

Below: The yellow circle is the Square's circumcircle. The purple circle is identical to the blue torso circles; its radius equals one side of the Square. Both circles pass a number of checkpoints; they join arcs and touch points on their passage through the engraving. The rose space between the circles contains engraved lines which seem to be split into two halves which rotate against each other. Coloring makes the lens in the middle stand out sharply. The psychological effect of protracted staring at the lens may be threedinensional apparitions of pyramids within. 
The KCircle






Abstraction of the Cone 


Setting the System Correctly There was a problem; the Frame units should be directly proportional to the unit circles forming the Cone. Instead, the unit circle's diameter came to 81 millimeters (with the image scaled 2 : 1). Yet, in view of the slight discrepancy between the actual and the ideal models, was it possible that if the Mother Star were adjusted to the existing Square, the two unit systems would become directly proportional? The scanned image of the engraving was then imported into a CAD drawing and the hypothesis verified  the unit circle's diameter worked out to 80 millimeters even. On a side note, there was remarkable synchronicity between my research needs and progress in the availibility of computer resources to the public. How to Derive the Square from the Image In order to let others reproduce and check my results, it's important to describe the process by which the Square is derived from the image, but first, allow me to make some cautionary remarks. I cannot overemphasise how desirable it would be to work with the original instead of a mere copy. Consequently, there is bound to be some uncertainty. However, this uncertainty can be mitigated by results which closely and consistently conform with distinct abstract ideas of an exact nature inherent in the image, and that is what happens in this case. Stéphane Lwoff's claim that he and his team went to great length to accurately copy the image from the stone tablet is clearly true. The frequent occurence of nearmicroscopic accuracy of exact ideas evident at substantial magnification of the (copied) image is truly stunning. Considering that there must be some deterioration due to copying, the original can only be better. What is the limit? A molecular version of 3D printing? 
Orienting the Square (It may seem strange that the xdiagonal is vertical here, but do recall that the first diagonal of the Square produced by the Mother Star was the xdiagonal; and the first diagonal (axis) is usually named 'x'.) As seen above and below, both the x and y diagonals are supported in a number of places in the image. However, results will differ somewhat depending on our choice of supporting points, although this is only visible upon magnification. After many trialanderror attempts, I settled on the following method: 
1) The entire
system is oriented by the ydiagonal, which is established rather
easily, and in a manner
guaranteed to produce practically the same result time and again: The ydiagonal is propped against two distinct curves found on the engraved crosses approximating the western and eastern corners of the Square. a) At the western corner of the Square located near Athena's back, the below cyan ydiagonal leans against the southern edge of an engraved curve. At this magnification the edges become somewhat hazy; so this is bound to create an error, but one so small as to be virtually insignificant. Once we have the Square's center, its size will be set by a circle drawn from the center so that it leans on the western edge of the engraved cross  the red circle line pointed to by an arrow in the image below. 
b) At the eastern corner of the Square located near Athena's bossom, the cyan diagonal leans against the northern edge of the curve in the engraved cross. Numerous examples encountered in the study show that this diagonal must be so close to the original plan as to render the difference invisible to an unaided eye. Consider that if you have a 24 inch screen, you are looking at approximately a 200:1 magnification here. 
In the diagram below, the western corner of the Square is visibly closer to the center, i,e, making the Square itself somewhat smaller (by a fraction of a millimeter). This is because the selection of the xdiagonal was done differently. 
Setting the xdiagonal After the ydiagonal is set, the angle of the xdiagonal is given automatically. Finding its actual position is not as easy. My selection from several similar possibilities was done by the trialanderror method. (Hypothetically, these other possibilities could function as alternate settings for the Square  in case, the Squre itself changes, pulsates?) For the best result, in my opinion, one should use the pair of points circled in yellow in the below diagram. 
