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1
It was shown in Article 20 that the I Ching table (Fig. 1) is a symbolic expression of the 3×3×3 array of cubes shown in Fig. 2a. Each of the upper trigrams
of the eight hexagrams along the diagonal of the table symbolises the three cube faces whose intersection defines a corner of a cube. The pairing of each trigram with a different one symbolises an association between the three faces defining a corner of the central cube and the three faces of one of the cubes surrounding it that is contiguous with this corner. Each corner of the cube at the centre of the 3×3×3 array of cubes (the grey one in Fig. 2a) is the point of intersection of the corners of seven other cubes. Three of
2 these (the violet, indigo and blue cubes in Fig. 2b) are at the same height as the cube and the remaining four (the green, yellow, orange and red ones) are above or below it. Let us express the pattern of eight cubes centred on any corner of the central cube as
where ‘1’ always denotes the
central cube, ‘3’ denotes the three cubes contiguous with it at the same height and ‘4’ denotes
the four contiguous cubes
above or below it. As we jump from corner
to corner in the top face of the central cube, the L-shaped pattern of three coloured cubes
in the same plane rotates (Fig. 3), as do the four cubes above them. Likewise, as we go from corner to
corner in the bottom face of the central cube, the three cubes in the same plane change, as
do the four cubes below them. Movement through all eight corners of the central cube
involves every one of the
3×3×3 array because they are all contiguous with its corners. Each corner of the central cube is the point of intersection of the three orthogonal faces (3) of each of the seven cubes that are contiguous with it (Fig. 4). The number of such faces generating the corners of the central cube is
Counting these corners in the same way as the pattern of cubes, that is, differentiating any corner from the three corners at the same height and the four corners below them, then
where
The three orthogonal faces consist of the face perpendicular to one coordinate axis (for convenience, let us choose the X-axis) and the two faces perpendicular to the two other axes. So
Therefore, 24 = [(1+3) + 4]×(1 3
where 3
The
where
and
Hence,
where
and
We see that the
4 Which set is associated with which group of corners is determined by whether the corner ‘1’ is regarded as an upper or a lower corner of the central cube. If the former, the 84 faces are associated with the lower corners. If the latter, the 84 faces are associated with the upper corners. For the present
Figure 6. 84 yods surround the centre of a 2nd-order tetractys.
purpose, no choice needs to be made for ‘1’.
The pair of 84 cube faces intersecting at the corners of the two opposite faces of the central cube is encoded in the inner form of the Tree of Life (Fig. 5). The first six regular polygons enfolded on each side of the central pillar of the Tree of Life can be divided into tetractyses of ten yods. There are 84 yods on the sides of each set of polygons. Each yod denotes a face, whilst the pair of identical sets of polygons corresponds in the cube to the two sets of four corners that form either a square face or a tetrahedron.
The Pythagorean significance of the number 84 is that the next higher order tetractys — the 2nd-order tetractys (Fig. 6) — is made up of 85 yods, that is, there are 84 yods surrounding its central yod. This means that 84 hidden degrees of freedom (here ambient cube faces) express the concrete form of a holistic system (here, the four corners of a face of the central cube), as symbolised by the yod at the centre of the tetractys, which corresponds to Malkuth of the Tree of Life.
Figure 7. The Klein Configuration of the
The Tetrad Principle
that is, 84 is the sum of the squares of
the first
Felix Klein (1849–1925) is well-known to mathematicians as one of the nineteenth
5 century’s great geometers. The one-sided, closed surface called the “Klein bottle” is named after him. In 1878, he discovered that the curve
has the 336-fold
symmetry of the group SL(2,7). This curve is known to mathematicians as the “Klein quartic.”
Klein showed that its Riemann surface is mapped onto itself (hence ‘automorphisms’)
by
The Riemann surface of
the Klein quartic can be represented by the so-called “Klein Configuration” (Fig. 7). It has
This composition can be represented as:
where
The numbers within each bracket denote the triangles in
the two slices of each of the seven sectors of the Klein Configuration. Each sector is made up
of two
where
and
Seven slices of one type have 84 triangles made up of 28
red, 28 yellow and 28 green triangles. The seven slices of the other type have 84 triangles
made up of Let us compare the 7-fold pattern of 24 coloured
triangles indicated by Equation 15, each denoting one of the 6 7 (according to Equation 6) of the seven sets of 24 cube faces: 24 = (1 Compare also the 84 triangles within the seven slices of each type (let us call them ‘A’ and ‘B’) given by Equations 16 and 17 with the two sets of 84 faces given by Equations 10 and 11. The following correspondences exist:
We see that the two
types of slices in the Klein Configuration correspond to the two opposite faces of a cube.
