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Nazca  Monkey & the Seal of Atlantis

 

"..the figures with their beautiful and regular curves, which could only have been produced in these giant sizes if every piece, being part of a circle, had a radius and a centre whose length and exact position were carefully laid out." (Maria Reiche on Nazca)

The Geometric Nature of Nazca

If one wants to know what the Nazca Lines are really about, the Nazca-monkey tells it all. This famous geoglyph from the plain of Nazca in Peru is a masterpiece of Science-Art. It's fairly easy to establish, how elements of the image had been constructed to express exact ideas. These ideas fall into two distinct groups which appear to be unrelated, however, because of utter lack of clues to the existing causality between them. By itself, the Nazca-monkey is
clad in impenetrable mystery; it is a practically unsolvable puzzle.
One needs to be kissed by lady Luck to make any progress on it. By an almost miraculous coincidence,
I had already solved a different edition of the same puzzle long before the monkey came to my attention. The 14,000 years old Athena-engraving, from the cave of La Marche, Lussac-les Châteaux, France,  is basically the same puzzle; and it comes with a complete set of clues for the basic solution.  

As for applicability of what I learned about the monkey to the rest of Nazca,  again it is Reiche making a key observation:  "This (monkey) drawing consists of no more than two elements. One is a wide line (or better geometric surface, being at the beginning twice as wide as at the end) with a stem which, almost a mile long, leads into the maze of lines at the edge of the pampa." 
Here we have a clear physical connection between the supposedly unconnected figures and lines (the animal and plant figures are generally attributed to the older pre-Nazcan people of the Paracas culture). This connection means that the monkey's hermetic depth may extend to all of the grand-design of Nazca. The entirety of the Nazca Lines should be scanned by our best instruments and with the greatest possible precision. The data should be digitized and fed into a geometric coding
recognition program.
                                                                           

In terms of sheer complexity, Nazca Monkey does not begin to compare to Athena; therefore, its unique ideas stand out more. Its most emphatically stated idea is a direct reference to a particular method of construction for the regular 5-pointed star. As far as I know, it is the second quickest such construction, as
it only takes thirteen steps.
Although sophisticated, that is no rocket-science. Still, a whiff of rocket-fuel hangs about, for, inspired by the monkey, we suddenly see this same idea clearly - and with extreme accuracy -  mirrored in the Athena-engraving.

Years later, I learned that the above-mentioned 13-step method of pentagram construction is essential to understanding the layout of the great Giza pyramids and to being able to independently recreate this layout (as surveyed by Petrie) with total accuracy. Despite over a century of efforts, there was no such complete solution before. Again, having prerequisite knowledge was decisive.    

A Big Question

What mysterious power looms behind this paradigm-changing global phenomenon of ancient sites mirroring each other's ideas? A name like "Agency" seems adequately vague for it. We don't know what it is, but over the span of 12,000 years or so, the Agency had left its inimitable signature at La Marche, Nazca, and Giza - advertizing  its power and quite posssibly offering an explanation to other historical enigmas like the Baalbek Terrace, vimanas - UFOs, Atlantis, nephilim, Giza pyramids, Abydos Helicopter, Saqqara Serapeum, etc.  

Present Status of  Discovery

Having evidence pass tests in the abstract realm of geometry leaves no doubt about Geometry in the evidence. All that remains is verification of its existence by objective and authoritative third parties. Unfortunately, none have been in sight. My findings go directly against firmly entrenched academic consensus on our past, yet, there is nothing wrong with my method.

Since the method cannot be faulted for the emergence of a hitherto unproven anomalous reality, it's time to summon Cognitive Dissonance..

The academia already has a recipe for handling hot potatoes like the one I served it; it hits back with a logically deficient over-generalised doctrine -- significant geometric or mathematical order can be found practically everywhere, in any random collections of lines, points, ancient art, cloud formations, and so on.
The doctrine shows its schizophrenia by omitting to take into consideration criteria by which it would be possible to distinguish naturally occuring patterns from willfully implemented order. That such criteria must exist is self-evident.

                      

  The Image

This study uses a copy of the Nazca monkey originally published  by  Maria Reiche,
 Nazca's scholarly guardian angel. I had it digitized by a service specialising in converting architectural and engineering plans into digital files, because back then in the early nineties it was utterly impractical to use large TIFF files as computers took too long to handle them.
Reiche had learned about the giant figure on the pampa from commercial pilots in 1952, some years after her arrival to Nazca. It became her favorite figure, and she ascribed it special significance; therefore, her measurements of it should be meticulous: 

"The monkey and surroundings would be an appropriate subject for a special study, as it is a complete unit and the pursuit of each line to its origin does not, as at the border of the pampa, lead unendingly from one thing to another." 

Unfortunately, Reiche never did carry out that special study..

Anyhow, it looks like my present copy is sufficiently accurate in preserving major aspects of the design. Still, having a highly accurate plan of the monkey, one that would map both line-edges, would be nice.


               
Manifest order in the monkey figure.   

an blatant balance of the Nazca monkey glyph

       Monkey's Orientation _ Standing Tall

The ubiquitous, and unflattering, stereotype which shows the monkey as if it were tumbling on all fours is incorrect.  Even a cursory inspection of the monkey glyph reveals evidence of  a measured effort.

In the diagram above, two long straight lines form a big X-shape; one of the two red lines is the vertical axis of symmetry of this X. It passes right between the monkey's feet.
The other red line shows clear horizontal alignment of the tail and hands bases, and the tops of the sixteen lines forming a zig-zag on the right.

Evidently, standing tall is one compellingly reasonable orientation for the monkey . 

Nevertheless, there is another, and just as strong, alignment in the monkey's figure  _  to the cardinal points. The dextrous monkey performs a highly intelligent geometrical pantomime.  

I believe that Maria Reiche had noted these two alignments.After all, we are working with her copy of the glyph; and she had taught mathematics in Germany. Why did she not mention these alignments specifically? 
My guess is that since she had no chance of learning that the two form a geometric set, there was no urgency in going into specifics. In contrast to the Athena-engraving, the Square which is behind the cardinal alignments is itself effectively invisible in the monkey. 


Science-Art


The two lines forming the big X hold the angle of visually perfect 36°;  ten of these angles side by side will form a perfect circle. This results in the below spectacular chain of ten monkeys.
The tail spirals around the head, and hands grip torsos with calculated effect. It's clear that I'm not the author of this amazing scene; I'm just the lucky finder.