From
the looks of it, these two points (line ends) are tasked with confining
the xdiagonal between them. The magnified versions, seen below, show
this very well. The cyancolored line is the xdiagonal, and it leans on the edges of the grey areas by each point. The solid black of each point is very much the same distance from the xdiagonal; so the xdiagonal is in the middle of the channel. Channeling as a method: Forming a narrow channel whose central axis proves to be the answer to a given problem seams to be a bona fide method. We can see this method used over and over in the geometry of the Giza ground plan. It works as follows: we get two solutions, two lines which are very close to a line given by W.F.M. Petrie, Giza's most accomplished surveyor. The alternate solutions form a narrow channel whose central axis then duplicates the line given by Petrie. Petrie gives states his measurements to the nearest tenth of inch (2.54 millimeter), so if our solution is within 1.27 millimeter of Petrie's  it is exact in the same terms. 
With both the diagonals
in place, we draw a circle from their center so it leans against the
outside of the engraved western corner of the Square (marked by the red
arrow in one
of the diagrams above). The south corner of the Square then lands on the edge of the engraved line, as seen in the digrams below. 
the Motherstar of
Athenaengraving The Square and the Mother Star form a set; therefore, given one element, the other one can be added correspondingly. In this case, we're adding the Mother Star to the Square, and deleting the Square for clearer view. There it is, the original star in the graphic below. Everything in the engraving can be traced back to this 5pointed star. Along with the star, the graphic also shows three small yellow squares and a rectangle extending horizontally through the upper half of the star  these are byproducts of the construction process which were assigned an important role. The φsquare Let's call the small yellow square 'φsquare', because it invokes the φratio (Phi). Its base equals the length of the central section of a 5pointed star's side. When the whole side is 5 units long, its central section equals 1.180339887.. , or (φ  1.5) x 10 
The Golden Column The first step in the star's construction (by the specific 13step method) was the line numbered 1 (diagram above) which then became the star's vertical arm. The circle, numbered 2, was the second step. Here, its diameter sets the thickness of the Golden Column (the horizontal rectangle) to its right. The column is divided lengthwise by the star into two upright golden rectangles with a square between them. Each combination of that square with the rectangle next to it forms a horizontal golden rectangle. The Golden Column and the little φsquare are crucial to understanding the geometry of Athena's head, yet I had missed this line of analysis until I became aware of their role in the ground plan of Giza. The plan's creation begins with the same 13step construction of the 5pointed star, so it was a clear directive to go back to the engraving and check for the Golden Column being unambiguously in evidence there. 
The
13step star construction method offers at least four ways
of projecting the φsquare to the top right
corner of the Golden Column. One of those seems to be given special
attention. The below shown position is based on the 13step method. The two Qcircles are unique to this method. Drawing either one suffices to complete the star, but drawing both of these circles automatically recreates the φsquare at the top left corner of the Golden Column. By intersecting extended sides of the green square these Q2circles recreate the four corners (1,2,3,4) of the original φsquare at this location. From there it can be projected to the other top corner of the Golden Column. Lines 'a', 'b', and 'd' in the diagram below give three more ways of projecting the φsquare to the top right corner of the Golden Column by golden diagonals emanating from key points on the original star. Below: with the Athenaengraving as the background, however, the projection pauses at the moment when one of the diagonals of the moving square enters the tip of the Mother Star, because of sudden harmony seen between the head and the little square. 
Harmony An instance of perfect symmetry  cap Athena's head with a 45° pyramid  and its vertical axis will fit right over the vertical axis of the φsquare. This appears completely accurate until about 6 : 1 magnification. The circle 3 shows that when the Golden Column is extended upwards by ½ of its height, its top side becomes an accurate marker for the top of Athena's head. It leads to the observation that half the Golden Column will form a well fitting frame for Athena's helmet (more on that later). 
Mother Star's Tip
Casts Rays Below: With our attention focussed on this position, other surprises emerge into view. The two antennae rising from Athena's helmet (a and x) are rays cast from the Mother Star's tip at the center of the red circle. Star angles All the lines drawn in the diagram, except 'x', hold exact Mother Star angles. That is a simple but farreaching fact. Lines 'a' and 'g' hold 72 and 54 degree angles respectively with the horizon. The line 'g' justifies its existence by being a microscopically accurate limit to two engraved lines pointed to by red arrows. If we also discover where to position lines 'b', 'd', and 'e' we'll be able to accurately redraw the main outlines of the upper part of the head. The line 'h' is another microscopically accurate limit to three engraved lines at once  also pointed to by arrows. Lines 'i', 'j', and 'k' lean on engraved points on the left, while on the right they lean on the bridge of the nose, the bottom of the nose, and the bottom of the chin. 
Five
Parallels Starting from below the chin and up to the line1, we se five parallel lines. The bottom three originate from major points on the Mother Star; therefore, the face from the chin  to the nose  to the bridge of the nose is stratified in accordance with the star. The next parallel up originates from the (encircled) center of the Golden Column. The last parallel up originates from as yet unknown source. 
Stars All the starangled lines can set up a number of stars. For example, in the below diagram lines 'b', 'c', 'e' and 's' create the star in view. The line 's' is a horizontal line drawn from the intersection of 'b' and 'c'. The engraving is impressively well fitted to this star. It is more or less unthinkable that the artist/scientist would be unaware of it. (apologies for the different lettering of lines) 
A Snapshot When the Golden Column is divided by axes into four equal rectangles, the two on the right are like a camera snapshot of Athena's face. A Portrait Adding an identical rectangle at the top produces a perfect fit with the top of Athena's hat (or helmet). The whole now looks like a classic portrait. The Hatbox The top two rectangles of Athena's portrait can be shifted directly west to become a nearly perfect containing rectangle for Athena's hat/helmet  the Hatbox. All of the white space inside Athena's helmet is inside the hatbox. Exact Coordinates Let's say that I want to draw Athena's face from memory using a computer . I know many exact coordinates for it in the context of the Mother Star and its products. I might begin from the 'Hatbox'  the rightside half of the Golden Column, which can be moved up by onehalf its height and then shifted horizontally to the left to enclose Athena's helmet just like in the diagram above. But how far left? The φsquare has its inscribed pentagram. Move the Hatbox left until the 45º diagonal drawn from its top right corner reaches the center of the inscribed star's horizontal arm. That shifts the original sides marked '1' into the position of the cyan lines marked '2'. The cyan line '2' on the right tells me exactly how far the face extends in that direction. Next, reduce the rectangle to a square whose right side is the cyan line marked '3'. This line is then an exact boundary marker for engraved lines in three places, each place marked by a yellow arrow, the tip of the nose being one of those. The exact width of Athena's face is now known (left side of the φsquare to the line '3'). Thus, step by step, we gather information on the position of individual elements of the engraving. For instance, the star tip on the upper right side is at the edge of the engraved line. Drawing a 72º line through this tip will duplicate the line edge. There is more relevant information to be gleaned from this diagram like the horizontal yellow line running along a line edge at the top of Athena's forehead. We see where it originates on the star; we can duplicate it. The reader will find an indepth analysis of Athena's head from the viewpoint of Mother Star angles in The Layout of Giza's Pyramids is at least 15,000 years old The analysis actually extends to the entire image and confirms the importance these angles bear in the image. 
A Suspiciously Fine Carving Technique The inside and outside lines of Athena's nose are unlike each other. While the inside edge progresses practically in a smooth straight line, the outside edge is somewhat rough. At the tip of the nose there is just one sharp turn in the inside line, compared to three turns in the outside one. Just below the tip of the nose the outside line edge turns towards the mouth in two long concavities while on the other side of the line the edge continues in an admirably straight line. Next, on the inside, the line of the nose base turns smoothly to descend towards the mouth while there is a bump in the same turn in the outside. Since lines in this engraving are on average less than 1 millimeter wide, it is hard to imagine how a single motion by the carver's hand would result in these effects. Indeed, it does look like the two edges were engraved separately and not by hand. 
Square's Domination of the Head Structure One reason why I missed the relationship between the Mother Star and Athena's head for so long was being already aware of the major influence cast over it by the Square. The diagram below is an excellent example of that. The green circle with an inscribed stare has its center in the north corner of the Square  lines 'a', 'b' are sides of the Square. The circle itself is the Square's φcircle, which I usually refer to as the golden circle. The star clearly sets comprehensive boundaries for the head's features. Examples: The star tip on the right  it connects to the top tip by a pentagon line, and to the far low tip by a pentagram line  these seem meant to set exact limits for the width of the face seen from that angle. The upper tip  its coincidence with the engraved features is simply spectacular. It is also planted in the most important point of the system. For confirmation, one only has to observe how regularly one sees this point, or its immediate area, involved with the designs throughout this account. Interestingly, this location is at the front center of Athena's frontal lobes. <health.qld.gov.au/abios/asp/bfrontal> quote: The frontal lobes are important for voluntary movement, expressive language and for managing higher level executive functions. Executive functions refer to a collection of cognitive skills including the capacity to plan, organise, initiate, selfmonitor and control one’s responses in order to achieve a goal. The frontal lobes are considered our behaviour and emotional control centre and home to our personality.> This is truly a telling selection because Athena's frontal lobes are thoroughly connected to all geometric elements of the system indicating her awareness and control over all of it. There is also an exact limit for the lowest point of the helmet. There is also an exact limit for the lowest point of Athena's chin. There is more order to be observed if one roams over the picture. 
Supreme Accuracy The circle shown below is centered in the star's tip, exactly positioned on the edge of a line, is an accurate limit to the three white areas pointed to by arrows. Considering the approximately 20 : 1 magnification of the scene, this is truly impressive. 
Convergence of Influences The outer line edges radiating from Athena's forehead conform with the Mother Star angles, rather than those of the Square. Already at this stage, the Mother Star and the Square combine to a create an ever more detailed map of Athena's head, 
Reappearance of Athena's Cone & Square Module in the Nazca Monkey The Cone & Square module is a sophisticated geometric engine. It powers the Athena Engraving. Its very existence, other than in my imagination, was never taken seriously. I was lucky and found some interesting coincidences  happens all the time. In 1992, after showing my findings to some people in Prague, I was given a small picture of the giant geoglyph of a monkey from Nazca, Peru by Mrs. Z, Hrubá. She wondered what geometry might be there to discover... Did her subconsciousness see in the monkey what I was describing in Athena? The monkey stands inside a large X shape (the XTree) whose angle begs checking out by resembling a 5pointed star. Moreover, the monkey's arms mimic a square. The long arms of the XTree are blotted out in the image below  by sides of 5pointed stars. If there were to be the Motherstar here, it should be the one centered on the monkey. 
The Cone part of the Motherstar runs parallel with the XTree sides. It is shown in magenta in the below image. 
Below: The Square part of the Cone & Square module is shown in yellow. Its lines are a perfect fit to the monkey. (see this chapter for details) Unlike in Athena, the same square cannot be accurately distilled from the image; it has to be positioned there by the Motherstar. However, the Square's diagonals seem to coincide with the world compass really well. 
Below: The containing rectangle for the monkey formed by lines in the directions of the Square's diagonals. The image comes with a manifest proof that this rectangle is a containing rectangle for the Golden Triangle. Also in sight below is a containing square for the monkey's feet. 
The
top right corner of the containing square on the feet connects to
the top and bottom corners of the Square closely imitating sections of
a 5pointed star. 
Extend B  C from the image above up to the horizontal diagonal of the Square. Next, mirror the A  B and extended B  C across the vertical diagonal of the Square. The result is the diagram below  an excellent facsimile of the regular 5pointed star. 
How excellent a facsimile it is can be seen upon merging it with a true exact star. The result is below. an Eye Opener In the lower corner of the Square, the Square's golden circle with an inscribed square forms an orderly harmonious position with the circled containing square for the feet (the Footsquare). This brings one thing to our attention  the golden circle and its inscribed square are meant to be exactly the same as the Footsquare with its circle. 
When
concentric, both circles and their inscribed squares look like the
diagram below. Although there are two separate circles/squares here, we
only see one. 
