Frame 3 - Advanced Geometry

* The Hex-Machine *

The Frame offers interesting geometry, as well, for instance, a self producing design comprizing three hexagonal stars.. The Tri-balance (EFGH) connected to the segment of 16 at B reappears as a unit again after having stolen the show in "The Frame", and "The Frame & Five-pointed Stars" chapters.

a) Rounded to the nearest degree, The angle F-G-H measures 120 degrees.
b) The line B-G then divides F-G-H into two 60 degree angles.
c) B-G passes through the center of the Square with faultless accuracy(diag).

d) The Square's center (the 0,0 point) connects to 'E' and 'G' by lines holding a 60 degree angle. In conjunction with the line passing through 'E' and 'G' (below), they create an equilateral triangle.

The Big Hexagon

To recapitulate, we were given some lines and an equilateral triangle, which could be part of a regular hexagon. Completion of the hexagon unveils considerable harmony between it and the picture. For instance, the hexagon's circumcircle and the envelope around the human figure fit in really well (the first diagram above) .

Because it is so obvious in the engraving, I had found the Big Hexagon before the others. However, its origin remains unclear unless one understands 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 in the illustration is an arm of a 60-degree angle (rounded out to the nearest degree), including lines from B to one of the four points in-a-row E,F,G,H. The angles in the below position are very accurate (visually exact).

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 staqrt with the line E-J. Now, if we rotate E-J 60-degrees clock-wise around 'E' - it turns out to be well supported by its passage through the engraving. Next, we complete the angle JEB into an equilateral triangle so that the third side rests on the point 'B' (diag. above). This side, too, looks supported by the image. Moreover, this triangle's circumcircle touches the point 'H'.

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 H happens to be just such a point.
Logically, H was created after B-E-J,

By extrapolation, the triangle based on the H-E-B-angle is second generation.

The decision to finish the triangles into full stars had paid off; I could see many lines passing through the other star's points. Some of the mutual intersections showed total accuracy; others were a perfect imitation of the slight inaccuracy of the circled intersection in the above diagram. By correcting the inaccuracy, as in the diagram below, the other points become perfect. Change the position of "H" on the circumcircle and the newly perfect points disappear.
Therefore the position in the engraving illustrates a special case. The design is shown in isolation from the engraving.

The special case occurs only when the line B-H ( or its mirror image ) passes through the circled point, as in the diagram above. (The point is at 1/4 of the circumscribing circle's diagonal, and 1/3 of the triangle's height).

Unfortunately, by having corrected the design into a perfect idea, the blue star below lost its fit with the point H. On the other hand, it taught me a valuable lesson in geometry. From that viewpoint, the existing design was a success.

Moreover, the corrected star (blue) finds itself in accurately looking relationships with the Big Hexagon (green), pictured below. Above all, the line from "B" to the blue star's center goes on to the point "G". This relationship is visually perfect; therefore, it is the means to a credible reconstruction of this diagonal of the Big Hexagon.

The Big Hexagon is the last of three generations

The Big Hexagon can only be reconstructed from the second, corrected star, as far as I know. Like the second hexagon is a child of the first, so the Big Hexagon is a child of the second hexagon.
It is sufficient to know that the line from "B" through 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, marked by big circles in the diagram, its reconstruction is a simple matter. Afterwards, the design is taken by the center of the Big Hexagon and placed into the center of the Square, and the B-G diagonal is aligned to its equivalent in the engraving.
Unfortunately, I haven't yet found any clues to the actual size of the Hex-machine, which would help the quest for identifying the position of the Frame points in the Cone & Square design.

The Big Hex also shares exact intersections (genetics?) with the first star, as shown below.

The Hex-Machine

The three six-pointed stars all together become the Hex Machine.

Note: Sorry, I realize that the same diagram ought to be shown over the engraving as well! This will be corrected whenever I reinstall CAD.

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