Piscis possessed an unbounded influence on the details of sacred architecture; and it constituted the great and enduring secret of our ancient brethren. The plans of religious buildings were determined by its use; and the proportions of length and height were dependent on it . . .’ Mr. Clarkson . . . considered that the elementary letters of the primitive language were derived from the same mystical symbol. He says that it was known to Plato and ‘his masters in the Egyptian colleges,’ and was to the old builders *‘an archetype of ideal beauty’ . . . In Heraldry the Vesica was used as the feminine shield . . . and was also figured as a lozenge or rhombus. In the East the Vesica was used as a symbol of the womb, and was joined to the cross by the Egyptians forming the handle of the Crux ansata.”4*

The Feminine Shield: Vesica with lozenge, or rhombus displayed

Stirling further elaborates upon the symbolism of the Vesica, especially its connection to the the feminine:

“Geometrically, the Vesica is constructed from two intersecting circles, so that it may be taken as having a double significance. Edward Clarkson says that it ‘means astronomically at the present day a starry conjunction; and by a very intelligent transfer of typical ideas a divine marriage,’ or the two-fold essence of life, which the ancients supposed to be male and female. To every Christian the Vesica is familiar from its constant use in early art, for not only was it an attribute of the Virgin, and the feminine aspect of the Saviour as symbolized by the wound in his side, but it commonly surrounds the figure of Christ, as His Throne when seated in Glory. As a hieroglyph the combination of Christ with the Vesica is analogous to the Crux ansata of the Egyptians.”5

The reference to the wound in the side of Christ conceals a deep hermetic mystery, one that is recapitulated in the Grail legend in the form of Lancelot falling upon his sword, resulting in a wound to his side that does not heal, and both are ultimately a reference back to the second chapter of Genesis wherein a wound in the side of Adam results upon removal of a rib by Yahweh in the allegory of the separation of the sexes, which leads, in due course, to the fall of man into mortality.

The frequent depiction of Christ seated within the Vesica conceals an astronomical correlation related to the great cosmic cycles, or ages of the world, the framework of temporal periodicity within which the great drama of the Fall of Man takes place. The symbolism of the Vesica as womb is epitomized by its role in geometry, as the fundamental figure from which all geometric form is generated. Early 20th century Masonic historian Sydney T. Klein wrote about this role of the Vesica and its connection with Freemasonry and the divine creative process.

“This Equilateral Triangle was the earliest symbol, in connection with the Vesica Piscis, we know of the Divine Logos and, as the Bible declared that the Universe was created by the Logos (the Word) so the form of the Lodge which represents the Universe was naturally created by means of the Equilateral Triangle. A great mystery this must have appeared to those who, like the Hellenic philosophers, postulated thateverything on earth has its counterpart in heaven, and who, in their religious mysticism, were always looking for signs of the transcendental in their temporal surroundings . . . But in what awe and reverence must they have held Geometry when they further found that the Equilateral Triangle was itself generated, as in the first problem of Euclid, upon which the whole Science of Geometry was therefore based, by the intersection of two circles.”6

Antiquarian, esotericist and author John Michell studied and wrote extensively on the meaning of the Vesica from the standpoint of archaic science, its relation to the canon of cosmic number and its use as a basis upon which to erect sacred temples. He wrote:

“The reconciliation of six and five and many of the other objects of sacred geometry may be achieved by means of the figure known as the vesica piscis. The ‘vessel of the fish’ is the simplest and most informative geometrical symbol, being the orifice formed of two interpenetrating circles, the centre of each lying on the circumference of the other: this is the womb from which are generated all the numbers and ratios of the Temple. Owing to its high reputation among the masonic builders of medieval cathedrals and to the Piscean associations of its name, the vesica has often been thought especially characteristic of the Christian mysteries. It recurs throughout theNew Testament as the esoteric symbol on which the holy myths are constructed . . . and its influence on architects and scholars at the Renaissance was considerable.” 7

The reconciliation of six and five refers to the potential to develop, from the Vesica as a starting point, polygons of six and five sides, as well as many of the other polygons, with all of their symbolic connotations and unique properties. The ‘vessel of the fish’ refers to the utilization of the Vesica as a symbol of the Piscean age, signified by the passage of the Vernal Equinox into the zodiacal sign of Pisces, the sign of the fishes, approximately 2000 years ago, coinciding with the birth of Christianity. Michell expounds further upon the meaning of the Vesica:

“Although the Vesica was particularly influential at the beginning of Christianity as it is at all such periods in history, it has been respected from the earliest times as a symbol of the sacred marriage, with the spiritual world of essences as the circle on the right penetrating the world of material phenomena on the left. Again, the length and width of the vesica, its longer and shorter axes, are considered respectively positive and negative

. . . The shape that is most characteristic of the vesica piscis is the rhombus of two equilateral triangles, again representing a duality, the upper triangle positive, the lower negative, as flower and root.”