Below: The design by itself. The containing square of the feet is the same as the square inscribed into the Square's golden circle. This position implies a specific 5pointed star construction, also shown below. Altogether, it takes 13 steps. 
Construction
of the 5pointed star in 13 steps The diagram above shows the first six steps. Step1 is a horizontal line, and already an arm of the sought after star. Construction of the 36degree angle step
7: Draw a
line between points C and 2. Construction of the regular 5pointed star _ steps 11,12,13: 
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. 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 TransAtlantic Connection & the Footsquare The Footsquare (containing square for the feet) 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 Athenaengraving's basic geometric system, I had to wonder if Athena's system included the Footsquare idea as well. It turned out that Athena gives this idea quite an indepth treatment. The Athenaengraving already has its Square; so we just add the 13step star with the Footsquare 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 Nazcamonkey, 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. The circle around the Footsquare (the Square's Goldencircle) clearly centers upon Athena's right leg below the knee. Whereas at Nazca the Footsquare 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 Footsquares. The Footsquare, along with the smaller squares inscribed within, is clearly customfitted 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 Footsquare I also tried moving the Footsquare 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 Footsquare; * 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 threetoed; so this 'coincidence' cements the special relationship between these figures. an unexpected "coincidence" For good measure, moving the Footsquare 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). the same square expanded into a golden rectangle The head area as a whole is a very good fit to the rectangular form based on the Golden Section: two side by side squares sitting on top of two upright golden rectangles. 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. 
So, overall, the experiment of testing the concept learned from Nazca to the Athenaengraving worked out beyond expectation. I had found three prominent instances of the Footsquare 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 Footsquare phenomena in Athena. I got the initial Footsquare 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 Footsquare is inscribed in the Square's goldencircle used in construction of a 5pointed star; so it's natural to experiment by also inscribing the circle around the Footsquare with a star of its own. The Monkeytree The experiment then presented a view of striking harmony between this star and the engraving. A detailed description here would be counterproductive; instead, I marked most, but not all, instances of it by arrows. Still, some of these correlations are simply in the mustmention 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 36degree cones. Lines b & c actually give two 36degree cones oriented tip to tip on the same axis. This is a dejavu of the Nazcamonkey's XTree! Along with Athena's threetoed boot, this reoccurrence of the XTree (call it Athena's Monkeytree) is overwhelming evidence that the two works come from the same source. Line 'c' of the Monkeytree is given by the pentagon inscribed into the golden circle. Accordingly, when assigning stars to the Monkeytree, let's make their sides the same length as on this pentagon (see above). Mirror these Xstars as in the above illustration. The new stars reveal the exact positioning of the second Footsquare. It begins with line 'e', which, by the way, the engraving also echoes; and which also passes through the immediate area of the Square's easternmost corner. Above: The small halfgreen circle on the right marks a point where star lines associated with the Monkeytree system intersect at the edge of an engraved line. It is the same as the midpoint of the right side of the second Footsquare which I first positioned there by eye. Now it is the true insertion point for the second Footsquare. For confirmation, a line drawn down at 45degrees from this point is an accurate boundary to the three toes of Athena's right boot, just like before. Above  a detailed look at the two Foot Squares: The direct connection between their centers is immediately interesting because out of two chances at becoming an accurate limit to engraved lines, it does just that. It limits the reach of the Monkeytree downwards on the right side; and it limits the reach of white space between the heel and the sole of Athena's right boot. Of course, two out of two is yet another attestation to the direct connection between the two Footsquares. The first time, it was the circled Footsquare which based entirely on the goldencircle of which it is a duplicate. The same principle applies the second time as well. The other Footsquare is entirely a product of the first. 
