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
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
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
Châteaux, France, is basically the same puzzle; and it comes with a complete set of clues for the basic solution.
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
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.
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
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
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
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,
Giza - advertizing its power and quite
posssibly offering an explanation to other historical enigmas like the
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
mathematical order can be found practically everywhere, in any random
collections of lines, points, ancient art, cloud
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.
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,
of it should be meticulous:
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
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.
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.
order in the monkey figure.
Orientation _ Standing Tall
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
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
noted these two alignments.After all, we are working with her copy of
the glyph; and she had taught mathematics in
Why did she not mention these alignments specifically?
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
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.
spirals around the head,
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 may be perceived
of 5-pointed stars
alignment. The question is: If
so, then what size are those stars?
The pentagram below the
The second longest line of the glyph -'c'
- cuts across the
extended line 'a'.
Let this cut set the size of the experimental star
Next, mirror this star above the X-point, then
the move to get the diagram below.
The star inside
the big star is remarkably
well centered upon the monkey. The entire star
well fitted to the X-tree.
( 'c' differs from a
pentagonal angle by an even
two degrees, good to know for reconstruction purposes )
big 5-pointed star above the
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
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.
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,
the purple star's tips actually mark the
monkey's head, hands, feet, and tail. This
will be the Monkey Star.
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
sinister - like a
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
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.
we see here is what we saw in the French
the design uses the same standard circles set by a 5-pointed
inner-circle is one of these standard circles (Triplets) of the
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
Its center is on the
vertical line b1 which descends from the Monkey-star's tip just above.
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
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.
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.
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
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.
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 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
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
Rounding up this number to seven decimal spaces will yield seven sevens in a row..
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
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
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.
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 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.
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
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
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
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
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?
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?
As seen in the diagram below, the experiment worked wonders.
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.
circles reveal the secrets of Cone's construction.
the Middle Cone circle
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...
are two ways to recreate it:
a) The upper intersection of the third row of unit
locates the exact mid-point on
the Cone's axis
between the Cone's bottom tip and the center of the big top
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
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
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.
three works are three chapters of the same book.
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
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
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
of the Great Pyramid's cross-section is certainly noteworthy, because
Pi is frequently discussed in studies and polemics
The curious fact leads to the observation that:
= 9.87 , and φ
- 1 times 16
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.
Placement of the Square on the Monkey
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.
of the four cones shares its central axis with the X-tree,
and its sides
parallel to X-tree's lines (diag. above). Therefore,
it is the first pick as the experimental carrier of 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
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
Adding the Square to the Cone was the obligatory next step,
and it drew back the curtain on extraordinary harmony
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
Being so important, was the idea also included in Athena, but hard to spot like the Square in the
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
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
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
Pyramid. The clarity of that fact is beyond crystal-clear. Therefore,
the Giza plan was already around some
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
Descartes "Cogito, ergo sum" - it is cogent; therefore it exists. It is
the mainstream historical science which has to reconcile itself to this
And so I addded the Square to the Cone. See
the two diagrams below. In the first one, the peach colored diamond is
Square, as set by the blue Cone. Already at the firs glance, there is much to be pleased with in the result.
diagonals are oriented to the cardinal points!
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
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.
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.
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
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.
master of squares
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
of the arms (East-West) = half
the monkey's height
of the feet (East-West) = half
the width of the arms = one-fourth of the monkey's
of the left foot (East-West)
= half the width of the feet
of the monkey's height
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
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).
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.
of the right hand
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.
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.
|The two stars are so
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.
of the 5-pointed star in 13 steps
The diagram above
shows the first six steps. Step-1 is a horizontal line, and already an
arm of the sought after star.
of the 36-degree angle
line between points C and 2.
circle centered in 'C' through the intersection of the blue circle-2 with the
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.
of the regular 5-pointed 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
equidistant to the points 3 and 4.
For the final two steps, draw lines from 3 and 4 through C to
the horizontal line from step 1, and the pentagram is
for steps 11,12,13:
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).
