Modelability is topologically conceptual in generalized
of size and time: ergo, conceptual modelability is metaphysical.
Conceptual formulation is inherently empirical and
as such is always special
case sizing and always discloses all the physical characteristics
of existence in time.
Synergetics is a book about models: humanly conceptual
conceptual models; primitively simple models; rationally
intertransforming models; and the
primitively simple numbers uniquely and holistically
identifying those models and their
intertransformative, generalized and special case, number-value
Model is generalization; form is special case.
The brain in its coordination of the sensing of each
special case experience
apprehends forms. Forms are special case. Models are
generalizations of interrelationships.
Models are inherently systemic. Forms are special case
systems. Mind can conceptualize
models. Brains can apprehend forms.
Forms have size. Models are sizeless, representing
Basic Disequilibrium LCD Triangle
The Basic Disequilibrium 120 LCD Spherical Triangle
of synergetics is
derived from the 15-great-circle, symmetric, three-way
grid of the spherical icosahedron.
It is the lowest common denominator of a sphere's surface,
being precisely 1/120th of that
surface as described by the icosahedron's 15 great circles.
The trigonometric data for the
Basic Disequilibrium LCD Triangle includes the data
for the entire sphere and is the basis
of all geodesic dome calculations.
As seen in Sec.
there are only three basic
structural systems in
Universe: the tetrahedron, octahedron, and icosahedron.
The largest number of equilateral
triangles in a sphere is 20: the spherical icosahedron.
Each of those 20 equiangular
spherical triangles may be subdivided equally into six
right triangles by the perpendicular
bisectors of those equiangular triangles. The utmost
number of geometrically similar
subdivisions is 120 triangles, because further spherical-triangular
subdivisions are no
longer similar. The largest number of similar triangles
in a sphere that spheric unity will
accommodate is 120: 60 positive and 60 negative. Being
spherical, they are positive and
negative, having only common arc edges which, being
curved, cannot hinge with one
another; when their corresponding angle-and-edge patterns
are vertex-mated, one bellies
away from the other: concave or convex. When one is
concave, the other is convex. (See
and drawings section.)
We cannot further subdivide the spherical icosahedron's
triangles into similar, half-size, equiangular triangles,
but we can in the planar icosahedron.
When the sides of the triangle in the planar icosahedron
are bisected, four similar half-size
triangles result, and the process can be continued indefinitely.
But in the spherical
icosahedron, the smaller the triangle, the less the
spherical excess; so the series of triangles
will not be similar. Each corner of the icosahedron's
equiangular triangles is 72 degrees;
whereas the corners of its mid-edge-connecting triangle
are each approximately 63
Geodesic Dome Calculations
When two great-circle geodesic lines cross, they form
two sets of similar
angles, any one of which, paired with the other, will
always add to 180º. (This we also
learned in plane geometry.) When any one great circle
enters into__or exits from__a
spherical triangle, it will form the two sets of similar
angles as it crosses the enclosing
great-circle-edge-lines of that triangle.
As in billiards or in electromagnetics, when a ball
or a photon caroms off a
wall it bounces off at an angle similar to that at which
If a great-circle-describing, inexhaustibly re-energized,
satellite ball that was
sufficiently resilient to remain corporeally integral,
were suddenly to encounter a vertical,
great-circle wall just newly mounted from its parent
planet's sphere, it would bounce
inwardly off that wall at the same angle that it would
have traversed the same great-circle
line had the wall not been there. And had two other
great-circle walls forming a right
spherical triangle with the first wall been erected
just as the resilient ball satellite was
hitting the first great-circle wall, then the satellite
ball would be trapped inside the
spherical-triangle-walled enclosure, and it would bounce
angularly off the successively
encountered walls in the similar-triangle manner unless
it became aimed either at a corner
vertex of the triangular wall trap, or exactly perpendicularly
to the wall, in either of which
cases it would be able to escape into the next spherical
area Lying 180º ahead outside the
first triangle's walls.
If, before the satellite bouncingly earned either
a vertexial or perpendicular
exit from the first-described spherical triangle (which
happened to be dimensioned as one
of the 120 LCD right triangles of the spherical icosahedron)
great-circle walls representing
the icosahedron's 15 complete great circles, were erect__thus
constructing a uniform,
spherical, wall patterning of 120 (60 positive, 60 negative)
similar spherical, right
triangles__we would find the satellite sphere bouncing
around within one such spherical
triangle at exactly the same interior or exiting angles
as those at which it would have
crossed, entered into, and exited, each of those great-circle
boundaries of those 120
triangles had the wall not been so suddenly erected.
For this reason the great-circle interior mapping
of the symmetrically
superimposed other sets of 10 and 6 great circles, each
of which__together with the 15
original great circles of the icosahedron__produces the
31 great circles of the spherical
icosahedron's total number of symmetrical spinnabilities
in respect to its 30 mid-edge, 20
mid-face, and 12 vertexial poles of half-as-many-each
axes of spin.
symmetrically superimposed, 10- and 6-great-circles
subdivide each of the disequilibrious
120 LCD triangles into four lesser right spherical triangles.
The exact trigonometric
patterning of any other great circles orbiting the 120-LCD-triangled
sphere may thus be
exactly plotted within any one of these triangles.
It was for this reason, plus the discovery of the
fact that the
icosahedron__among all the three-and-only prime structural
systems of Universe (see Sec.