                   The Big X


The big X may be perceived as two tips of 5-pointed stars in a symmetric tip-to-tip alignment. The question is: If so, then what size are those stars?

 The pentagram below the X-point

The second longest line of the glyph -'c' - cuts across the extended line 'a'.
Let this cut set the size of the experimental star below the X-point.

Next, mirror this star above the X-point, then repeat the move to get the diagram below.

The star inside the big star is remarkably well centered upon the monkey.  The entire star pattern is well fitted to the  X-tree.


( 'c' differs from a pentagonal angle by an even two degrees, good to know for reconstruction purposes )


 



the big 5-pointed star above the X-point

We base a second experimental star on the length of line 'a' above the point-X, but 'a' ends in a curve, leaving two basic choices for its length: 

The first choice is set by a circle from the X-point touching the end of line a; the second one is shown below: we unfold the curve in which 'a' ends and add it to 'a'.
                     
 


        Two Methods _ Identical Results


Unfolding the curve is the correct choice.
The resulting pentagram is the same as the one set by line 'c'.

                                   

If we set the size of the big pentagram from 'a' without straightening the curve, in which 'a' ends, and superpose the result over the previous one, it looks like the diagram above. The Φ relationship holds, but there is now some tiny separation at the top; its accuracy could be better.

While the top of 'a' is ambiguous, the cut of line 'a' by line 'c' is straightforward.
So, let it set the star's size. It will remain our standard throughout this study.



the Nazca-monkey's star

There are two basic stars which fit the set size; one points up, the other one down. The other star is shown in purple color below; it has a cone whose sides run parallel to the original cone here.
While the yellow star's tips do not coincide with the monkey's body, four of the purple star's tips actually mark the monkey's head, hands, feet, and tail. This will be the Monkey Star.

                      the purple Monkey-star is the one which the square isderived from




        the 60° Grill 
    


The sixteen roughly parallel lines, forming a grill-like pattern on the bottom right of the glyph, average out to 60° held with the long line crossing the grill near its midle. Of the sixteen lines, fully a half comes very close to the perfect 60°. If completed into a system of equilateral triangles as in the diagram below, an extensive geometric harmony is established between that system and the system of 5-pointed stars. This harmony really catches the eye because of how good it looks at this magnification, leaving no doubt about its planned nature.



Another visual proof that the angle of the Big X is 36°

Let's array the upper part of the Big X, (including the monkey) five times around the center of the Monkey-star.
Supposing we didn't know what the angle was, the result would seem strange - Five times two lines (of the Big-X) equals ten lines, but we only see five lines (one 5-pointed star). That means lines overlap two at a time, so the angle of the cone must be 36°.
Judging by the way the five monkeys entwine, the centre of the Monkey Star is the right pivotal point, indeed. So, the idea repeats - we get another chain of monkeys. But quite in contrast to the other chain which has rather comical overtones, this rendition appears sinister - like a free-for-all battle.
The hands, feet, and heads all meet in one spot. For instance, at the top right of the image, the green feet press the light brown tail against the purple head, which is held by the blue hands, one of which is pushed into the head by the dark brown tail.  A fascinating picture!





The Monkey's Head, Hands, and Feet Standardize on the Monkey-star's Inner-circle   

With great accuracy, these elements fit into the X-star's inner-circle. However, the head does so in its own way. It fits into the inner-circle's inscribed pentagon (see below).
The inner-circle fits both the hands and the feet to within three millimeters on each side. The method for reconstructing the position of all three circles is given in the Appendix.  
What we see here is what we saw in the French Athena-engraving _ the design uses the same standard circles set by a 5-pointed star. 
The inner-circle is one of these standard circles (Triplets) of the Athena-engraving




Reconstruction of the Head, Hands, and Feet Circles







Despite fitting the original Head-Hand-Foot circles to the image by eye, at first - their positioning with regard to the Monkey-star tuurned out easy to define in simple geometrical terms resulting in a neat blueprint - another key to the monkey's reconstruction.

 Hand-circle's Exact Coordinates

a:

Its center is on the vertical line b1 which descends from the Monkey-star's tip just above.

b:


Pentagon No. 2 (diag) is a direct projection of the inner pentagon of the Monkey Star.
Its rotation about the star's center describes a circle; and the Hand-circle is tangential to it. (diag. below). We draw a circle of a corresponding radius from the star's center, and it intersects the line-b at the Hand-circle's center. 

Now, we can reconstruct the Hand-circle, and with it the line-1, which is the laser-like line of sight from the center of the Monkey Star through a pointlike aperture between the hands. We can also reconstruct line 3 - a continuation of line-1 in a different direction.


Foot-circle's Exact Coordinates

a: elevation

Line-3 originates at the same point, at which Line-1 exits the Hand-circle.
Line-3 is then tangential to the top of the Foot-circle, giving us its elevation. 

 







Second coordinate of the Foot-circle:
The pentagon we see inscribed into the Foot-circle is a direct projection of Pentagon No. 2 downwards and parallel to line "b".

With two coordinates, we can draw the Foot circle.

The star lines within it then help to locate some of the main parameters on the feet. For instance, the small toe of the left foot is delimited by these lines from three sides.

Special Effect 
Two distances involved measure 17.9999.. X-Star meters - almost a perfectly round value: These are the distances of the centers of both the Foot-circle and the Monkey Star to the nearest corner of the other circle's pentagon. 

Head-circle's exact coordinates 

a)
A line from the Head Circle's center perpendicular to Line-1 is a tangent to the inner Monkey Star circle. And the line drawn from the center of the Monkey Star as a tangent to the Head Circle will be perpendicular to Line-1. 
b)
The second coordinate is given by the Square, not seen in the diagram above. It involves a major line of the Square's grid (through the 1/4 point of its y-diagonal.
                                   * 
The resulting distance between the centers of the Head-circle and the Cone's Key-circle ( see the "seat1.htm" for details on the Cone)  is also very interesting 

                      
                           
11.777,777,67... X-Star meters.

Rounding up this number  to seven decimal spaces  will yield seven sevens in a row..






Engraved arcs translate into three symmetrical circles which set the Magic Cone puzzle

in the beginning

I was thrilled by fantastic imagery percolating from subliminal perception to my mind. A lot of it seemed significant. For instance, lines inside the lady's torso made me think of the Giza pyramids; and the torso itself looked like a flying saucer with a cupola on top. In fact, the entire image could be disassembled into what looked like Hi-tech transport craft. Could it be that this image held secrets of prehistoric pyramids, civilisation, even Aliens?
Disappointingly, when I tried to share my impression with others, I ran into a wall of skepticism. The proof was lacking..
Were the Ancients just teasing those like me? Mocking from the shadows could be ominous. On the other hand, if these Ancients were nice, they might furnish the proof by formal design on a high scientific level. Thus was born the initial hypothesis - an assumption based upon ethics.