` 
Athena's
head & the Footsquare  the Halocircle 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 Halocircle and combine to give us the insertion point for the Halocircle. 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 Halocircle/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 Halocircle 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 As we all know the credit for the column's idea goes to the Nazca Monkey. This column has some surprizing qualities. The Square's 'y' diagonal sets the column's width. Its height is also derived from the Square: Mirror the top half of the yellow Square upwards, and its top line will mark the highest reach of the (white) internal space in Athena's head  the top of the column. The bottom line of the Footsquare sets the bottom of the Square's column. There are two more columns to look at here  one created by the golden circles centered in the top and bottom corners of the Square, and one created by the squares inscribed in those circles. The engraving is visibly attuned to all the three columns. ABDE in the diagram marks a golden rectangle, one of several in view. 
Virtually Exact
Starmaker 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 24hour 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 motherstar, 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 rectangle's long axis crosses the describing circle; it creates an angle with that axis which is exactly half the angle that axis holds 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 headtotoe 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 5pointed and five 6pointed stars will be in sight in the end. Monkey's Impact Without help from the Nazca Monkey, as long as there was just the engraving, I was being accused of inventing its Cone & Square spirit. Therefore, the Nazca Monkey coming to the rescue was like a miracle. 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. Moreover, it guided me to some of the engraving's secrets, which I had been oblivious to up till then. Any
number of designs claiming descent from a particular
fivepointed star can be brought to the same scale and unified for
comparison
with the others by that star. 
Geometrical
Modules in the Torso Most of the lines in the torso create significant angles with the Square's diagonals x, y. These lines together with the Square's diagonals create rational order, a geometric module. The straight lines 'c' and 'a' subtend irregularly curved engraved lines. Drawing subtending lines presents a way of finding a simple meaning for complex multipurpose lines. In contrast, the straight lines 'e', 'b', and 'd' stay within the engraved originals. From left to right: Line c holds a 30 degree angle with the xaxis. Line e holds a 36 degree angle with the xaxis. Line a holds a 36 degree angle with the yaxis Line b holds a 36 degree angle with the yaxis. Line d holds a 30 degree angle with the yaxis. Line f holds a 36 degree angle with the xaxis Lines a and b create a triangle with the Square's yaxis, which is found on a regular 5pointed star (36x36x108 degrees).
I
had used compasses to
develop the
triangle into a 5pointed
star, utilizing the blue circle from diag. 19 below.
Right away, what was a simple though artificial idea, advances into the complex category.

This pentagonal side of the Pyrostar coincides with the 36 degree angle constructed from 'D', and is a tangent to the so called Golden circle centered in the opposite corner of the same square (see below). The Pyrostar, and the Square's minor Golden Rectangle This
Golden Rectangle (actually white in
the diagram) has the length
of the Square's diagonal, and the height of the Golden
circle's diameter.
It is safe to say
that it fits the engraving beautifully, see the second diagram
down. 
The
big circle in the diagram below divides the Square's horizontal diagonal
in the
Phi ratio, and sets radius of the Golden Circle, It is
centered at the 3/4 point of the Square's horizontal diagonal, and
passes through two corners of the square. segment R00 / seg. 00G = seg. 00G / seg. GL = seg. GL / seg. LM = 1.6180339887.. = Phi The torso's width is well contained within the band of the Golden Rectangle's width (the height of the Golden circle). This is clearly visible. 
Nested Circles 
Above
we can see a strong indication by the engravers of the Golden
Section in process within the Square. The big circle is one of the four such possible circles (one for each quadrant), and it definitely fits the engraving, in at least twelve places. Refer to it as the 12point circle. Concentrical with the 12point circle, is the small brown circle in the diagram below. Its diameter forms the Golden ratio with the diameter of the Golden circle itself, as seen from the star in the diagram above. Like the 12point circle, this small circle also fits the image neatly. It is wedged between the lines a and b (which create the Pyrostar), and lines d and g. So far, the engraved line 'd' translates into a straight line passing entirely within the engraved line. But, one diagonal of a Golden Rectangle originating from the point D of the square, subtends the engraved line 'd' with good accuracy. An engraved line of the Torso is symmetrical with the line 'd' across the xaxis. So, the enclosure of the small brown Golden circle is consistency itself. One Golden circle and four lines, each having to do with the Golden Section within the Square. 
The Square's major Golden Rectangle This is the Golden Rectangle, whose height equals the diagonal of the Square. 