__required the least energetic, vectorial, structural
investment per volume of
enclosed local Universe, that led to the development
of the Basic Disequilibrium 120 LCD
Spherical Triangle and its multifrequenced triangular
subdivisioning as the basis for
calculating all highfrequency, triangulated, spherical
structures and structural subportions
of spheres; for within only one disequilibrious LCD
triangle were to be found all the
spherical chord-factor constants for any desired radius
of omnisubtriangulated spherical
In the same way it was discovered that local, chord-compression
be islanded from one another, and could be only tensionally
connected to produce stable and predictably efficient
enclosures for any local energetic
environment valving uses whatsoever by virtue of the
approximately unlimited range of
frequency-and-angle, subtriangle-structuring modulatability.
Because the 120 basic disequilibrious LCD triangles
of the icosahedron have
2 l/2 times less spherical excess than do the 48 basic
equilibrious LCD triangles of the
vector equilibrium, and because all physical realizations
are always disequilibrious, the
Basic Disequilibrium 120 LCD Spherical Triangles become
most realizably basic of all
general systems' mathematical control matrixes.
Omnirational Control Matrix: Commensurability of
Vector Equilibrium and Icosahedron The great-circle
subdivisioning of the 48 basic equilibrious LCD triangles of the
vector equilibrium may be representationally drawn within the
120 basic disequilibrious LCD triangles of the icosahedron,
thus defining all the aberrations__and
their magnitudes__existing between the
equilibrious and disequilibrious states, and providing
an omnirational control matrix for all topological,
trigonometric, physical, and chemical accounting.
Properties of Basic Triangle
Subdivision of Equilateral Triangle: Both the spherical
equilateral triangles may be subdivided into six equal
and congruent parts by describing
perpendiculars from each vertex of the opposite face.
This is demonstrated in Fig.
where one of the six equal triangles is labeled to correspond
with the Basic Triangle in the
Positive and Negative Alternation: The six equal subdivision
the planar equilateral triangle are hingeable on all
of their adjacent lines and foldable into
congruent overlays. Although they are all the same,
their dispositions alternate in a
positive and negative manner, either clockwise or counterclockwise.
Spherical Right Triangles: The edges of all spherical
triangles are arcs of
great circles of a sphere, and those arc edges are measured
in terms of their central angles
(i.e., from the center of the sphere). But plane surface
triangles have no inherent central
angles, and their edges are measured in relative lengths
of one of themselves or in special-
case linear increments. Spherical triangles have three
surface (corner) angles and three
central (edge) angles. The basic data for the central
angles provided below are accurate to
1/1,000 of a second of arc. On Earth
These calculations are therefore accurate to one inch
of Earth's arc.
|1 nautical mile ||= 1 minute of arc
|1 nautical mile ||= approximately 6,000 feet
|1 second of arc ||= approximately 100 feet
|1/l,000 second of arc ||= approximately 1/10 foot
|1/1,000 second of arc ||= approximately 1 inch
The arc edges of the Basic Disequilibrium 120 LCD
Triangle as measured by
their central angles add up to 90° as do also three internal
surface angles of the triangle's
|BCE =|| 20° 54' 18.57"|| = ECF
|ECD =|| 37° 22' 38.53"|| = DCE
|DCA =|| 31° 43' 02.9"|
| || 90° 00' 00"
The spherical surface angle BCE is exactly equal to
two of the arc edges of
the Basic Disequilibrium 120 LCD Triangle measured by
their central angle. BCE = arc
AC = arc CF = 20° 54' 18.57".
Surface Angles and Central Angles: The Basic Triangle
ACB can be
folded on the lines CD and CE and EF. We may then bring
AC to coincide with CF and
fold BEF down to close the tetrahedron, with B congruent
with D because the arc DE =
arc EB and arc BF = arc AD. Then the tetrahedron's corner
C will fit exactly down into
the central angles AOC, COB, and AOB. (See Illus.
As you go from one sphere-foldable great-circle set
to another in the
hierarchy of spinnable symmetries (the 3-, 4-, 6-, 12-sets
of the vector equilibrium's 25-
great-circle group and the 6-, 10-, 15-sets of the icosahedron's
31-great-circle group), the
central angles of one often become the surface angles
of the next-higher-numbered, more
complex, great-circle set while simultaneously some
(but not all) of the surface angles
become the respective next sphere's central angles.
A triangle on the surface of the
icosahedron folds itself up, becomes a tetrahedron,
and plunges deeply down into the
congruent central angles' void of the icosahedron (see
There is only one noncongruence- the last would-be
hinge, EF is an external
arc and cannot fold as a straight line; and the spherical
surface angle EBF is 36 degrees
whereas a planar 30 degrees is called for if the surface
is cast off or the arc subsides
chordally to fit the 90-60-30 right plane triangle.
The 6 degrees of spherical excess is a beautiful whole,
excess. The 90-degree and 60-degree corners seem to
force all the excess into one corner,
which is not the way spherical triangles subside. All
the angles lose excess in proportion to
their interfunctional values. This particular condition
means that the 90 degrees would
shrink and the 60 degrees would shrink. I converted
all the three corners into seconds and
began a proportional decrease study, and it was there
that I began to encounter a ratio that
seemed rational and had the number 31 in one corner.
This seemed valid as all the
conditions were adding up to 180 degrees or 90 degrees
as rational wholes even in both
spherical and planar conditions despite certain complementary
transformations. This led to
the intuitive identification of the Basic Disequilibrium
120 LCD Triangle's foldability (and
its fall-in-ability into its own tetra-void) with the
A Quanta Module, as discussed in Sec.
Copyright © 1997 Estate of R. Buckminster Fuller