Frame 3 - Advanced Geometry

* The Hex-Machine *

The Frame offers exquisite geometry, as well. Its showpiece is a self producing design made of three
hexagonal stars. The Strong Connection - (CDEF) connected to the segment of 16 at L - reappears as a unit.
The same group absolutely stole the show in the preceding chapter.

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.

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 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.

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

Continue
Previous Top Menu Gallery