Global Connections This chapter gives more examples on how one geometrical system unites two ancient designs, each from a different continent and age. An unidentified prehistoric agency has had global influence over our planet and created art and architecture as mathematical puzzles for future generations: We look at: a) the monkey figure from the world renowned Nazca Lines, Peru b) a 14,000 years old engraving from the not so wellknown grotto of La Marche, France The elements common to Athena and the monkey tend to lurk at a different depth in each. When observing a given property in one work which escapes notice in the other, going back to check for its presence there often works out. The same is true about going back to these two works with the Giza layout  it was a revellation. Therefore, the three are best analyzed together. A star brings us together The monkey from Nazca spans a hundred meters, while the engraving from La Marche tangles its lines over a portable A4 sized stone tablet. Each had gone through being a 5pointed star in its embryonic stage; duly, the two can be scaled 1 : 1 by scaling the works until their motherstars become the same size.The two works can be merged for comparison by fusing their motherstars into one. This idea is implemented in the image below. Out of the three Giza pyramids only the Great Pyramid si shown, in order not to clutter the view, The original star is divided into left and right halves by a central axis. Athena's head and the Great Pyramid are to the left of that axis; the monkey's head is to the right, and is away from the pyramid. 
There is some duality in view: The heads are at approximately the same elevation, and a tip of the star is inserted in each head. Face Merging These attributes are reflected across the stars vertical axis  so, why don't we mirror the entire monkey across this axis to see what happens? The outcome is spectacular in that Athena's face is now almost onehundred percent inside the monkey's face, and her right eye now doubles as the monkey's right eye! The spiral tail looks quite well fitted to Athena's feet. 
In the next experiment; without any rotation whatsoever, we move the monkey's head over the head of Athena to see how the two fit together. A Perfect Fit We could not have hoped for a better result! It looks like the Agency had built in confirmation of our scaling procedure. 
Except for a tiny sliver, Athena's face is once more completely contained in the monkey's head. Its hands, esppecially the left one, are clearly attuned to the engraved lines of Athena. The space around the star's center can be seen as a dial divided into multiples of 9 degrees  angles created by the pentagram. In that case, a rotation of 54 degrees is one of the expected possibilities. As well, the right side of the Xtree  both its straight part and the triangles to the right relate to Athena's lines in a significant manner. 
The Best Circle Expressing the Arc of the Monkey's Cranium A parallel to a side of the Motherstar drawn from the Handcircle's center draws attention because its passage through Athena looks nonrandom, as it passes through the corner of the pyramid pictured in the torso and rests on a couple of arcs. the last of which belongs to Athena's helmet. Since this parallel looks to have some dignificance and cuts a golden triangle out of one tip of the star, I experimented by completing that triangle into a smaller star. One of its arms then merges with a line of the torso, and the engraved line next to its right passes right through the small star's center. 
But the same line is already a side of another star, as seen in the image below, so this is interesting. Moreover, the same line also passes right through the center of the monkey's head. The same line serves as a line of centers for three circles: a) the newly created star's circumcircle b) the same size circle is the best fitting circle for the circular top of the monkey's head c) a smaller circle expressing the arc of the cupola on top of the torso disc The same circle, best expressing the arc of the monkey's head, is seen in greater detail below. 
More fantastic than sciencefiction, the high level geometric union between
the Athenaengraving and Nazcamonkey is a fact of life. Yet, it would
not be complete without the ground plan of the three great pyramids of
Giza. The three are one  a trilogy. This trilogy could not exist
without having a shared backgroundof advanced knowledge and technology. 
Index Page  
Exact Reconstruction of Petrie's Giza Ground Plan 



Testing the Nazca Monkey for Connections to the Great Pyramid  
A Long Prehistoric Message in Thirteen Numbers  The Frame  
The Frame  the HexMachine  a family of three hexagons  
Abydos Helicopter & the Golden Section  
Hesire's Tomb Door  
Giza Pyramid Temples & the Golden Section  
Next  Nazca Monkey Report 