Connect the Dots
Drawing the external boundary for the Stone Age Athena-engraving is rather easy due to the existence of clear main peripheral points. These thirteen points are then connected by lines.
This is the
'Frame' - as simple a step as can be
checking the image for hints of planned layout.
It was 1985, at
the end of
Stone-Age for computing masses. I was one of the primitives with no
scanner, nor computer to help me with graphics. Life-size
copies of the picture were too small for me to
do accurate work on with the
classic ruler and compasses, and so I got a
sheaf of 2:1
blow-ups from the nearby printers. Did the ancient agency foresee that
someone will resort to just that one day, due to limitations of human
vision? One way or another, it's reasonable that the units should
At double-size, the
engraving's units of length equal
the metric system.
my readings were identical to the designer's intended values!
measurements were rounded to the nearest
millimeter, the finest detail available on my ruler.
The Game of Quotes
At the first glance, the
thirteen whole numbers ranging from 16 to 175 - the distances
between neighboring points of the Frame in millimeters - are no big deal. But then one starts learning that these numbers are simply ideal for the purpose of quoting Pi, Phi, and rates of Equinoctial
Precession as many times as possible, and as far as the
- rates match today's
state of the art measurements.
I'm convinced that no other set
of thirteen whole
numbers in a comparable range can rival the Frame at this type
of communication. Clearly, its
designers had to be highly sophisticated, and also
of astronomical instruments at least equal to what we have
Could the Frame just occur by chance?
In that case
many possible combinations are there of thirteen
the range from about 10 to about 180, as long as their total
falls somewhere between 1,000 and 1,300, and numbers can appear more than once?
Given such conditions, any mathematician realizes that
the odds against coming across the Frame by accident are
mind-boggling. There are sextillions of such groupings of numbers.
But the order in which segments appear in those groupings matters as well. Each of the
sextillions of sets of thirteen segments can be arranged in 6,227,020,800
the Frame's functions cannot be improved by rearranging
and replacing some, or even all of its
well, then the Frame must be the best solution in its category among Nonillions
(10 to the 30th power) of competing combinations possible.
This truth no naysayer dares to challenge, facing the disconcerting reality of being dumber than
alleged 'blind chance'.
Game Rules and Pieces
can scramble our Frame into some 4,000 unique combinations of
segments, but we can break it up into only 156 unique
between two Frame points (a piece is either a
single segment, or a
sequence of neighbouring segments).
That's where we
ought to look for
In addition, there is
something called 'the Strong
between Frame points B and G. This connection proves of great
in a number of ways. Consequently, it counts in with
additional Frame sequences. This, I
believe, adds another 96 combinations.
If so, there are 252 total combinations
sequences of segments make sense when subjected to simple
manipulation like addition, subtraction,
multiplication, or division.
The rule is that any gamepiece
for the next move must be either a part of the present gamepiece, or
immediately adjacent to it. Segments connected across the Frame by the
points B and G (16 & 175 across to 113 & 146) are considered immediately adjacent, as well ( as if through
subspace, a worm-hole ).
We can look at the Frame's numbers in at least four ways:
1) as is
2) ordered by segment size
3) ordered by unique segment size
4) as the segments appear on the 'Wheel of 113' (a pie-chart of 113's moduli)
Opening Moves (beginner
The Frame serves
as the doorway into the picture, but there must be something
to entice us into entering, some
attention-getter, a sign above the door. Sure enough,
no one can miss
Section of Regular Proportions
The section of five segments spanning the points
M exhibits regular proportions!
1-2-4-3-1, as well as the actual
numbers - all multiples of 9 - look deliberate.
The Frame lends itself to
transformation into a
81 & 27
The four segments to the left of 27 add up to 270, and the eight
segments to the right of 81 add up to 810 - two segments and their tenfold multiples side by side.
The twelve segments involved add up to 1080, a tenfold multiple of 108,
which we have twice (108 & 81 + 27), twenty-fold of 54, and
forty-fold of 27 - all these distances are inside the stretch of 1080.
Below: And then we see more regular proportions: two neighboring sections
measure 340 each - those are followed by two sections of 108 each. A
middle segment in the section of three (80-113-146) equals one third of that
section, etc. Who'd guess that looking into ratios among these segments will be so entertaining?
Pi & Phi
The number 16 could represent the first two digits of Phi, the acclaimed Golden
and it is the base of the hexadecimal
A check of the next door neighbors of 16, reveals that
they total 314. Of course, those are the first three digits of
the best known ratio of all.
Phi is embraced by
Pi here. Is this a sign of things to come?