The Cone & Square Formation - a brief history of its discovery

In the image above, the Square itself is not shown, but you can see the lens shape on its inside, given by two circles centered in diagonally opposite corners of the square and passing through its adjacent corners. Only the bottom circle is fully shown. You can also see its diagonals.
The Square was the first abstract element I had identified in the engraving; its presence is truly conspicuous there, in contrast to its obscure presence in the monkey glyph. The Cone followed after, because its presence in Athena is as unclear as the Square is in the monkey.

The Cone & Square formation is the basic design platform for both Athena and the monkey. It is extremely sophisticated. We'll see yet how the designers  are second to none in understanding the geometry involved. 

                                                     The Abstraction

the big circle

First, one has to identify the three circles forming the Cone. The big circle at the top of the Cone is conspicuous as it presents symmetry through the center between two arcs, one of which is beautifully regular. Its center is at the edge of an engraved line. Seen together, the two strong abstractions, the Square and the Cone's top circle, imply even more planning as the circle locks onto two main points of the Square, its center and its top corner (diagram).

I had the first substantial confirmation of my initial hypothesis _ the Stone Age engraving would prove to be scientific, somehow, in order to confirm the strong artistic impression of showing advanced technology. I began believing that this impression was based in reality.



a view of the Cone and the Square together and separately




the middle circle

As seen in the graphic above, the other two circles are not as defined since each is based just on a single arc. However there is strong confirmation for the existence of the middle circle: There was a stage when all apparently circular arcs in the engraving had been converted into circles, and quite a few of these turned out extremely close in size to the middle Cone circle. The middle circle stands confirmed, because it is a member in a whole class of circles. Evidently, a circle which is ubiquitous is some sort of a unit circle.

the third circle

a) The engraving validates the principle that when the center if the center of a given circle appears somewhere near an engraved line - then it is to be placed at the edge of that line.
b) The central axis of symmetry of the third circle is quite clear.

The combination of these two factors gave me the third circle. Unfortunately, the reconstruction of this small circle in the abstract geometrical position still eludes me.

three symmetric circles

I noticed that these three circles center on the same line. Then I noticed their mutual symmetry, and the Cone was born.

Next, the
Cone may imply a 5-pointed star, because the angle between its sides is 36 degrees.

But, what would the star's size be?

If the middle circle of the Cone were indeed a unit circle, it might provide the answer. How does describing the Cone with it work out?


the Solution

As seen in the diagram below, the experiment worked wonders.

pentagram arms are divided into five units

It gave a perfecr proof that like the X-tree of the monkey glyph, the Cone derives from a pentagram. In this prehistoric system, the arms of a 5-pointed star are five units long and each unit serves as the radius of a circle.

These circles reveal the secrets of Cone's construction.

the Middle Cone circle

the position of the Middle circle on the Cone explained



The upper intersection of the fourth row of unit circles on the Cone shows where to center the Middle-circle. It looks identical with the unit-circle, but it is almost imperceptibly smaller:
If the unit-circle's radius is 1, the Middle-circle's radius is 0.9975...


the Top Cone circle

There are two ways to recreate it:

a) The upper intersection of the third row of unit circles locates the exact mid-point on the Cone's axis between the Cone's bottom tip and the center of the big top circle.
The unit circle concentric with the top circle is then an exact tangent to the nearest unit circles.

b) A circle whose radius is 2 units and is centered in the fifth row of the unit circles then intersects the cone's axis at the exact center of the big top circle
.

the Square
     
  
                                   

The Square is entirely based on the Cone.  Its  reconstruction is truly simple.

a) the horizontal diagonal of the Square rests on two of the Cone's unit circles, the 5th one on the left, and the 4th one on the right, as seen in the above diagram.

b) The point at which the Cone's big circle crosses this newly made diagonal will be the Square's center.

c) The vertical diagonal from this center then crosses the big circle at the Square's top corner.

The rest of the Square is added from there.




The Cone comprizes the cross-section of the Great Pyramid.

This observation is the latest in a series showcasing direct connections among the Giza pyramids, the engraving, and the monkey glyph.  The three works are three chapters of the same book.


magic cone - the Great Pyramid crosssection

the Cone shows the Great Pyramid's cross-section twice


The two red triangles in the diagram are truly special:

1) The length of half the base of the smaller triangle reads out directly the exact value of minor Phi-ratio (φ - 1) or 0.6180339887...

2) The ratio of each triangle side with half the base is the exact φ - the Phi ratio.

3) The angle held by each side with the base is 51.82729237°, which rounds out to 51°50'.

A quote from Petrie:"On the whole, we probably cannot do better than take 51°50' ± 2' as the nearest approximation to the mean angle of the Pyramid.."

Our result of 51°50' complies to the lower ± limit set by Petrie.

Therefore, the Cone presents the cross-section of the Great Pyramid and does it twice for good measure.

the π curiosity

If we double the base of the top triangle to eight-times φ - 1, the triangle height will be 3.14(460).
This direct statement of a close Pi approximation on a model of the Great Pyramid's cross-section is certainly noteworthy, because Pi is frequently discussed in studies and polemics regarding this pyramid.
The curious fact leads to the observation that:
π2 = 9.87 , and φ - 1 times 16  = 9.89

The ratio between the two values is 1.002

It follows that π2 divided by 16 will be a close approximation of  φ - 1. Indeed, it is a thousandth off the true value of p;φ at
0.617.
                 

       


the purple Monkey-star is the one which the square isderived from

                
                  Placement of the Square on the Monkey Figure

Of the 5-pointed stars offered by the X-tree, the one of most interest to us is the star that covers the figure of the monkey (diag. above).  It offers ten cones, and the tips of four of those actually mark the monkey's head, hands, feet, and tail. All four cones belong to the purple star; I called it the Monkey Star.
One of the four cones shares its central axis with the X-tree, and its sides run parallel to X-tree's lines (diag. above). Therefore, it is the first pick as the experimental carrier of the Square.