In regards to the connection between the Vesica, the fish, the origins of Christianity and ultimately to its use as an architectural device, Joseph Gwilt, in his monumental Encyclopedia of Architecture writes:

“The Greek word ΙΧΘΥΣ, [Icthus] signifying a fish, seems to have been in early ages a mystical word, under which Christ was denominated. ‘Eo quod in hujus mortalitatis abysso, velut in aquarum profunditate, sine peccato esse potuerit, quemadmodum nihil salsedinis a marinis aquis pisei affricatur;’ that is, Because in the unfathomed deep of this mortal life he could exist without sin, even as a fish in the depths of the sea is not affected by its saltness. The term, too, at a very early period, furnished an anagram,whose parts were expanded into the expression, 9Ihsoy§ Xreisto§ Qeoy Yio§ Svthr . The initials of these words were, in their turn, expanded into a long acrosticon the day of Judgment, said to have been delivered, divino afflatu, by Erythrean Sybil .

. . From an early time the triangle seems to have been associated with as much mystery and veneration as the number 3. Without here touching on symbolism, in its use, whether equilateral or isosceles—we cannot but perceive, both in one and the other, a tendency to the production of the pointed arch.” 8

Gwilt here discloses the important role of the Vesica in the derivation of the pointed, or gothic arch, so prevalent in the great cathedrals of the 12th and 13th centuries. The universality of the Vesica, and its correlate, the equilateral triangle, as an architectural device is underscored by the remarkable fact that its use has apparently been found in ancient North America as a template upon which to base some of the oldest monumental earthworks known to archaeologists.

In northern Louisiana are found three mound complexes, Caney Mounds, Watson Brake and Poverty point, all of which display either the equilateral triangle or the Vesica in their design. Watson Brake, a series of 11 earth mounds laid out on a vesica pattern is estimated to be nearly 5,400 years old!9 What primeval race of man bears responsibility for this geometric effigy that many millennia ago is not known. In an extraordinary little work on the kabbalistic method known as Gematria, the cipher code by which certain sacred writings can be translated into number, and from number into geometry, the authors, architect Bligh Bond and the reverend William Simcox Lea, reiterating Joseph Gwilt’s comments, relate the symbolism of the of the Vesica to the mysteries of early Christianity:

“Icthys, the Fish, is well-known as a frequent symbol of the Christian Faith, occurring in early inscriptions. As a natural type, it has an allegorical meaning, which may be rendered thus. The Fish, though living in the in the salt sea, is not penetrated by saltness. Even so, Jesus, incarnate in a world full of sin, is Himself free from sin, and His followers are also made free, those who are caught up in His net being saved from the salt waters. The Gematria of the Icthys has survived in the Sibylinne Acrostic, Ἰησοῦς Χριστός, Θεοῦ Υἱός, Σωτήρ”, (Iēsous Christos, Theou Huios, Sōtēr), the initials of which spell ΙΗΣΟΥΣ.But in the orthodox Gematria it is now clearly seen to have a prominent place, and its teaching is reinforced in the language of geometry and mathematical symbol. There can be no doubt that it is the same Mystery of the Icthys which is traditional in Sacred Art and Architecture, in the form of the Vesica Piscis. In this geometrical guise it is everywhere met with in ecclesiastical buildings of the mediaeval era, both as latent in the plan and proportions of Christian Temples, and apparent in the ornamental detail, where it is often associated with sculptured figures of Christ or of the Virgin.”10

Reproduced below are two variations on the Icthus symbol.

Writing back in 1912, author J. W. Norwood discusses various symbolic meanings associated with the Icthus symbol and describes its relation to the creation of heaven and Earth.

“This symbol had a mystical meaning to theologians as well as guild masons as it formed the “Womb of the Logos,” in that with two strokes of the compass, a figure appeared making possible the construction of the equilateral triangle of perfection, representing the sacred delta or “Word.”In Milton’s Paradise Lost, this idea is brought out when God is said to have cut out the world with one stroke of the compasses and the heavens with another stroke. After creation only, came the appearance of man, his fall, and subsequently, as in Paradise Regained, his redemption by Him that was called the Logos.”11

You may notice that by a slight manipulation of the arcs forming the Vesica, shifting them along the line of centers, the result is the figure representing Pisces, the twelfth and last sign of the Zodiac. Here we see the two arcs representing heaven and Earth bound together by a horizontal line, or cable, recalling the Egyptian depiction of Pisces in the Denderah zodiac of two fishes bound together by a cord in the shape of a V. A number of variations on this symbol were utilized by guild masons as a mark, or sign, to designate their work, signifying the nature and purpose of their labors as the uniting of heaven and Earth.