The four corners of a face that include ‘1’ are contiguous with (4×3×7 = 84) faces of
surrounding cubes in the 3×3×3 array and correspond to slice B. The four corners of the
opposite face are contiguous with (4×3×7 = 84) faces of surrounding cubes and correspond to
slice A. The 4:3 splitting of the triangles in the heptagons mirrors the fact that there are
four contiguous cubes above or below any given corner of the central cube and three
contiguous cubes at the same height. As a cube face is either positive or negative depending
on whether it faces the positive or negative direction of a coordinate axis, there is for
each of the
one of opposite polarity.
This means that there are
Fig. 8 displays the isomorphism between the 24 = 4
8
This shows that the
Configuration are isomorphic to one another. It was argued in Article 20 that the degree of correspondence between the first two is too high to be dismissed as coincidence. The same conclusion applies to the lattermost two. The 1:3:4 pattern of cubes associated with the central cube corresponds in each sector of the Klein Configuration to the cyan hyperbolic triangle of the
9 central heptagon, the three red, yellow or green triangles in one half of a sector and the four triangles with these colours in the other sector. The existence of this pattern in the archetypal pattern of 27 cubes symbolised by powers of 3 in Plato’s Lambda reflects the pattern of Sephiroth of Construction in the Tree of Life (Fig. 9). ‘1’ represents the starting point (any corner of a cube) and corresponds to Daath (Knowledge), from which objective manifestation emerges. ‘3’ represents the three other corners of a square — the halfway stage in the creation of a cube. They correspond to the triad of Chesed, Geburah and Tiphareth. ‘4’ denotes the remaining four corners of the cube. They correspond to the four-fold emanation of Netzach, Hod, Yesod and Malkuth, which signifies the material aspect of the Tree of Life, the element ‘Earth,’ whose particles the ancient Greeks believed has the form of a cube.
The Klein Configuration can be embedded in a 3-torus because the latter is a compact manifold of genus 3. The 3-torus is topologically equivalent to a cube whose opposite faces are imagined to be glued together (Fig. 10), so that movement along each coordinate axis is periodic, like motion around a circle. This means that all eight corners of a cube represent the same point of a space that has the topology of a 3-torus. However, suppose that they were tagged or labelled in some way that differentiated them. As the cube has three 4-fold axes of symmetry, namely, rotations of 90°, 180° and 270° about the X-, Y- and Z-axes, there are nine possible configurations of labelled corners into which the original one can be rotated, that is, there are ten different configurations of corners connected by these rotations about the coordinate axes. They form the tetractys pattern shown in Fig. 11. The cubes at the corners of the tetractys are the central cube after rotation through 90°, 180° and 270° about the X-axis. The two triads of cubes that are connected by dotted lines are the result of similar rotations about the Y- and Z-axes. Figure 12. Equivalence of the Tree of Life and the tetractys. As geometrical representations of the ten-fold nature of holistic systems, the Tree of Life and tetractys are equivalent (Fig. 12). The Supernal Triad of Kether, Chokmah and Binah corresponds to the three points at the corners of the tetractys. The next two triads of Chesed-Geburah-Tiphareth and Netzach-Hod-Yesod correspond to the two triangular arrays of points whose intersection forms a Star of David. Malkuth at the base of the Tree of Life corresponds to the centre of the tetractys. The ten rotational configurations of a cube constitute a Tree of Life pattern because the set of three rotations through 90°, 180° and 270° about an axis perpendicular to a face of the cube (say, the X-axis) corresponds to the Supernal Triad, whilst the pair of three-fold rotations about the Y- and Z-axes corresponds to the pair of triads of Sephiroth of Construction above Malkuth, this being the last Sephirah of Construction that corresponds to the non-rotated cube.
The 3×3×3 array of cubes
has (4×4×4= 10 number of faces defining
the corners within the six faces of the array = 6×4×4×3 = 288 = 1
faces of cubes. Three
faces define each corner of the array itself. The number of cube faces defining corners
along the sides of the array = 3[12×(2+2) + 8] = 3×56 =
648. They consist of
(648/3= 11 4! of these faces belong
to the cube itself and 7×4! belong to the
These numbers are also the numbers of combinations of,
respectively, one, two, three and four objects, so that A 168 because it is the sum of the 12 odd
integers after 1 that can be arranged
along the sides of a square (symbol of the number 4),
Articles 17 and 18 showed that the average distances (in 1/10 AU) of the planets Venus-Uranus from the asymptotic centre of the logarithmic spiral-shaped planetary nebula that spawned them are the Bode numbers: 12 13 The average distance of
the Asteroids from the asymptotic centre is 24. Uranus, the last member of the first octet
of planets, has an average distance of
This is confirmed by the
fact that the
As discussed in Article
18,
The cube has 13 axes of symmetry that are of three types:
1. three 4-fold axes
(C 2. four 3-fold axes
(C 3. six 2-fold axes (C2) of rotation of 180° (therefore, 6 rotations): 14 4. the identity (one rotation of 360°).