                                    The Moment of Truth

Adding of the Square to the monkey was critical; I anticipated the moment of truth. Though I felt that my presentation of Athena's Cone & Square system was solid, it was rejected outright by all those that mattered, with the sole exception of the mathematics professor Jiří Fiala of Prague's Charles University.  He bravely rendered his opinion in an official letter, in which he stated that to be proven, my theory would have to identify the same system in other artefacts.
Suddenly, there was the monkey with its own variation of the Cone. Fantastic! But, was the Square built in as well? Since it was the Cone's main product in its La Marche version, why would it not be reincarnated in the monkey and be as dominant there as it is in the Athena Engraving?

Adding the Square to the Cone was the obligatory next step,

and it drew back the curtain on extraordinary harmony between the monkey and the Square. Moreover, the monkey's Square is clearly designed to bring up a certain important geometric idea, one I did not observe in the Athena Engraving before.

Being so important, was the idea also included in Athena, but hard to spot like the Square in the monkey? 

Testing revealed that Athena actually explores this idea to a much greater depth than the monkey. It gave me invaluable new insights into the engraving, which I would not otherwise be able to discover as easily, if ever. This was the first case of Nazca Lines and La Marche engravings forming a two-way street for movement of ideas, and a road-map to some fantastic results. This is an unending story; it doesn't stop at La Marche and Nazca.

A day came when I tested the same idea against Petrie's survey of Giza's big pyramids; and I was able to independently duplicate his Giza ground plan. No one had ever done that before. It left no doubt that there was a Giza plan prior to the three pyramids building. Yet, a skeptic will comment that the fact in no way proves that Giza is directly connected to La Marche and Nazca. The skeptic is correct; it could theoretically be a rare coincidence.

There is just one way to eliminate that objectiion. Did the Agency support the researcher and include the Giza plan in Athena? Did I miss it because it was hard to see?

Testing of Athena and the monkey for the Giza plan then showed its clear presence there. Above all, Athena's head is intensely focused upon various aspects of the Great Pyramid. The clarity of that fact is beyond crystal-clear. Therefore, the Giza plan was already around some fifteen-thousand years ago.

Little by little, step by step, we witness a long march of coordinated ideas which create an autonomous reality, completely at odds with official history. This reality does not need to be conciliatory with anything. To paraphrase René Descartes "Cogito, ergo sum" - it is cogent; therefore it exists. It is the mainstream historical science which has to reconcile itself to this unapologetic reality.

The test 

And so I addded the Square to the Cone. See the two diagrams below. In the first one, the peach colored diamond is the Square, as set by the blue Cone. Already at the firs glance, there is much to be pleased with in the result.


   



the monkey integrates with the Square

                           The Square's diagonals are oriented to the cardinal points!

Of course, my "imaginary" Square could be oriented to the cardinal points by chance, but here it is just a small part of the whole story. Knowing the whole story equals knowing that  the designers are as conscious of this particular square as we are. 
The monkey seems to know it too, as its arms pantomime a similarly sized square whose sides are parallel to the Square's diagonals.


a list of other obvious planning in the image

Three corners of the Square are anchored in the monkey's figure _  the spine, a  knee, and a hand.

As for the fourth, the top corner, a horizontal line resting on this corner limits the southern reach of the monkey figure. It is one of the four lines containing the Square at 45 degrees to its sides _  forming an enclosing square. All four lines are strategically positioned with respect to the monkey figure:

The spine is a tunnel for the vertical line from the left corner of the Square.

One of the monkey's thighs is a tunnel for the horizontal line from the bottom corner of the Square.

The vertical line from the right corner of the Square divides the wrist from the rest of the right arm, while the hand of the other arm, except for a thin sliver of one finger, is entirely to the west of this line.

The Square's vertical axis tunnels down the upper right arm. At the top, it divides the arc of the left ear from the skull. At the bottom, except for a thin sliver of one heel, the feet are entirely to the west of this line.

The red vertical border lines "a" and "b"  show exact spacing derived from the Square:  "a" extends the Square's diagonal by one-fourth exactly, and it is the exact border for the right hand from the west; "b" doubles the diagonal length exactly and  is the exact border for the glyph here. This is an interesting position; this part of the glyph shows three triangles that have been alleged by some to symbolize three pyramids. The diagonal passes through the apex of the largest one; the sides of this pyramid hold an angle of 60 (59.5) degrees. Moreover, the base of this pyramid also equals the diagonal length of the Square.

All these regularities bespeak planning. I was impressed by these initial results even without knowing that the Square is also oriented to the cardinal points. ( I learned that important fact later, when I came across Reiche's drawing of the monkey with the world-compass included. ) 

                       

 

   



                             Monkey's containing rectangle

For more persuasive evidence that the monkey’s layout with respect to the cardinal points is premeditated, we simply enclose it by East-West and North-South oriented lines _  x,y-diagonals of the Square. This is its containing rectangle. Adding central axes to the rectangle shows that:

•  the spine divides the monkey in half neatly along the East-West axis

• with great precision, the lower right forearm divides the monkey in half along the North-South axis

Therefore, axes of our containing rectangle are the two ‘great divides’ which clearly govern the monkey’s layout.

                       Goldent triangle's containing rectangle

A Golden Triangle is an isosceles triangle, where the ratio of one side to the base is 1.618099...,  the famous φ-ratio, also known as Golden Ratio, Golden Mean, Golden Section, Sweet Proportion, etc.

The monkey's containing rectangle in the above image has two inscribed triangles which are almost indistinguishable from each other, because their directions differ by only 1/10  degree, 18 and 17.9 degrees.  One is the Golden Triangle itself.

The below photograph shows some serious damage to the eastern portion of the spiral tail. Obviously, Maria Reiche had faced a difficult challenge in establishing the exact contours of the lines in this damaged area, and she did a pretty good job of that, as the Golden Triangle was ostensibly the idea behind the containing rectangle.

      
                                                           


|




More Frames


Results like the one above encourage more experiments with containing rectangles: in the monkey's body-language, its arms signal two squares. In the first case, two sides of the Monkey Frame and its horizontal axis in combination with a vertical line from the turn of the elbow and on through the outside of the upper right arm form a perfect square.

•  width of the arms (East-West) = half the monkey's height

•  width of the feet  (East-West)   = half the width of the arms =  one-fourth of the monkey's height.

• width of the left foot (East-West) = half the width of the feet =  one-eighth of the monkey's height


The vertical limits of the feet, the horizontal line at one-fourth height of the Monkey-frame, and the bottom line of the Monkey-frame combine into the crucially important Foot-square. 
  master of squares

The monkey is also a template for yet another square. This time, the top side of the square rests on the southern-most points of left ear and elbow. Marked by small red circles, the points align in the East-West direction.
Two more sides of the square are attached to the western and northern limits of the hands.
The eastern side rests on the point at which the line of the head takes a sharp turn and changes into the neck (line).
The south-east corner of this square is inside the line of the head. 


 


More order in the orientation of the monkey to the cardinal directions
:
The same diagram above also shows that according to my CAD program, the width of the four fingers of the left hand very much equals the height of the right hand.

                               height of the right hand = 10.04244144
                                  width of the left hand = 10.02699519

Because the radius of the Monkey-star's inner circle = 10.04057079, the right hand's height differs from the radius of the inner-circle by less than 2 millimeters! Of course, the real lines have a width, and it's impossible to tell from my position how this would work out on the spot. Nevertheless, both hands hint strongly that the inner-circle of the Monkey-star really is the matter of these facts.


A Perfect Clue

  .......            
a clue which leads to revolutionary progress in the case

The position of the Foot-square in combination with the Square has a truly inspirational solution.
To begin, the top right corner of the Foot-square connects to the top and bottom corners of the Square by lines closely approximating angles found on a regular 5-pointed star.

I took it as a clear indication that a star should be drawn here. 

So I extended the line through B and C to where it met the Square's horizontal diagonal. Then I mirrored lines A-B, and B-C across the Square's vertical axis.

The result was a really great looking facsimile of a 5-pointed star.

Next, I drew a second, and mathematically exact, star in the same place, i.e., from the top of the Square, with the horizontal diagonal of the Square serving as one of the star's arms. 
        

Nazca monkey's clues here create an excellent facsimile of the regular 5-pointed star

The two stars are so close to being identical that they pass for a single star when seen like this.

While drawing the regular star, I saw that the instrumental 'golden-circle', centered in the bottom corner of the Square, seemed as big as the circle drawn around the Foot-square. On the right of the diagram, the two circles are drawn from the same center-point, and they do look like a single circle. Their radii differ by mere 2 centimeters in my CAD.dwg of the monkey -  virtually nothing on the scale shown.




Above - the same geometry by itself.

 
The Foot-square is an intriguing addition to the otherwise standard position. Why did the ancients set it up?

I surmised, wrongly, as it turned out, that it was a reference to the quickest
straight-edge and compasses construction of the regular 5-pointed star. It is probably the second quickest

At the time, I had no way of knowing that this method was the key to solving, i.e., being able to reconstruct, the ground plan of the three pyramids at Giza as given by the meticulous survey of W.F. Petrie, plus, establishing who and when had created this plan.

Construction of the 5-pointed star in 13 steps

first six steps to a construction of the regular pentagram by the compases & straightedge method
The diagram above shows the first six steps. Step-1 is a horizontal line, and already an arm of the sought after star. 

Construction of the 36-degree angle

             
construction of a regular pentagram in 13 steps

step 7: 
Draw a line between points C and 2.  
step 8: 
Draw a circle centered in 'C' through the intersection of the blue circle-2 with the new line.
steps 9&10: 
Draw lines from the top of circle-3 through points P1 and P2, which are the intersections of circle 'C' (yellow) with circle-3 (green). These lines are tangents to circle 'C', and the angle betwen them is exactly 36 degrees. They represent two more sides of the star.



Construction of the regular 5-pointed star _ steps 11,12,13:

13 step construction of the pentagram
The points 1 and 2 are there from the previous diagram. Now it can be seen that a circle from the point C, through 1 and 2, shall be equidistant to the points 3 and 4.
For the final two steps, draw lines from 3 and 4 through C to meet the horizontal line from step 1, and the pentagram is complete.   



              point Q gives four points of the star - two tips - two corners of the inside pentagon


alternatives for steps 11,12,13:

Since the horizontal line will serve as one arm of the star,  the point 'Q' circled in green will be equidistant to points numbered 1, 2, 3,  needed to complete the star (Q could be on the other side as well).

The Trans-Atlantic Connection & the Foot-square

The Foot-square theme seems to be the culmination of monkey's geometry. Not forgetting that success of my analysis was due to the hypothesis that the monkey's geometry duplicated the Athena-engraving's basic geometric system, I had to wonder if Athena's system included the Foot-square idea.
                                                                                    *
The Athena-engraving already has its Square; so we just add the 13-step star & the Foot-square to the template. We have to rotate it 90 degrees; however, so it points towards Athena's feet (diagram below).

By the way, unlike the barefoot Nazca-monkey, it's clear that Athena has footwear on. To me, it looks like a heeled boot with a stirrup on her left foot, and something not so easily identifiable on her right. Whatever it may be, it is startling in that it has three toes just like the monkey.


feeding back the position from Nazca to La Marche shows mighty correlation
 
The circle around the Foot-square (the Square's Golden-circle) clearly centers upon Athena's right leg below the knee. Whereas at Nazca the Foot-square covers both feet, here it covers just one, because the designers had availed themselves of the greater complexity of Athena and expanded the Nazcan idea into a system of two Foot-squares.

same foot-squares fit both feet


The Foot-square, along with the smaller squares inscribed within, is clearly custom-fitted to the lower leg and the foot; with the exception of its left side, all the other lines define the foot in some way. For instance:

* the bottom line of the square does exactly the same thing here as in the monkey glyph - it limits the right foot from below and does it with extreme accuracy;
* its right side is a perfect limit for the middle toe;
* one side of the smaller inscribed square forms such a limit for the heel.

The other sides of this smaller square, the diagonals, and the axes all relate to the engraving in a meaningful manner. I skipped listing the numerous instances of it for the sake of brevity.

the other Foot-square

I also tried moving the Foot-square over Athena's other foot just by eye, to see how it fits there. The move was sound:

* the width of the left boot including the stirrup, is the width of the Foot-square;
* counting from the top of the left leg, the line of the left side of this square relates meaningfully to the engraved lines in the area fully six times;
* the horizontal axis also correlates with the engraving strongly;
* the extended right lower side of the inscribed square is a perfect boundary to the three toes of Athena's right boot. As a rule, neither monkeys nor humans, not to mention boots, are three-toed; so this 'coincidence' cements the special relationship between these figures.


an unexpected "coincidence"

For good measure, moving the Foot-square over Athena's head results in an amazingly precise fit. It is shown below at about 2 X lifesize (inscribed in the Square's golden circle along with a star).






So, overall, the experiment of testing the concept learned from Nazca to the Athena-engraving worked out beyond expectation. I had found three prominent instances of the Foot-square in Athena. Along with the previously discovered material, it was enough confirmation for the initial hypothesis. Satisfied, I went on to other things.

