Top   Previous

## Frame's Pentagrams

What mathematical meaning can a hunter-gatherer bestow upon a set of thirteen  2-D points, the Frame? Can he accidentally outdo a professor of mathematics?

The gif above is scanned in from a drawing on paper I did long ago. When connected by lines, eight of the thirteen Frame points, six points in-a-row,  C, D, E, F, G, H, and K, L, form pentagonal angles with one another. However,  lines a and b hold a fairly accurate angle of 18 degrees between them, but are inaccurate in relation to the other lines. Even these inaccurate angles still round out to accurate values, i.e., they are within half a degree of the desired value.
Down to the Pixel - A good indication that all these pentagonal angles are deliberate is found in the harmonious passage of their arms through the engraving. The lines c, d, e, f, and g do so to an especially good effect. To see this harmony, the reader has to follow each line along for him/herself.

### The Tri-balance

I find the Tri-balance combination (E,F,G,H) absolutely stunning in its intelligence; it serves so many functions in the context of the Frame analysis.   Below,  lines through L-K, L-F, and L-G create two triangles with E-H , which subtends the Tri-balance. The position can completed in several ways into pentagons and 5-pointed stars falling over the image.

The Blue Star
One way of completing the star is to take the length 1-K-L as one side of the star's pentagon. This is the blue star in the diagram below.

The Red Star

Another possibility is letting the star's arm-length be set by the length from 1 to E.
This is the red star in the diagram below.
The same star can also be set from the midpoint of the distance between K and L.

Harmony

We get a big surprise - the blue and the red star systems actually constitute a single larger system! This is clearly visible in the details below. The harmony is in no way forced. Slide the size of the red star along the line E-H-1 in either direction, and the harmony will go Poof!

Some of the Exact Geometric Relations Between the Two Stars

The E point of the Tri-balance, the tip of the red star, does the Golden Section on the distance between the tip (8) and the inside corner (7) of the blue star.
The circumcircle of the red star is tangential to the right side of the blue star's pentagon.
The upper tip of the small blue star is the same as the red star's center; its other tips are on the lines of the red star.
Line 4 is at once an arm to both the red and blue small stars.

*

Most of the points we've seen set the 'TriHex', inclusive of the Tri-balance, are now involved in a new system, dealing with a unique pentagonal (Golden Section) grid. Therefore these points are common to both grids, hexagonal and pentagonal.
The Frame is certainly deep, so it would be pig-headed to propose that it is random, a work of chaos. From those standing on the cathedra of authority, it'd be lynching of reality. Yet, such behaviour is expected; it is the academic norm, being lost upon exposure to good evidence of the so called Lost Science.

*

While I was able to draw visually accurate angles (using a pencil, a protractor and a ruler) over an undigitized copy of the engraving, scanning the whole thing later - the same points in the two scanned versions produce slightly less accurate angles. Yet, the results are more than satisfactory. I believe that the original engraving should produce the most accurate geometry..
Because the Frame points fit a number of exact designs, it is self-evident that all these exact designs are interconnected. What my research has not yet uncovered, is how to derive the Frame points from the Cone & Square formation. However, judging by numerous indications, solving the Frame's construction is within the realm of possibility.

Ther diagrams below develop the star pentagonal structure in more detail and under a greater magnification.

In many cases, we see the star-lines running the same course, as the image-lines, or delimit each other's boundaries. A strong example of such improbable passage is given by the E-H segment: This segment finds ten points out of eleven engraved points available! The Tri-balance has a long history, it brandishes steady purposefulness, it breaks the constraints of chaos; it is unmistakably artificial.

(Figuratively speaking, from the lofty sophistication  level of the Tri-balance, spontaneous cases of geometric order  in art appear as formidable as the Seven Dwarfs playing basketball against the Lakers, and as dashing as dachshunds playing Frisbe :)

Area of the top point of the Square (red circle)

The idea repeats

The same idea repeats from the very next point down. The Tri-balance points F and G create the same angles with L's neighbour, the M point.

: