Frame 3 - Advanced Geometry

* The Hex-Machine *

The Frame also records well thought-out geometry. In its showpiece, an equilateral triangle leads to another,
and both are extrapolated into two seemingly unrelated hexagons. The pair then produces a third hexagon. The three share a wealth
of exact points of intersection, in other words, the composition is a geometric masterpiece.
The Strong Connection, a group of segments that absolutely stole the show in the preceding chapter,
reappears as a formation - E-F-G-H cross-connected to the segment of 16 at B, with D adjoined.

diagram1

a) Rounded to the nearest degree, The angle F-G-H measures 120°.
b) The segment line B-G splits F-G-H into two 60° angles.
c) B-G passes precisely through the center of the Main Square (cross-hairs in the diagram).

d) The Main Square's center (the 0,0 point) connects to 'E' and 'G' by lines at a 60° angle.
These two lines create an equilateral triangle with the line passing through 'F' and 'G', as below.

The Big Hexagon

To recapitulate, we were given some lines, and an equilateral triangle, which could be part of a regular hexagon.
Adding the rest of the hexagon in is natural, (diagram1 above). The experiment proved rewarding. For instance,
the big hexagon's circle surrounds, and frames the image very well. The same can be said for the envelope shape
fitting the human figure.

The Big Hexagon was the first hexagon I found, due to being so obvious. Its origin had remained a mystery
until observation of the other 60° angles in the Frame, and how they function.
The basic lineup of the 60° angles on the Frame is as below. The six points, five of them in a row, encode
the entire Hex-Machine idea. To me, it seems most elegant, and beautiful, but this page is seldom visited.

I apologize, letter assignment on some diagrams below is different. The material is pre-cambrian.
(will correct ASAP)

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). Below: there are some visually exact angles in this position, too.

the birth of the equilateral triangle, from which derives the Big Hexagon

the birth of the equilateral triangle, from which derives the Big Hexagon

Let's base our triangle 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). These sides harmonize with the image heavily.
The circle around the triangle also finds 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.
Here is the design. Perfect points between the stars are circled in.

A critic might comment: So what, those intersections are natural to the position".

But, in control experiments changing position of "F" on the circumcircle, great many
of those perfect points go missing! Therefore 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 circle's diagonal, 1/3 of the triangle's height).

The Hex-Machine - the third generation

<>

The Big Hexagon is third-generation!

Comparison of the first hexagon found on the Frame - the Big Hexagon - to the other two shows
points, at which they all coincide. Like the second hexagon is a special case of the first hexagon,
the Big Hexagon is a special case based on both of them.

In the diagram above, we see how the Big Hexagon fits 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 simultaneously 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 one 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 basic position. It is intelligent, while the rest is random.

********************************** Discussion Forum

Continue
Previous Top Menu Giza Ground-plan - Explanation & Exact Reconstrucxtion