    

  Following the script

Years later, and after much progress on other fronts, I returned to these Foot-square phenomena in Athena. I got the initial Foot-square by construction, but the other two by just shuffling it about the figure. Yet, I knew that there had to be a correct way to do it  by exact design because that's how the Ancients did everything else.

The Foot-square is inscribed in the Square's golden-circle used in construction of a  5-pointed star; so it's natural to experiment by also inscribing the circle around the Foot-square with a star of its own.


           


The Monkey-tree

The experiment then presented a view of striking harmony between this star and the engraving. A detailed description here would be counter-productive; instead, I marked most, but not all,  instances of it by arrows. Still, some of these correlations are simply in the must-mention category.
The passage of star lines a & c through the image is the most blatant correlation as these merge with the engraved lines over long distances, long enough to set the legs' basic directions.
Lines a & b  and b & c create 36-degree cones.

Lines b & c actually give two 36-degree cones oriented tip to tip on the same axis.

This is a deja-vu of the Nazca-monkey's X-Tree!


Like Athena's three-toed boot, this X-Tree (c
all it Athena's Monkey-tree) is another sign of a common source for the two works.

Here we see an analogy to Nazca-monkey's 36-degree X-tree, two cones tip to tip.


Line 'c' of the Monkey-tree is given by the pentagon inscribed into the golden circle.  Accordingly, when assigning stars to the Monkey-tree, let's make their sides the same length as on this pentagon (see above).
             
           

Mirror each of these two stars as in the above illustration. While these stars are definitely helpful in mapping of the engraving, they also reveal the exact positioning of the second Foot-square. It begins with line 'e', which the engraving also echoes.





Above: The small half-green circle on the right marks a point where star lines associated with the Monkey-tree system intersect at the edge of an engraved line. It is the same as the midpoint of the right side of the Foot-square I positioned there by eye. So now it is the unsertion point for the sought Foot-square. Just like before, a line drawn down at 45-degrees from this point is a precise boundary to the three toes of Athena's right boot; so it looks like we got this right.   




Above - a detailed look at the two squares - note the green line connecting their centers.  Along the way, the line gets two chances to be a limit to engraved lines; and both times it performs wonders.  It is a precise limit for one of the lines of the Monkey-tree. This seems to be the latest attestation to the direct connection between the two Foot-squares - again.
The first time, it was the circled Foot-square which based entirely on the golden-circle of which it is a duplicate. The same principle applies the second time as well. The other Foot-square is entirely a product of the first.




     
                                


Athena's head & the Foot-square - the Halo-circle

Is that a halo around Athena's head? It's an exact duplicate of the Square's golden circle, and its location is given by a method much like the one we used to position the two squares on Athena's feet. Namely, each of those was derived entirely from the nearest golden circle.
Here, it is almost the same case. The nearest golden circle is centered in the nearest corner of the Square, and it does participate in the process.

Lines 'a' and 'b' combine to give the line 'c'.
Lines 'd' and 'e' combine to give the line 'f'.

Lines 'c' and 'f' belong to the star and the square inscribed in the Halo-circle and combine to give us the insertion point for the Halo-circle.

Below - the golden circle centered in the nearest corner of the Square:

Lines 'd' and 'e' are parts of the star and the square inscribed in this circle;
line 'a' is a starline originating from the center of this circle; see that it is parallel to 'f'.

The line 'b' is the only one to come from elsewhere - we see that it is the extension of a side of another square covering the upper part of Athena's head. This other square is absolutely key to understanding the head's architecture. The reader may be surprized to hear that it actually represents the base of the Great Pyramid.

 
           

This other line 'b' only became available after exact recreation of Petrie's ground plan of the three big pyramids of Giza from our familiar Cone & Square configuration. Testing the validity of such method called for its importation into Athena and its geometry, as well as the monkey, of course. This is the subject of other chapters.
Yet, there is another way to place the Halo-circle/square. It is shown below, and it involves a star whose height equals half the Square's diagonal. You can see that the middle of its base rests on the Square's corner. However, this recreation of the square within the Halo-circle is a tiny fraction of a millimeter lower than the square fitted to the head by hand, whereas the first method does recreate it with microscopic perfection
.








From Head to Toe - the Square's Column
 
Surely, without inspiration from the Nazca monkey, this idea would never occur to me.
           The figure of Athena is engineered to fit a configuration based upon the Square
The Square's diagonal sets the column's width. Its height is also derived from the Square; it has some surprizing qualities.

columns's upper limit 

Mirror the top half of the yellow Square upwards, and its top line will mark the farthest reach of the (white) internal space in Athena's head in this direction.

column's bottom limit

The bottom line of the Foot-square sets the bottom of the Square's column.

The vertical lines set by the golden-circle also set the golden ratio across the column. The engraving is visibly adjusted to them.
A-B-D-E in the diagram marks a golden rectangle, one of several in view.





Virtually Exact Star-maker Template

The Square's Column has an amazing property: the angle of its diagonals with the horizontal is 24.0000356.. degrees, which is supremely precise.
The difference from perfection is a little less than 1/8th of a second of a degree, or approximately 1/10,000,000th of the circle. ( a second of a degree in decimals is 0.00027777.. , or
30.864 meters in planetary terms.
 24 degrees comprise 86,400 seconds - just like a 24-hour period. If this were a mechanical watch, it would be off one second every eight days - the best ever.

For comparison, the Swiss watchmaker Zenith has unveiled a watch in 2017 called the Defy Lab which it claims is the most accurate mechanical watch in the world. Its precision rate of just 0.3 seconds per day far exceeded the standards for COSC chronometer certification.

(An interesting tidbit, in units set by the mother-star, the circle around the column has an area of 66.66.. )

The diagram becomes an illustration of the following rule valid for all inscribed rectangles:

Draw a line from a rectangle's corner to where the extended rectangle's axis crosses the describing circle; it creates an angle with that axis which is exactly half the angle of that axis held with the rectangle's diagonal.

Moreover, our rectangle automatically produces the meaningful angles of 12; 24; 36; and 48 degrees. Hence it allows the division of its circle into 30 equal parts (diag. below).
 
                                 

This
head-to-toe rectangle set by Athena's figure is a practical template for inscribing a number of regular pentagrams and hexagons into its circumcircle:

Connecting every sixth point on the circle creates a pentagon, every fifth point a hexagon. With thirty points on the circle, six 5-pointed and five 6-pointed stars  will be in sight in the end.
   
