#22 Hexagram and 5-fold representations of the superstring structural parameters 2016 and 1680

In my book "Foundations of Sacred Circular Geometry" (to be published in 2026), I derived the mathematical condition for isomorphism to exist between well-known examples of sacred circular geometry, e.g., the Seed of Life and the Flower of Life, and examples of sacred geometries that are composed of straight lines, in particular, the outer and inner forms of the Tree of Life. The way in which information about the natural order — both physical and superphysical — is encoded in the latter kind of sacred geometry is through conversion of its polygons into Type A, Type B, Type C, etc. This turns a sacred-geometrical object into an integer that is the sum of the numbers of various types of yods (corner yods and hexagonal yods) needed to construct the object. It has not been obvious how sacred circular geometries can represent archetypal numbers (I.e., numbers that parameterize holistic systems such as superstrings) and so be more than just geometric symbols of metaphysical concepts, which is how they have been regarded for centuries. I have now solved this problem, enabling these geometries to be raised out of the realm of religious speculation and to be analysed in a mathematically rigorous way that reveals their true, sacred character.

The most natural way whereby a circle can embody a number is through the length of its circumference. The condition for sacred geometries composed of circles to be equivalent to sacred geometries composed of straight lines is:

all circles present in the sacred circular geometry must have diameters D that are integer multiples of 168: D = 168n (n = 1, 2, 3, etc.), where 168 is the number value of Cholem Yesodoth, the Mundane Chakra of Malkuth, which is the tenth Sephirah in the Tree of Life, signifying its outer, material form.

This means that the smallest circle must have a diameter of 168 units (what unit is chosen is, of course, immaterial — they could be Angstroms, centimetres, miles or light years). A basic circle of circumference 22/7×168= 528 is two joined semicircles, each of length 264 when pi is approximated by 22/7. This compares with the 264 yods in each mirror-image half of the inner Tree of Life. In other words, every unit of length in all the arcs making up basic circles in sacred circular geometries corresponds to a single yod in the inner Tree of Life constructed from tetractyses. The fact that the lengths of 528 and 264 are not exact because only an approximate value of pi is used to calculate them means that numerical information is embodied not — as one might have thought — in the exact lengths of circles and their arcs but in their values based upon taking pi as 22/7. Any basic circle in sacred circular geometry has to be thought of as being divided into 528 similar arcs, each corresponding to one of the 528 yods in the two separate halves of the inner Tree of Life, each half corresponding to a semicircle. It is not the exact size of their lengths in terms of the diameter of the circle that count, only that implied by approximating pi as 22/7. This is only one of its many approximations known to mathematicians. Archimedes proved that it is the upper limit to the exact value of pi. Sacred circular geometry assigns a similar, unique status to this approximation.

Because of the hexagonal symmetry of their overlapping circles, sacred circular geometries such as the Flower of Life are composed of circles that are each divided into six equal arcs of length 528/6 = 88. These basic arcs are the building blocks of this type of sacred geometry. The Vesica Piscis formed from two basic circles overlapping centre-to-circumference is composed of four such arcs and therefore has a boundary 4×88=352 units long. A hexagon circumscribed by a basic circle has six corners that mark out the six arcs. Each of its sides is 84 units long. The rhombus circumscribed by a Vesica Piscis has four equal sides of total length 4×84 = 336. When the smallest circles present in sacred circular geometries have a diameter of 168 units, their overlap generates various Vesica Piscis that individually embody through their circumscribed rhombi the fundamental superstring structural parameter 336 that has been discussed in previous pages, as well as in numerous sections of this website! This is both remarkable and highly significant, because it indicates that these geometries can measure the basic oscillatory form of the whorls in both the UPA described by Annie Besant & C.W. Leadbeater and its dark matter counterpart observed by Ron Cowen (see picture above). One is realised on paper, the other in space-time. But both express the same signature of God.

The hexagram inside the Seed of Life comprises 12 equilateral triangles with 6×4=24 sides of total length 24×84 = 2016. The length 336 of the boundary of each of its six rhombi corresponds to the 336 circular turns that I predict make up each of the six revolutions of each whorl of the particle described by Cowen. The Seed of Life embodies the very number of circularly polarised oscillations that I predict in each whorl of the basic unit of dark matter. A pentagram array of five Vesica Piscis generates the number 5×336 = 1680 as the total length of the boundaries of their rhombi. This is the number of turns that Leadbeater counted in the five revolutions of each whorl of the UPA when he remote-viewed the sunatomic world. The fundamental, physical meaning of the Vesica Piscis is that it embodies the structural parameter 336 of the subquark state of the E₈×E₈′ heterotic superstring and its equally basic E₈-singlet state. The Vesica Piscis is like an egg whose embryo is the straight-sided rhombus —the starting point for the development of all sacred circular geometries and whose manifestation in 4-d space-time is the cyclic sequence of 336 circularly polarised oscillations made in every revolution of a whorl of the E₈×E₈ʹ superstring around its axis of spin. That is the scientific meaning of the Vesica Piscis.

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