The
octahedral group O
It was
pointed out in Article 11 Figure 17. Plato’s Lambda and its tetrahedral generalisation. and
that this tetractys is but one triangular face of a tetrahedral array of 20 integers
(Fig. 17) has been demonstrated
15 As this article has
shown that the 3×3×3 array is isomorphic to the Klein Configuration, we see a remarkable
continuity between the mathematical algorithm that Plato said God used to design the
universe and the Klein Configuration. This represents the
Fig. 19 shows the way in which the helical form of each whorl of the
UPA/superstring is represented by the I Ching table and by the Klein Configuration. It
demonstrates that the hyperbolic triangles of the Klein Configuration and the yin/yang lines
of the I Ching table signify fundamental, physical phenomena manifesting in the subatomic
world as circularly polarised oscillations or waves that propagate around the string-like
whorls of
16 the superstring.
The
The distinction between
the 32 exterior corners of the array and the 32 interior ones is mirrored in the inner Tree
of Life as the two sets of seven enfolded polygons, each set having 32 corners other than
those coinciding with Sephiroth on the side pillars (Fig. 20). The number Table 1. Number values of the Sephiroth in the four Worlds.
The first hexagram,
Ch’ien (The Creative, Heaven), corresponds to the corner located at Daath and the last
hexagram, K’un (The Receptive, Earth), corresponds to the corner located at Tiphareth. As
the largest of the integers in the generalised Lambda (see Fig. 17),
These examples of the appearance of the gematria number value of a particular Kabbalistic Mundane Chakra in widely diverse and seemingly, unconnected contexts demonstrate that a hidden harmony made visible through number pervades phenomena, whatever their space-time scale. * 1 AU (Astronomical Unit) is the average distance of the Earth from the Sun (approximately, 93 million miles), this being defined as the arithmetic mean of its least and furthest distances. 17 This unity, which could
have no rational cause if it were the miraculous product of coincidence, exists because
these phenomena are all manifestations of the mathematical archetypes of the Divine Mind,
which — whether expressed in mystical traditions or in pure mathematics — possess isomorphic
features because they embody the same idea. -
Numbers in **boldface**are the number values of the Hebrew names of the ten Sephiroth of the Tree of Life, their Godnames, Archangels, Orders of Angels and Mundane Chakras. They are listed above in Table 1. They are calculated by means of the ancient practice of gematria, wherein a number is assigned to each letter of the alphabet, thereby giving a number value to a word that is the sum of the numbers associated with its letters. -
Phillips, Stephen. Article 1: “The Pythagorean Basis of Superstring and Bosonic String Theories,” (WEB, PDF). -
Hurwitz, A. “Über algebraische Gebilde mit eindeutigen Transformationen in sich,” Math. Ann. **41**(1893), 403–442. -
Murray Macbeath, A. “On a theorem of Hurwitz.” Proc. Glasgow Math. Assoc. **5**(1961), 90–96. -
Murray Macbeath, A. “Hurwitz groups and surfaces.” *The Eightfold Way*. MSRI Publications, vol. 35, 1998, 11. -
Phillips, Stephen. Article 15: “The Mathematical Connection between Superstrings and their Micro-psi Description: a Pointer towards M-theory,” (WEB, PDF). -
Phillips, Stephen. Article 18: “Encoding of Planetary Distances and Superstring Structural Parameters in the I Ching Table,” (WEB, PDF), pp. 15–16. -
Phillips, Stephen. Article 11: “Plato’s Lambda — Its Meaning, Generalisation and Connection to the Tree of Life,” (WEB, PDF). -
Phillips, Stephen. Article 12: “New Pythagorean Aspects of Music and the Connection to Superstrings,” (WEB, PDF). -
Phillips, Stephen. Article 20: “Algebraic, Arithmetic and Geometric Interpretations of the I Ching Table,” (WEB, PDF), pp. 19–22. -
Phillips, Stephen. Article 17: “The Logarithmic Spiral Basis of the Titius-Bode Law,” (WEB, PDF).
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