                

Monkey's Impact

Without assistance from the Nazca Monkey, as long as there was just the engraving, its Cone & Square spirit was unanimously credited to my imagination.
Therefore, the Nazca Monkey coming to the rescue was like a miracle. It guided me to some of the engraving's secrets, which were otherwise too hard to see.
Its intervention removes the burden of responsibility from my shoulders as it decides the issue of who is the creator of the Cone & Square system in favor of the prehistoric agency.

Any number of designs claiming descent from a particular five-pointed star can be brought to the same scale and unified for comparison with the others by that star.

Of course, this principle only applies if such stars exist in reality rather than in imagination.
The Athena Engraving and the Nazca Monkey glyph are vastly different from each other in size. In the image below, the two were scaled and positioned so that their respective 'mother' stars merged and became one and the same. Both works were thus presented on the mutually proper 1 : 1 scale.




Merging of the Athena Engrasving with the glyph of Nazca Monkey on a 1 to 1 scale



     First impressions

The heads of both subjects are similarly positioned across the mother star's vertical axis.
Both heads are planted in the other subject's midsection at a similar depth.
The monkey's left hand fingers seem purposefully poised on Athena's lines.
This also seems true for the rest of the monkey's body, as well as the right side of its X-tree.  To see this phenomenon follow the monkey's contours.

Aware of the other - the plot thickens

The engraving gives no clue to the existence of the Hand-circle. Bringing it into the common picture yields instant benefits, as it lets us learn some interesting facts about the heads of both the monkey and Athena.


 We can see that a parallel to one of the star's arms drawn from the Hand-circle's centert passes through Athena in an interesting manner, and exits as a tangent to Athena's helmet. This parallel also creates a golden triangle with one of the star's tips; so I completed it into a full star.
and saw one of its arms merge with a line in the torso. This is meaningful because seven of the other lines within the torso were already lettered, in the first illustration from the top of the linked arcticle,as parts of a star system integrated with the Square. The center of this star is on a straight engraved line which serves as a line of centers for three circles; each circle has the radius of the new star's circumcircle. One of these circles is now the best fitting circle possible for the circular top of the monkey's head.


Without any rotation whatsoever, next, we move the monkey'head to Athena's head to find the best mutual fit. It turns out that the fit is extraordinarily good! Also, the entire monkey head is contained within Athena's head, and its chin actually completes the ovoid suggested by Athena's head.

This and the above experiments confirm the idea that it is possible to size both Athena and the monkey to the 1 : 1 scale for comparison in positions predetermined by shared geometry.

Change the size of either head even a little bit, and the just seen accurate fit will worsen; the more such change, the more the fit will worsen. 



Unbelievably good fit


For starts, let's go back to the familiar idea of mirroring, and mirror the monkey across the vertical axis of the original star.

Mirroring the monkey across the central axis of the X-tree


 The faces of both Athena and the Nazca Monkey occupy the same space.
But for a small sliver, all of Athena's face is inside the monkey's head.
As seen below, the monkey now seems to have a detailed face of its own, in which Athena's right eye is also its right eye. 

Athena's head is entirely contained in the monkey's head
 

Rotate the monkey around the original star 54 degrees to the left (the 54 degree angle is one of the angles found on the star). But for a tiny sliver and a speck of space, the entire face of Athena is once more entirely contained within the monkey's head. Moreover, the arms, and especially, the hands of the monkey correspond strongly with Athena. The second illustration down shows it in magnified detail.


Rotate the monkey around the original star 54 degrees to the left (the 54 degree angle is one of the angles found on the star). But for a tiny sliver and a speck of space, the entire face of Athena is once more entirely contained within the monkey's head.

Below:

The monkey's hands are completely attuned to Athena's lines; it's as if they were in control of an intricate instrument.



On a lighter note, about the subject of pareidolia _ the space enclosed by the monkey's arms may be seen as a grimacing, devilish face with a bitter smile.





The Seal of Atlantis

The Peruvian "Nazca Monkey" is identical to the 14,000 years old "Athena" engraving from La Marche, France in that both images are instances of the same geometric engine, the Cone & Square. Using this engine, we have just successfully completed a round trip from the Athena-engraving to the Nazca-monkey and back.  
The Nazca-monkey accentuates the Cone element, and so it was easy to see at a glance. But the impulse to look for it came from La Marche, where it was really hard to discover. In turn, the monkey gives practically no clue to the Square. The impulse to check out the monkey for the Square came from the engraving, as well. However, once we add it into the position, its becomes perfectly evident that the Square already was there in spirit. I saw the monkey as heaven sent help in proving the presence of the Cone & Square on both sites.


Since there was a flow of ideas from La Marche, France to Nazca, Peru, an experiment was in order to test the flow going the other way, too. The monkey suggests experimenting with the Foot-square idea in the engraving. Performing the experiment then rewards us with in-depth treatment of the theme in Athena. You have just witnessed the fact. The hypothesis was borne out.
                                                                                  *
Imagine my bewilderment at discovering, many years later, that the "13-step" method of pentagram construction makes possible an exact reconstruction of the ground plan of the three great pyramids of Giza, as surveyed by Petrie!  In turn, that fact called for testing both Athena and the monkey for the presence of the ground plan of Giza. And again, this backchecking brought highly positive results, as you will see.


Next : The Giza Plan

    Appendix

       





Reconstruction of the Monkey Frame


The Golden Triangle

The triangle inscribed into the rectangle in the diagram above is a pretty good facsimile of the Golden Triangle. The angle at its tip is 35.8 degrees, i.e., it is very close to being like any of the five yellow triangles on the inscribed pentagram in the above image.
The scale model of this situation is very reproducible from memory This is the second such scale model we have for the monkey.

             

The idea seems to be that the Monkey Frame's intended height should equal the  Foot-square's perimeter.  (4x one side of the square inscribed into the Golden Circle) 
The idea is easy to reproduce (see above), because we know the position of the Foot Square. We get the southern and northern lines of the Monkey Frame, plus the horizontal axis. 
                                                         *
Next, we need to determine the East-West position of the Monkey Frame:  
It seems that the lower line of the triangle pointing west passes through an inside corner of the Monkey Star. That point is marked by a small yellow circle in the diagram above. So, we try this idea. See the reconstructed Monkey Frame below, where the inscribed triangle is exactly 36 degrees at the tip.  