Without these three segments, the entire rest of the Frame forms several Pi proportions. In efffect, the above quote of Pi represents an answer to the question posed by those proportions!
Twice 340 (680), next to twice 108 (216) The
arrangement is shown below.
is the ratio between the groups?
The first three
digits are those of Pi.
That's two approximations of Pi
(680/216, and 340/108) in one fell swoop.
The second 340-section (clockwise) is next to
the segment of 80. Their total of 420 also has a factor pair of 3 x 140,
without the multiplication symbol.
the Question & the Answer
The open ended Pi
sequence of ten
consecutive segments adjoins
the segment of 139 on one end, and
the segment of 175 on the other. Those two segments total 314,
the first three digits of
360 degrees of order
After the above mentioned 314 comes 16,
the last segment left in the circuit.
16, or 31416
- Pi rounded to five digits. The entiree Frame circuit is devoted to Pi!
Quoting Pi to twenty-eight decimals
Just by themselves, the last three segments (139 & 16 & 175) enable a somewhat refined result.
175 + 139 = 314
175 - 16 = 159 314 159 the first six digits of Pi
The arangement is planned, and there is a lot more confirmation still to come. The immediate vicinity of 16 & 175 is rife with Pi relations.
First of all, there is yet another way here to arrive to the above
result, and beyond. We worked with 16 & its short wings, now
we begin from:
& two segments to its left
16 & 175 & two segments on each side
27 & 139 millimeters translated to centimeters in whole numbers becomes 3 & 14
314 __ the first three digits of Pi
Of course, 16 is next: 3-14-16..
Pi rounded to four decimals, but we are after a greater prize.
After 27 & 139 (3 & 14), there come the Strong
Connection segments 16 & 175. The difference between the latter is 159:
3 14 159 the first six digits of Pi
After 16 & 175, we have the pair of 113 & 147, which totals 260 millimeters, or 26 centimeters.
3 14 159 26
the first eight digits of Pi
Composite Numbers and Phi
Before going on with Pi, let's take a look how the very same location also delves into Phi. Subtracting
from 175 on the left helped
us with the Pi's first six digits. Now, subtracting
from 175 on the right (175 - 113) leaves 62 - the fraction of Phi rounded to two decimals.
Does Phi find additional
support here? The two
to the left of 16 made a nice play on Pi. Could the two segments to
the right of 16 possibly make a
nice play on Phi?
113 = 288 = 16 x 18
So, 16 is contained 18 even times in the next two segments.
The Ancients seem to be trying to bring our
composite numbers. Although 288 breaks down into various multiples of
whole numbers, the presence of 16 before 288 indicates that if
look at 288 here as a composite number, it should be in the
16 * 18 Without the multiplication sign:
first four digits of Phi!
175 millimeters as a whole number in centimeters is 18.
In that case we have 16 followed by 18 ___ 1618 again!
Back to Pi - the Wings of 260
Being on 260
(113+147) we have 175 on the left, and 80 on
We just saw 175 work with 113 (and 16) as part
idea of showing a composite number as a pair of factors: 288 = 16 x 18
175 works by itself too, as a pair of factors (35 x 5) in the idea 355/113=3.141592...
Of course, this pair of factors can be reversed to 5 x 35, instead.
= 5 x 35 (535)
Pi = 3.1415926535..
The right wing of 260 is
80 millimeters or 8 centimeters long.
A second pair of factors in the progression
The three Pi digits after 8 are 979.
To the left of 80, from 113 to 16:
These nine segments
units each (9 x 79)!
113 + 146 + 27 + 54 + 108 + 81 + 27 +139 + 16 = 711 = 9 x 79
In reading Pi out this far, to fifteen digits, we have
entire Frame, once again.
To extend the Pi sequence by three more digits, we need a 323. Curiously, 711 translates as either 9x79,
and no other composite factors. That would be six digits of Pi in a row, 979 323. Unfortunately, the
seventh digit, 7, is off by one; so this doesn't count.
Is there a legitimate way to get the digits 323?
Right now, we are parked on 16, at the
point B. We've been here before. This is also a crossroads, where the Strong
Connection cuts acrosss
the Frame (image) to G, as if through subspace, a wormhole. See more on
Strong Connection below - use this anchor.
The image below shows the B-G line connecting through the central point of the entire
engraving - the
origin point of the square, and apparently, of the blue pyramid's vertical axis.
This is how the Frame acquires another loop, and 16
& 175 become directly connected to 146 & 113, and
likewise, to the 339-sequence of 146 & 113 & 80.
The Strong Connection and the 339-sequence are going to be instrumental in all the upcoming cases of further Pi digits.
It's only fitting that in the pie-chart shown below, 16 rises symmetrically over the 339
The symmetry between the two is
the best possible (the intervening spaces are 436 and
So we ferry 16 over to the opposite side via the Strong Connection, in order to diminish the 339-sequence.
- 16 = 323 ( Later, 339+16 also plays a role.)
Next Pi digits are 8 462 64 33.. It just so happens that 80 millimeters, or 8 centimeters, part of
the 339-sequence, is once more the immediate continuation.
We still stand on the 339-sequence; next up are its wings.
On the Wings of the 339-sequence
The three segments wedged on the right between the
339-sequence and 16, with the segment on the left of the 339 sequence,
add up to 462.
And again from the 339-sequence, 80, its part, minus the 16 from across the S. Connection gives 64.
Now from 16, add its two wings of 175 & 139 to it: 314 + 16 = 33(0)
We could get the same 33 within the 339 sequence by subtracting 80 from 113.
A third pair of factors
Next to 139 is 27; together, the two make 166, whose factors are, among others, 83 x 2.
Our readout of Pi reaches twenty-eight Pi decimals. IThe
Frame is also a mnemonic aid which enables me to remember twenty-eight decimals of Pi. That's
a dramatic benefit from what naysayers allege are wild scribbles by
the Frame in both Phi & Pi
We did more than a couple of
complete circuits of the Frame dealing with Pi, and
on top of
that we got started with Phi as 1618.. The last Frame part
we used was 113 (in 175+113=288 = 16 x 18).
Does 113 have anything to do with Phi?
= 1.6180 339 887..
where we find 113 in
= 113 x 3 and
887 = 1000 - 113
Judging by the above, the answer is a conditional yes.
Fascinating, two Frame parts are 113s, and the Frame
less those equals 1,000
On the Strong Connection: 16 & 146
+ 113 = 259
259 / 16 = 16.18..
the first four digits of Phi
In this position, it's hard to miss
that the Strong
Connection pair of 146 & 113 has wings of 27 & 54 totaling 81 on
the left, and 80
on the right, whose total is :
the first three digits of Phi
Of course, 259 divided by 161 is 1,6.. the first two digits of
Our readouts here seem repetitious.
Although these 81 & 80 segments are not direct neighbors, they
enclose a section which addresses the same subject as they do - Phi. Moreover, the step taken is just the
first in several steps occurring in this immediate area, which produce
a quote of Phi to ten decimals.. In my eyes, these facts legitimize taking
this step of proceeding from a shorter quote of Phi to a more elaborate
So, let's write 161
Of course, 80,
on the right of the Strong Connection, is part of that 161.
Write down 161 & 80 ________ 161 80
80 is part of 339
Write down 161 & 80 & 339 ______ 161 80 339
the first eight Phi digits
339 is surrounded by the rest of the Frame _ 887 (The Frame totals 1226)
161 80 339
887 _____ the first eleven digits of Phi!
Some may favor a different method of arriving at the same quote, one in
which we descend through the progression:
We start with 887 & 339 ____ the Frame divided into two sections.
In the 339-section, we focus on 80, which is part of 339. That 80
with 81 to the left of 339 gives 161. The latter method seems
161 Read from the bottom up 161 80 339
887 eleven digits..
Once we get to 16180, namely 80, we note that it has a wing of 340 on
its left; and as for itself, it's part of another 340 section, a direct
neighbor of the first. Is 34(0) somehow significant in this context?
1618034 gives Phi rounded to seven digits. 161 80 34
There is an obvious inner
divide in the image, it's the line subtending
both the 339 and 887 sections.
Take a good look, from E to H,
whenever this line passes near a point in the image, it then finds the point - eight consecutive times out of nine
possible, and the miss is by only a little bit.
This time, the already familiar 339-section performs feats of geometry. In
the image above, the
girl is shaded by a yellow umbrella - otherwise, the 339-section. The
angle FGH in this section is very close to being 120 degrees. Look
at the shaft of the umbrella between B and G - the Strong Connection - a perfect arrow through the
- the red
crosshairs - the center of the Square . The connection divides the 120 degree
the middle into two 60 degree angles.
Below: Connect the center of the Square to E, the start of the 339-section. This line then creates an equilateral
triangle with the lines through BG, and FG.
How nice, the head rests on a triangle's basis, and
is about equidistant to
the other two sides.
simply begs experimental completion into a hexagon centering in the 0,0
point, the center of the Square.
Thist hexagon then fits the figure
from head to toe.
the containment of the figure within the envelope shape. The center of
the first equilateral triangle falls neatly onto Athena's forehead.
And later we learn that the hexagon is the offspring of two other