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
much like a
regular lens, the overlapping area of two circles of the same
size. This was hard to see, because the right portion of the lens
is neatly sliced off.
I saw some points (line crossings), which seemed very nearby to
the hypothetical circle centers. There was also a point, where the arcs
of the torso might meet, if
extended past the cut-off. 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 within the short lines on
the torso, or trace their edges (contours)
or
thread themselves through breaks in lines,
or
limit lines,
or
pass through points (crossing lines).
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By my count, there are altogether 25 instances of the arcs interacting with the torso, out of which
22 fit one of the above categories. Such powerful harmony is hardly random. Do check it 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 an axial
cross. The lines and extended diagonals of this square harmonize
with the engraving
rather ostensibly. Onwards in this study, the square is referred to as the "Square".
The Square's diagonals are the main axes of the whole image.
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A small
shift of the lens creates one more case of undeniable harmony
between it and the torso. The truly snug fit shown in the
diagram below confirms that we are on track. Each arc of the lens
runs with, or within a long engraved arc on the torso. It
is important that even under a 2:1 magnification, the fit remains
picture perfect. The evidence has to be as strong as the study is
extraordinary..
So, evidently, the lens
within the Square has moved, and
rotated as well. After being under construction in phase-one, the lens
got solid body, and has moved out. Such dynamism of
the design has much promise.
By the way, virtually every engraved line within the torso sets some highly
significant whole angles with the diagonals (axes) of the first lens, such as 30° or 36°. Altogether these lines form a rational pattern, which integrates neatly into the Square. |
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The yellow circle below is the Square's
circumcircle.
The purple circle is the same as the blue torso circles. Its radius
equals one side of the Square.
Both circles snap onto arcs and hinge onto the engraving
in a number of places.
The concentrical circles are like waves pushing parts of the picture before them.
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The
Key-Circle (K-circle for short)
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 on the girl's
hat.
How good is this
symmetry? It is a good example of the art's
technical perfection. Look at the magnified details below.
The two arcs are 1) concentric
2) symmetrical through their common centre
3) are parts of one and the 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
K-circle and the two engraved arcs is remarkable, especially
so with the free-floating arc. It makes for quite a sight.
Once we get around to regenerating the K-circle, it 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 K-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 (like the tip of a 5-pointed star) with
the x-axis of the Square. Coincidence? Well, no, because it too is part of the greater
plan. It 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 K-circle
retro-engineered from the Square, and the related free-floating
engraved arc of the original K-circle.
This indicates that the Square remains constant, while
its progenitor - the Cone - undergoes subtle changes, inducing
corresponding changes in some other figures. |
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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.
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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.
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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.).
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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.
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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.
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Next: a high level solution to this constructional masterpiece. |
Discussion Forum
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