The Monkey Frame turns out slightly higher, and slightly narrower. We can just see  daylight between the Foot-square's base and the lowest point of the foot. The Monkey Frame fits especially well on the western (right) side, to within a couple centimeters. Everything else in the reconstruction below, like the Foot-square's width, the axes, and the Arms-Square, turns out very exact. Note, how the straight horizontal line of the right forearm is completely blotted out by the Frame's horizontal axis. The same line on the upper arm is similarly blotted out in its straight part by one side of the Arms Square
 
Diagram below:
Another view of how well the geometrical template fits over the monkey figure. Note, how the Big-X lines almost disappear without trace under the star lines.  



                                                 Nazca Lines theories

The entire Nazca plain and some of the surrounding Andean foot-hills host a world-unique, spectacular panorama of numerous lines, trapezoids, animal and plant figures. Not surprizingly, Nazca theories are dime a dozen, often on the borderline of reason. Of this merry company, Daniken's astronaut theory has provoked  most scornunanimously relegating it to the lunatic bin. Yet, Daniken was right in assuming that an advanced civilisation had somehow played a role at Nazca.  But, even the far more serious and methodical Maria Reiche, the saviour of Nazca from being plowed into fields, had her theory about astronomical alignments subjected to analysis and dismissed. That has cast a shadow in which Reiche's observations on the geometry within the figures are simply overlooked.

Johan Reinhard notes that an extraordinary proportion of trapezoids trace the course of geological faults bearing water from aquifers. It throws some light on those trapezoids and triangles.
Anthony Aveni's popular theory about the religious-magic significance of the Nazca lines presumes that because of the dry microclimate, water was uppermost on the collective mind of Nazcans; so, they had made the lines for walking as a form of rain-dancing. For support, he cites a mystical experience he had when walking the contours of the hummingbird figure.
Clearly, explaining the lines by the animal figures, as Aveni does, yet denying connections between the two, is a blatant faux-pas. And if Nazcans had indeed adopted Paracan magic practices of decorating the desert, why did they create an entirely new style and treat the older figures as if they weren't there?
Perhaps, a line crossing over a figure doesn't make the line ages younger, and irrelevant to the figure. 
Aveni says that Nazcans had traditionally associated spiders with water, hence the giant spider glyph keeps in character with the aqueous aspect of Nazca -- I'm rather incredulous on that; my memories of trying to rescue spiders fallen into bathtubs; most drowning in under a minute, associate spiders with dry places (such as Nazca). Soliciting rain by walkathons within a symbol of love for dryness in a place meant by higher powers to be rainless seems counter-intuitive.
There are other ways to speculate, too: After a couple of generations, Nazcans would learn that rain over the Nazcan desert, one of the most arid places on Earth, is a perfect non-factor. They would see the old figures unchanged over decades, whence the yearning to add their own drawings to an eternal gallery. Naturally, such drawings must be heavy in symbolism, and have hermetic significance. Since practically everything at Nazca is interconnected, the glyphs may be telling a very long and detailed story.





Microscopic

The Square with its products visibly controls the engraving's layout in this orientation. Below, we have a close-up of the situation at the top of Athena's helmet. 
The blue vertical line is the Square's diagonal extended to exactly one-and-a-half times. Its end-point is positioned with microscopical accuracy at the edge of the engraved line.
The green horizontal line extending from that point is microscopically accurate as the limit for the height of white space inside Athena's head.
The red lines belong to the Foot-square (which, by the way, is also derived from the Square) placed over the head. The mid-point of its top side (red lines), is also at the very edge of the engraved line.  This type of exact fit reoccurs frequently throughout our analysis.


points exactly at the edges of a line

Meanwhile, some others are trying reconstructions of Nazca figures in their own style: 
http://www.onagocag.com/nazca.html
The reconstructor, Joe Nickell, chose primitive methods to emulate the ancient Nazcans. He does not think Nazcans could measure angles!  "... there appears to be no evidence that the Nazcas had such a capability" he wrote.


If you'd like to contact me, or weigh in with an opinion,  I am at Yahoo.com. Just use Jiri Mruzek without the space on the left of Yahoo
.




 


For example, the small yellow star in the graphic above is part of the star occupying the lower half of the Monkey-tree. Vertical lines y & z drawn down from points on this star are absolutely precise  limits for the engraved line at the front of Athena's right foot. The line x is a perfect boundary set by the bigger star; it limits the white space on the lower foot in this direction, and it runs along the edge of a line section further above. The distance between lines w & z equals a side of the square inscribed into the Foot-square. An arrow points to where this line acts as a boundary for an engraved element. 
These boundaries can be called exact because the error is so small that measuring it becomes impossible while working with a paper copy rather than the original. The "tif" file, serving as the background in my CAD drawing, has a grey band of uncertainty at line edges under a very high resolution. The thickness of this band is roughly one or two hundredths of a millimeter, or ten to twenty microns.



the same square expanded into a golden rectangle








































The bottom line of the golden rectangle nestles neatly atop an engraved area  (arrow).
Its left bottom corner is likewise at the edge of an engraved line.

The top of the head to the face                   1 / Φ  = 0.618..
 is as the face is to the entire head       Φ / (Φ+1)  = 0.618..

Overall, the height of the head is Phi + 1  ( 2.618..), and its width is 1 + 1.

The vertical distance from the bottom of the chin to the top of the head is about 4.5 centimeters. It probably is a lot more than that on your screen, I hope, for the sake of having a good view of the accuracies.





The top corner of the Square is situated on Athena's face. Therefore, I wanted to see what the golden circle (of the Square) would do when centered in this corner and inscribed with a pentagram and a square. The star shown below was the one most spectacular result.




The horizontal line "a" is from a rotation of this pentagram. It's included because it's a perfect boundary from below for Athena's helmet (or whatever else it may be). An arrow points to the lowest point on the helmet. The other arrows do the same thing. They bring our attention to more such perfect strategic points.  For instance, line "x" is a horizontal line through an inside corner of the star and sets a perfect boundary for the face from below. You can see just how perfect it is in the close-up below. Keep in mind that in lifesize, the engraved line is just 1.2 millimeters thick. 

Close up example of the image's accurate coordination with exact geometry
____________________________________________________________________________

Below:
The upper tip of the star is at the top of Athena's forehead, which only protrudes through the star's outer circle by 0.02 mm (in lifesize).  That's three to four times thinner than an average human hair.
The star lines from this point flow with the engraved lines there.
The green circle drawn from the star's tip forms a perfect limit for three white areas, at once. They reach exactly as far as the circle. That's three out of three chances to be either exact or somewhere in the vicinity.
   


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