Frame 3 - Advanced
Geometry
* The Hex-Machine *
The
Frame offers exquisite geometry, as well. You are about to see a self producing design made of three
hexagonal stars. The Tri-balance (CDEF) connected to the
segment of 16 at L reappears as a unit again. This is very
consistent, because the same group absolutely stole the show in our
previous analysis of the Frame for Pi.

a) Rounded to the
nearest degree, The angle F-E-D measures 120 degrees.
b) The line E-L then divides
F-E-D into two 60 degree angles.
c) E-L
passes through the center of the Main Square with faultless
accuracy(diag).
d) The Main
Square's center (the 0,0 point) connects to 'E'
and 'C' by lines in a 60 degree angle. These two lines create an
equilateral triangle
with the line passing through 'E' and 'D', as below.

The Big Hexagon
To recapitulate, we were given some lines, and
an equilateral triangle, which could be part of a regular
hexagon. To fill in the rest of the hexagon next
is
absolutely natural. The result, or
better said -
the resulting harmony - is below - for your admiration. Our
operation was justified, because of the great fit between the big
hexagon, and
the picture. For instance, the big hexagon's circle surrounds, and
frames the image
very well. The same can be said for the envelope around the human
figure.
This
Big Hexagon was the first hexagon I found, because it is so obvious.
However, its origin remains unclear until we understand two other 60
degree angles in this position, and the unobvious ways, in which they
work together.
Several experiments have to be carried out. The basic lineup of the 60
degree angles on the Frame is as below.

Each line from L to one of the
four
points C,D,E,F, serves as an arm of at least one
60-degree
angle (rounded out to the nearest degree). But, there are some
visually exact angles in this position, too.

Let's base on the line C-G. Now, if we rotate C-G 60-degrees clock-wise
around 'C' - it turns out to be a supported line, judging by its
passage through the engraving. Next, we complete the angle GCL into an
equilateral triangle so that the third side rests on the point 'L'
(diag. above). This side, too, looks supported by the image. Moreover,
this triangle's circumcircle touches the point 'F'.

<>
All of a sudden, the position turns into an
illustration of a geometrical theorem:
Any
point on the
circumcircle
of an equilateral triangle connects to that
triangle's
corners by lines holding 60 degrees, when above the basis, and
120
degrees, when below the basis.
The
point F
happens to
be just such a point.
Logically, F was created
after
L-C-G,
By
extrapolation, the triangle based on
the F-C-L-angle is second generation.

We see a
number
of perfect mutual intersections (circled), when finishing the triangles
into full stars.

Special Case
Here is the design and the perfect points without interference of the
engraving's
background.

A critic will comment:
" So
what, those intersections are
natural to
the position".
But, in
control experiments changing position of "F" on the circumcircle, great many of
these perfect points go missing! This
mandates that the diagram above illustrates
a special case

which
occurs only when the line F-L ( or
its mirror image )
passes through the circled point, as in the diagram above. (1/4 of the
circumscribing
circle's diagonal, 1/3 of the triangle's height).
The
Hex-Machine - the third generation
<>
The Big Hexagon is third-generation!
Comparing the first hexagon found on the Frame - the Big
Hexagon - to the other two hexagons shows Frame points, at which they
coincide. Could it be that like the second hexagon is a special case of
the first hexagon, the Big Hexagon is a special case based on both?
In the diagram
above, we see how the Big Hexagon
fits together with
the orange star - the Hex Machine's first generation.<>The
diagram below shows the Big Hexagon and the Hex
Machine's second
generation, and their points of exact coincidence.
We have seen enough to attempt reconstruction of the Big Hexagon from
the two parent hexagons. It is sufficient to know that the line from
'L' to the center of the second hexagon is at the same time one of the
Big Hexagon's diagonals. That gives us one line's position and
orientation. Since the Big Hexagon passes through several other known
points (C,F,L), its reconstruction is a simple matter.
<>
All three hexagonal system
together become the Hex Machine (below)

It is that time again to review the
standard
argument against all research like
mine that we can always find
some wonderful geometric order in anything - bicycles, dimensions of cereal
boxes, pieces of wind-blown newspaper,
etc. So, there is a modicum
of omnipresent, irrepressible order everywhere. Great!
The Frame exemplifies a
different kind of order, however. It occurs in only one configuration out
of all the possible ones in
the given basic position. It is intelligent, while the rest is random.
One can only hope that people will eventually notice that this
important
research is
being held back by fallacious arguments.
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