U.S. O.T.O. Grand Lodge
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The Scarlet Letter
Volume VII, Number 1 | March 2002
Esoteric Geometry
By Fr. Dionysos Thriambos
with diagrams by Fr. Phillip X Dick

Perhaps the most prominent figure in esoteric geometry is the vesica pescis. The name means “fish’s bladder,” and alludes remotely to the ability of the figure to represent the vagina—”fish” having been a common circumlocution for female genitals in the ancient Near East. The most venerable and precise description of the vesica is given in Euclid’s Elements I:1. The figure is formed by the intersection of two circles so that the periphery of each touches the center of the other. The vesica is the two-pointed shape formed by the area common to the two circles. (Figure 1)

Figure 1The entire figure with the two circles has been considered an important symbolic representation of the principle of the Trinity by Christian esotericists. Each circle intercepts exactly one third of the circumference of the other, so that the sum of the two edges of the vesica is equal to the circumference of each circle remaining. Furthermore, the rectangle defined by the major and minor axes of the vesica is capable of being divided into three rectangles, each of which is proportionally identical to the original rectangle, and capable of being similarly subdivided. (Figure 2) This process can be carried on indefinitely, and the property is unique to this particular rectangle.

Figure 2Of course, neither the notion of the Trinity nor the symbol of the fish is exclusively of interest to Christians. But Christianity is tied especially to the precession of the equinoxes through the sign of Pisces, the Fish. And the vesica has been used as a Christian insignia and the basic form for a wide variety of seals and ornaments since apostolic times. It has been particularly used as a symbol of the Virgin and in reference to the wound in the side of the crucified Savior. Elaborated with an additional crossing of the lines at one end, the vesica is used to directly depict a fish, and by implication indicates the Greek formula ICQUS (“fish”). Given the synthetic character of early Christianity, that formula may have had a previous significance in one of the many mystery cults popular at that time. But the official Christian notarikon of the word became the Greek phrase for “Jesus (I) Christ (C) God’s (Q) Son (U), Savior (S).” The modern vulgarization of this elegant formula is Figure 3the word “JESUS” in a cross-tailed horizontal vesica plastered to the rear end of an automobile. (A more lucid updating might instead display the English word “FISH” as a notarikon for “Fashionable Idiots Still Hope,” in reference to the doctrinal perspective of the sort of people who put those things on their cars. (Figure 3)

Figure 4The form of the vesica pescis permits the perfect inscription of two equilateral triangles in the relationship that is designated as the Hexagram of Air in the practice of ceremonial magick. (Figure 4) The two triangles of the hexagram are generally attributed to the human and the divine, and their conjunction in the hexagram is representative of the Great Work.

Figure 5The vesica can in turn be perfectly inscribed within an elongated hexagram which is a flattened image of the double cube, the typical shape of the altar used in ceremonial magick. (Figure 5) The double cube, formed by stacking one cube on top of another, represents the universal decad because its outer surface can be divided into ten equal squares, with Kether on top and Malkuth on the bottom.

Furthermore, overlapping the two figures just described will result in the overall geometrical basis of the qabalistic Tree of Life, perfectly proportioned, with only a few paths out of place. (Figure 6)

Figure 6

This demonstration is only one of many ways in which the vesica might fulfill the reference to it in Liber VII (VI:2) that “We made us a temple of stones in the shape of the Universe, even as thou didst wear openly and I concealed.”

Figure 7The Tree of Life glitters with examples of another key figure of esoteric geometry: the right triangle. This triangle is first and foremost symbolic of knowledge. It will be seen that two right triangles are required to cover the area of a single equilateral triangle. (Figure 7) Where the equilateral triangle may represent the complete human or deity, the half covered by a right triangle is that which is known about the subject, and the half uncovered is that which is unknown.

For the entirety of the Osirian Aeon, the study of right triangles, or Trigonometry, was the basis of virtually all of the quantitative sciences. As Thomas Paine explained in his Age of Reason,

Trigonometry […] when applied to the study of the heavenly bodies, is called Astronomy; when applied to direct the course of a ship on the ocean, it is called Navigation; when applied to the measurement of any portion of the surface of the earth, it is called Land-surveying.

