On the Double-Triangle
By Stan Tenen
Director of Research
Meru Foundation
The forms of sacred geometry do not represent things. They represent processes.
Pictures and models of things, after all, are all frozen. They are not at
all alive. For sacred forms to be sacred, we must see them as snapshots
of the processes of life. A triangle is just a triangle when it sits
alone. When we see it as one snapshot in an animation storyboard
of snapshots that illustrates how a point becomes a line, a triangle, a
tetrahedron, and a hyper-tetrahedron (a 5-cell) - which then becomes a new
point which starts the whole process over, we are looking at a model of
life; we are looking at the sacred.
When we can feel within ourselves the sacred path of a living point (a seed)
that transcends itself by becoming a line (a tree), a surface (a leaf),
a volume (a bud), and a hypersphere (a fruit with new seed inside itself),
these same feelings can enable us to taste the transcendent for ourselves.
Among the many geometric forms that have been traditionally used as snapshots
of the sacred, the triangular forms are very important. This is because
we model wholes as circles, spheres, hyperspheres or cycles. In two dimensions,
it takes three points define a particular circle. These three points take
the form of a triangle. In three dimensions, it takes four points to define
a sphere (a hyper-circle). These same four points also define a tetrahedron.
In 4-D, a hypersphere is specified by a hypertetrahedron (a 5-cell) of points.
The same rule, and the same "triangular" forms, are continued
as we go up in dimension. There is a triangle for every circle and a tetrahedron
for every sphere.
The Meru Foundation LAH Swath illustrates how the sequence of simple geometric
forms, repeated over four dimensions, is part of a Great Flame of Life.
LAH, which means flame in Arabic and Hebrew, is the second letter of the
second Sura of the Quran. It is also understood as the traditional source
of the Arabic Letters.
The LAH Swath takes the form of a special spiral vortex. This is the same
vortex as is illustrated on each face of Meru Foundation's Shushon Flower
(at the lower left in the Lah illustration.) Thus, the LAH Swath is embedded
in the LAH Swath just as new seed is embedded in the fruit that grew on
the tree that sprouted from its seed. This is the process of life: Acorn,
Oak-Tree, Acorn, Oak-Tree, backwards and forwards in time as long as the
Acorn/Oak-Tree life survives.
The LAH Swath illustrates
the description in Genesis I.11:
"Fruit tree yielding fruit whose seed is inside itself."
by showing in sacred geometric metaphor (see the Meru video, Geometric
Metaphors of Life) how a seed point unfolds into a fruit with new seeds.
- A seed-point is thrown out of the Shushon Flower. (This seed is not
actually a point, it is itself a whole because it contains within itself
the potential to grow. That is why we will later consider it to be a tetrahedral
"triple-point.")
- The "zero"-dimensional seed-point grows by extending a line
below (its root) and a line above (its stem)
- The one-dimensional (line-root and) line-stem grows as three branches.
These become two-dimensional triangular-leaf surfaces (I.).
- The triangular-leaf surfaces, combined with the triple stems and the
seed-point, define three-dimensional tetrahedron-volume buds (II.).
Now out of the plane, the sacred, living system of geometry continues to
unfold in three dimensions. It is paralleled by the Platonic solids (and
the cube-octahedron).
- The 8-vertices of the double-tetrahedron also define a cube (III.).
(The original seed-point is actually tetrahedral. It combines with its projection,
1.-4., above, to produce the double-tetrahedron.)
- The 6-faces of the cube also define the vertices of an octahedron
(IV.).
- The octahedron "jitterbugs" (R. Buckminster Fuller's term)
into first an icosahedron (V.), and then
- into a cube-octahedron (VI.). Fuller
calls the cube-octahedron the vector-equilibrium because it holds twelve
radial vectors in equilibrium around its center. This unique twelve around
one pattern is the "Thirteen Petaled Rose" referred to in the
introduction to the classic kabbalistic work, the Sefer Zohar, the Book
of Splendour.
- The cube-octahedron is the simplest 3-dimensional form that can pull
through itself like a torus. When it does, it becomes encased in a dodecahedron,
which splits open and throws out new seed to restart the process.
