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The Seal of Atlantis
 

      
           Discovering Geometrical Elements in the Athena Engraving


In late 1985, after weeks of admiring the mysterious Stone-Age engraving, I put it to the first test for geometrical ideas. I wanted to verify my impression that the torso of the human figure was kind of like a regular lens, an overlapping area of two circles of the same size. 
I saw some points (crossing lines), where I felt the circle centers might be. There was also a point, where the arcs should meet, if extended past the present cut-off line. This point is seen on the right side of the torso, in the diagram above, or below. 
Using a pair of compasses, I drew a circle from each point to see how well it would mesh with the contours of the torso. From the result we see: 
         both arcs run either with, or within the short lines of the torso,
         or thread themselves through breaks in the lines of the torso,
         or limit other lines,
         or pass through points (crossing lines). 
                                               *
Altogether, there are 25 instances of the circle arcs interacting with the torso,
out of which only three seem random. Do check this fact out. 

       The Diamond - The Square

The centers of the torso circles, and the points where the circles
intersect -  mark out four corners of a squaare.

The square is oriented like a diamond, and its diagonals form a  cross. The lines and extended diagonals of this square harmonize with the engraving rather  ostensibly. Onwards, its name will simply be the Square. In the course of our investigations, this square reappears regularly, even in the figure of the monkey from Nazca, Peru. Its axes are the main axes of the overall design.



 
 
 
 
The regular lens, which we have established on the torso, can be shifted
a little to a new snug fit with the torso, shown in the diagram below. It is
simply faultless. Each arc of the lens passes entirely within one of the two
long engraved arcs of the torso. Evidently, the lens has moved, and rotated -
so, some dynamics are at play, here. 
Of course, a question can be asked if this second torso lens did not actually
come first. It did not. The precedence of the first torso lens is firmly posited
by the sheer amount of  geometry which is integrated with it. For instance,
virtually every engraved line within the torso sets some highly significant whole
angles with the diagonals (axes) of this the first lens,  such as 30° or 36° .



 

    The yellow circle below is the Main Square's circumcircle.  The purple circle is
    identical to the blue torso circles, and its radius equals one side of the Square. 
   Both circles snap onto arcs and hinge onto the engraving in a number of points. 
   The concentrical circles are like waves pushing and pulling the picture outwards.

    

 

    The K-Circle
 
 
The top corner  of the Square is a point on a beautifully engraved arc. This 'random scribble',  as experts on prehistoric art would say, is visibly symmetrical with an arc of the girl's hat. 
How good is this symmetry?  - It is a good example of the art's technical perfection. Below, we are looking at a greatly magnified detail. 

The two arcs are 1) concentric

                         2) symmetrical through their common centre 
                         3) lay on the same circle

The circle implied by the two arcs is onwards called the K-circle. 
This circle had existed as a template for the arcs,  before one arc was given a  zig-zagging end,  and the other arc was bent a little. 
The K-circle plays a key role in the picture's system. For now, note how the K-circle seems to pass through both the top corner,  and the center  of the Square.


 
The agreement between the Key-circle and the two engraved arcs is remarkable, especially so on the freefloating arc. This arc is quite perfect.
Once we get around to regenerating the K circle from the overall design, the new K circle proves the engraved arc's quality by faithfully tracing  the contour of its inside edge.
Anyhow, part of the order between the Main Square and the Key circle is that this circle passes both through the square's center, and its top corner.
Next, as in the diagram below, the line (k) subtending the lower arc of the Key- circle holds 36 degrees (the angle at the tip of a 5-pointed star) with the x-axis of the Square. Coincidence? Well, no, because it is part of the greater plan. If it were a mere coincidence,  it would have led us up a blind alley, as is usual in such cases. Instead, this setup implies the KX-stars,  based on the Golden  Section within the Main Square



Incidentally, in the diag. below, we can see the agreement between the aforementioned Key circle retro-engineered from the Square, and the related free-floating engraved arc of the original Key circle.
This is part of the proof that the Square remains constant, while its progenitor - the Cone - undergoes subtle changes, inducing corresponding  changes in some other figures.
 


 

     A Constructionist Masterpiece

Some arcs in the engraving seem so perfectly circular, one starts wondering if the artist had used geometric tools of some kind. The arcs are like sockets inviting placement of corresponding circles. 
The Hip-lens:
The (blue) circle through Athena's arching hip, and her right leg, is the same as the circle through the arc it faces - on a pants' pocket. The overlap forms a symmetrical lens. The line of circles' centers points through the Square's center, and straight to the center of the K-circle!

Diagram below:
The other axis of the hip-lens then passes through the intersection of the y-axis with the lower circle of the Torso-lens. 
A line from the center of the lower circle of the torso lens perpendicularly to 
Athena's body axis will pass through the center of the blue circle of the hip-lens 
(see diagram below).
There are two triple intersections of Athena's body  axis with the two circles just mentioned. 
These descriptions may be tedious, but they describe a spectacular design. Given the Square and the K-circle, we are now able to recreate the hip-lens, as below. Thus,  the indication is that the hip-lens is  the product of these figures. 
Yet, we do not know, how to derive either the K-circle from the Square, or the Hip-lens. The possibilities are endless. As beautiful as this design is, we are missing the point.

 

 
 
The below diagram of the hip-lens is a considerably blown up scan of  an old experiment of mine with a pair of compasses, and a fine example of the engraving's technical excellence. Each arc of the hip-lens fits the same circle. These circles have a line of centers, which is one and the same with the line of centers K-circle - Main Square. 

 

 
 
 
After converting every distinct arc into a test circle, it was apparent that among the new circles,  there were two groups with auspiciously similar radii. The right leg produces three such standardized circles. Let's focus on the two color circles to the right of the diagram below, given by the arcs  of the calf and of the shin. These two circles virtually install themselves in  the picture.
The circles' centres, their points of mutual intersection, plus the points, at which these circles intersect their line of centers - form a set of four equilateral triangles! (see diag.).




 
The red line in the diagram above connects the centers of  the two circles, which
together create a set of equilateral triangles. It deserves particular attention, as it
exposes one of the major balances in the engraving. One only has to follow this
red line to see it. This particular balance is very strong, resembling the balance of
the monkey from Nasca with the V shape it is wedged into. No wonder, this line
makes the list of obviously deliberate balances in the engraving. 


           An Ideal Four Way Balance

    The Key-circle from the head, the lens of the torso, the lens of the right hip,
    and the diamond of the right leg together create a perfectl balance. One can
    say that because this balance is easily described in general terms, such as
    perpendicular' or 'all four centers fall on the same (straight) line'.


   


We can see this ideal balance implemented in the figure below.

 
Two sides of the diamond arising from two arcs on Athena's lower 
right leg are perpendicular to Athena's body axis. One of  these lines
extending from the diamond passes through the centers of two circles,
the other merges with the long axis of the hip-lens, and continues on, 
to where the Square's vertical diagonal and one of the torso circles
meet at a point.
                                              *
Next:  a high level solution to this constructional masterpiece.

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