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Frame 3 - Advanced
* 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 HexagonBecause 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.
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) .
Each line in the illustration is an arm of a
angle (rounded out to the nearest degree), including lines from B to one of the
points in-a-row E,F,G,H. The angles in the below position are very accurate (visually exact).
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.
be just such a point.
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
circle's diagonal, and 1/3 of the triangle's height).