The dodecahedron, the Greek model of the universe, consists of a swirl of
10-tetrahedra. These are the new seed-points that start the process over
again. (As can be seen on the LAH Swath drawing, the letters of the first
verse of the Hebrew text of Genesis line up on the vortices on each face
to specify the tips of the 10-tetrahedra.)
The specially designed vortex forms on the LAH Swath indicate how the geometric
forms unfurl and connect to each other. (It is not logically consistent
for these to be based on Golden Mean or Fibonacci spirals, for example,
as bastardizers of the Meru Foundation work often claim.) They represent
the path of growth of the process of life - just like the whole LAH Swath
itself. These are the "lilies among thorns" also described in
the introduction to the Sefer Zohar.
Meru Foundation has designed a three dimensional sculpture which we call
"The Light in the Meeting Tent," "Naked Recursion,"
and "First Hand" that consists of our special spiral vortex in
a tetrahedral vessel. This Meru sculpture is also intended to represent
a "lily" in a "thorn." The vortex is the "lily"
and the tetrahedron is the "thorn." (An actual rose thorn is similar
to a tetrahedron.) Shadows (2-dimensional views) of Meru Foundation's specially
designed three dimensional spiral vortex sculpture produce Meru Foundation's
"Flame-Letter"
font(s) of the Hebrew, Greek and Arabic alphabets.
The key to the entire process of the LAH Swath is contained in two related
geometries. The double-triangle and the double-tetrahedron. Together they
symbolize the process of dimensional unfoldment. The double-triangle is
a hyper-triangle; it brings 2-dimensions into 3-dimensions. Likewise the
double-tetrahedron is a hyper-tetrahedron; it brings 3-dimensions into 4-dimensions.
The whole process takes nine "triangular" stages. Another sacred
form that expresses this same process of life is the Sri Yantra. The Sri
Yantra is the creation mandala in the eastern traditions. It consists of
9-interpenetrating triangles, representing the union of the male and female
principles. In the center of the Sri Yantra is a "bindu", toroidal
point. The perimeter of the Sri Yantra is said to lie on a hypersphere.
The Sri Yantra can be thought of as the Adamic grand-mother/father of both
the double-triangle and the double-tetrahedron. The conventionally drawn
Sri Yantra can be understood as an elaboration of the "Star of David"
pattern made by the double-tetrahedron. Using a 4-symmetry (octahedral)
form instead of the usual pentagonal 5-symmetry form to define the frame
of the Sri Yantra, it is possible to transform the normally flattened Sri
Yantra into a full 3-dimensional "meru" (world mountain) configuration
by pulling it up from its center. (This is like turning a doughnut through
its hole.) These is a sculpture of the "Octa-Yantra"
on the Meru Foundation website also.
In Hebrew, the "Star of David", the double-triangle pattern, is
known as the "Mogan David", which is usually translated to mean
"star" or "shield" of David. But we can analyze the
meaning of the Hebrew words to obtain a more precise meaning. The word translated
as "star" or "shield" actually consists of two parts.
"MoG" and "-aN". In Hebrew the root MoG is the root
that means magician, as in Magus/magi, and it indicates "manifestation."
The letter "M" means "source of" (M=Mem="sea/expanse")
and the letter "G" (G=Gimel="camel") means "action"
or "carrying." The final "N" means "to go on forever"
(as in the English, "go on to the Nth degree".)
David, spelled Dalet ("D"), Vov ("F"), Dalet ("D")
in Hebrew, means to repeatedly do (Vov, "F") what happens at a
"delta", the English and Greek meaning of the Hebrew word "dalet."
David represents dispensing and giving out. (A line-like river fans out
into a two-dimensional plane as it dispenses itself into the sea at a delta.)
Thus, in Hebrew, the double-triangle, known as the Mogan David represents
the "dispensation of ongoing manifestation." In other words, the
double-triangle represents the way one dimension pours itself out into the
next dimension just like the way a seed pours itself out until it becomes
a fruit. This is the sacred process and the sacred path. It is an animation
of the "hero's journey" expressed in a sequence of geometric snapshots.
This is the true meaning of the double-triangle.
It models the critical point of life, the process of self-unfurling from
generation to generation, endlessly.
Stan Tenen,
Director of Research,
Meru Foundation
© 1985, 1997 Stan Tenen / Meru Foundation