So, in the Vesica we have a powerful composite symbol, representing in a general sense a womb, or gateway between orders of Being, in a more specific sense as the womb of the Virgin from which the transcendent Christ emerges into physical form in his role as redeemer of the World. In the depiction of Christ within the Vesica we have the basic idea of duality, with a unity emerging from the integration or marriage of two circles each endowed with contrasting meaning.

We have the symbolism of the Fish and all that it connotes, including that of the Piscean Age as one of the cosmic hours in the immense wheel of time known by the title The Great Year and an implication regarding the existence of an early Christian gnosticism. We have the function of the Vesica as the generatrix of all geometric form, with its attendant symbolism according to the tenets of Sacred Geometry. We have its use by builders throughout the ages as a matrix upon which to develop their plans and elevations for all manner of sacred structures, vessels of transmutation, instruments for the uniting of the heavens and the Earth. And, we have a key to compositional harmony in the ratio of the length to the width of the Vesica.

Finally, as the Vesica is a doorway between worlds, so it represents a doorway between Ages of the World, this being the compelling message of the great medieval Masonic builders, who enshrined the Vesica and its symbolism over the doorways of magnificent Gothic cathedrals, those textbooks in stone of ancient Hermetic wisdom, and, by this means bestowing on future generations a powerful and profound key to unlocking the Mysteries of Time and Space, Matter and Energy, and the ability to read the blueprints of the Great Architect, while revealing to the sincere and inquiring mind the nature and purpose of the Great Work and the means to the fulfillment of Man’s destiny on Earth.

So mote it be.

-Randall

If you are interested in learning the divine science of Sacred Geometry from independent scholar Randall Carlson please visit the following link

Sacred Geometry International online Sacred Geometry Classes

References

1 Achad, Frater (1925) Anatomy of the Body of God: Samuel Weiser, Inc. (1969) p. xii – xiii

2 Achad, Frater (1925) p. 6

3 Fletcher, Rachel (2004) Musings on the Vesica Piscis. Nexus Network Journal, Vol. 6, No. 2 pp.

95 – 98

4 Stirling, Willian (1897) The Canon: An Exposition of the Pagan Mystery Perpetuated in the Cabala as the Rule of all the Arts. Elkins Publishers, reprinted in 1981 by Research Into Lost Knowledge Organisation, dist. By Thorsons Publishers Limited. pp. 11 – 13

5 Stirling (1897) pp. 13 – 14

6Klein, Sydney T. (1910) Magister-Mathesios: Transactions of Quatuor Coronati Lodge, vol. xxiii, pp. 107 – 151

7 Michell, John (1972) City of Revelation, David Mckay Company Inc. Republished in 1973 by

Ballantine Books pp. 75 – 76

8Gwilt, Joseph (1867) The Encyclopedia of Architecture: Historical, Theoretical and Practical. Republished by Crown Publishers, 1982. pp. 968 – 969

9 Clark, John E. (2004) Surrounding the Sacred: Geometry and Design of Early Mound Groups as Meaning and Function. In Signs of Power, ed. by Jon L. Gibson & Phillip J. Carr, University of Alabama Press, pp. 162 – 213

10Bond, Bligh & Lea, Thomas Simcox (1917) Gematria: A Preliminary Investigation of The Cabala: Republished by Research Into Lost Knowledge Organization in 1977, distributed by Thorsons Publishers Limited. p. 53

11 Norwood, J. W. (1912) Fish and Water Symbols, The Open Court, Issue 11, pp. 662 – 672

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Fascinating information. That tree of life diagram is spectacular.

Thank you! Glad you enjoyed the material 🙂

do you think you could tell me what this shape in the middle is if anything please?

I believe it is the Sri Yantra. http://en.wikipedia.org/wiki/Sri_Yantra

When is a book coming out?

Randall is working on it now. In the interim. Please scour through the articles here for a glimpse of things to come. Best wishes – Astromonk

the joining of the 5 and 6 to produce 7, as aliphas levi intended is not as shown, as they are not of equal diameter. to solve the riddle one has to know the goddess of 15.