Trans-Atlantic Connection & the Foot-square
Foot-square theme seems to be the culmination
geometry. Not forgetting that success of my
was due to the hypothesis that the monkey's geometry duplicated the Athena-engraving's basic geometric system, I
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
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.
The circle around the Foot-square (the Square's Golden-circle)
clearly centers upon Athena's right leg below the
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.
The Foot-square, along with the smaller squares inscribed within, is
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
* 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.
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.
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
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
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"
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).
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
previously discovered material, it was enough confirmation for the
hypothesis. Satisfied, I went on to other things.
Following the script
Years later, and after much progress on other fronts, I returned to
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
it's natural to experiment by also inscribing the circle around the Foot-square with a star of its own.
experiment then presented a view of striking harmony between this
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
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'
Lines a & b and b & c create 36-degree cones.
& c actually give two 36-degree cones oriented tip to tip on
This is a deja-vu of the Nazca-monkey's X-Tree!
Like Athena's three-toed boot, this X-Tree (call it Athena's Monkey-tree) is another sign of a common source for the two works.
'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
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.
The small half-green circle
on the right marks a
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
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
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
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
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 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
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
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.
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.
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.
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
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
| 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
and saw one of its arms merge with a line in the
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
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
For starts, let's go back to the familiar idea of
mirroring, and mirror the monkey across the vertical axis of the original star.
| 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.
|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.
hands are completely attuned to Athena's lines; it's as if they were in control of an intricate instrument.
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.
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
came from the engraving, as well. However, once we add it into
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
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.
my bewilderment at discovering, many years later, that the
"13-step" method of pentagram construction makes possible an
reconstruction of the ground plan of
the three great pyramids of Giza, as surveyed by Petrie! In
that fact called for testing both Athena and the monkey for
presence of the ground plan of Giza. And again, this backchecking
brought highly positive results, as you will see.
of the Monkey Frame
triangle inscribed into the rectangle in the diagram above is
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.
scale model of this situation is very reproducible from memory This
is the second such scale model we have for the monkey.
idea seems to be that the Monkey Frame's intended height should equal the
perimeter. (4x one side of the
square inscribed into the Golden Circle)
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
where the inscribed triangle is exactly 36 degrees at the
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
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
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
are dime a dozen, often on the borderline 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
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
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
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
as if they weren't there?
Perhaps, a line crossing over a figure
doesn't make the line ages younger, and irrelevant to
Aveni says that Nazcans had traditionally
associated spiders with
water, hence the giant spider glyph keeps in character with
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
by walkathons within a symbol of love for dryness in a place
meant by higher powers to be rainless
There are other ways to speculate, too: After a
of generations, Nazcans would learn that rain over
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.
everything at Nazca is interconnected, the glyphs may be telling
a very long and detailed story.
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
The blue vertical line is the Square's diagonal extended to exactly
one-and-a-half times. Its end-point is positioned with microscopical
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.
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
Meanwhile, some others are trying reconstructions of Nazca figures in their own style:
Joe Nickell, chose primitive methods to emulate the ancient Nazcans. He
does not think Nazcans could measure angles!
appears to be no evidence that the Nazcas had such a capability"
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
Monkey-tree. Vertical lines y & z drawn down
on this star are absolutely precise limits for the
engraved line at the front of Athena's
foot. The line x is a perfect boundary set by the
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
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
CAD drawing, has a grey band of uncertainty at line edges
a very high resolution. The thickness of this band is roughly
or two hundredths of a millimeter, or ten to twenty microns.
square expanded into a golden rectangle
|The bottom line of
the golden rectangle nestles neatly atop an engraved area
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..
the height of the head is Phi + 1 ( 2.618..), and its width
is 1 + 1.
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
screen, I hope, for the sake of having a good view of the
|The top corner of
is situated on Athena's face. Therefore, I wanted to see what
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
|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
points. For instance, line "x" is a horizontal line through
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
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.
The Horse Debate
The Giza Plan