This general potency of Trigonometry derives from the theorem attributed to Pythagoras, the founder of the ancient mystery school that bore his name. Euclid included the theorem as the forty-seventh proposition in Book I of his Elements.

Trigonometry rests on the application of the Pythagorean theorem to triangles inscribed in a unit circle, i.e. a circle with a radius measured as “one.” Typically, this circle is shown with rectangular axes positioned on its center, thus presenting the mystical emblem of the Rosy Cross. The hypotenuse of the right triangle, i.e. the side opposite the right angle, is given a constant measure of one, permitting the proportional application of the Pythagorean theorem to the measurements of the angles as well as that of the sides of the triangle.

The Pythagorean theorem states that the sum of the squares of the legs of a right triangle are equal to the to the square of the hypotenuse. Thus the length of one side can always be determined if the other two are known. Trigonometry expands the principle to apply it to measures of the angles (which must vary in proportion to the sides that they oppose), and to combinations of sides and angles.

In keeping with the Pythagorean theorem, the right triangle also represents the principle of generating a resultant (the hypotenuse) from complementary terms (the legs). It is a geometric expression of the Hegelian dialectic of thesis, antithesis, and synthesis. Thus esotericists have repeatedly attributed the legs of the right triangle to Isis (base) and Osiris (perpendicular), and the hypotenuse to Horus. The right triangle has also been used as an illustration of how matter and spirit (the legs) combine in human life (the hypotenuse).

Figure 8To return to the Tree of Life, note that there are only two right triangles that can be drawn for which each side is a whole path, and one path only. Other attempts require that a side consist of a partial path or of two paths run together. Both of these special triangles share a hypotenuse in the path of gimel, or the High Priestess running between Tiphareth and Kether. (Figure 8)

On the side of Mercy, with its right angle stationed in Wisdom, the triangle has legs of heh (the Emperor) and aleph (the Fool). The letters gimel, aleph and heh spell the Hebrew word for “became powerful, grew high.”

On the side of Severity, with its right angle stationed in Understanding, the triangle has legs of zain (the Lovers) and beth (the Magician). The letters gimel, zain and beth spell a Hebrew word for treasure or wealth.

The conic sections of ellipse, parabola, and hyperbola were first related to the Tree of Life in Appendix B of Crowley’s Book of Thoth. He attributes them to the “three Veils of the Negative,” but the “Diagram I” which purports to give the specifics appears to be incorrect. In light of the explanation of the Naples Arrangement given on page 32 of the same text, the ellipse should be Ain Sof Aur—as the Veil most proximate to Kether. Thus Ain Sof is still represented as the parabola; and the hyperbola becomes the symbol of Ain, the absolute qabalistic zero.

Certainly the hyperbola seems at first to be a more fit glyph of “absence of extension in any of the categories.” The little parabola that we use to represent zero is after all a boundary distinguishing inside and outside; thus it is a one, defining two, and becoming a third, far from zero. But the hyperbola itself is a distinction, though not a closed boundary. Like the parabola, the curve of the hyperbola is unlimited in extent, and distinguishes a space inside the curve, from one outside it.

A more surprising feature of the hyperbola does not appear in Figure I of the Book of Thoth appendix. In analytical geometry, hyperbolic functions are seen to describe two disjunct curves, similar in form and opposite in direction. Thus the “inside” of the curves is the “outside” of the space between the two—highly suggestive of the 0 = 2 formula.

The same progression of conic sections can be used to illustrate the transition from the Aeon of Osiris to that of Horus. The ellipse is the Egg of Blue in which is the babe Hoor-par-kraat, whose minister Aiwass delivered the Law. The parabola is the arch of the heavens, Nuit, across whose body the Equinox precesses, entering the sign of Aquarius and leaving the sign of Pisces. The hyperbola is itself the two curves of the glyph of the astrological sign of Pisces, the fish which became associated with Jesus/Osiris partly through the auspicious symbol of the vesica pescis.

With the return of the vesica, this essay comes full circle, and the contented Geometer sets down his compass.

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