Connect the Dots
It is rather easy to draw a boundary
of straight line segments between the main peripheral points of this Stone-Age engraving from the rock-shelter of La Marche.
This is the
'Frame' - as simple a step as can be
towards checking the image for hints of planned layout.
Beginner's luck
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.
It just so happens that at double-size, the
engraving's units of length equal the metric system. Now,
my readings were the same as the designer's intended values. The
measurements were rounded to the nearest
millimeter, the finest detail available on my ruler. Was this a
phenomenally lucky coincidence, or, was it because the designers knew
the human limitations, and because we inherited the prehistoric
units of length?
The Game of Quotes
At the first glance, the thirteen whole numbers - the distances between points of the Frame in millimeters - are no big deal. Yet, if our goal were
to show off to keen observers one's advanced knowledge
of Pi, Phi, and Equinoctial Precession, then these numbers would be the ideal choices, presented in their ideal order.
The Stone-Age designers made the search for secrets in the Frame into a bona-fide
logical game of numbers. As such it also has set rules. Among the
objectives - to quote Pi, and Phi, and rates of Equinoctial
Precession as many times as possible, and as far as the following:
Pi =
3.141592653589793238.. Eighteen decimals
Phi =
1.6180339887.. Ten decimals
Equinoctial Precession - rates match today's state of art measurements.
This set of only thirteen whole numbers ranging from 16 to 175 accomplishes all that. Clearly, its designers had to be highly sophisticated, and in
possession
of astronomical instruments at least equal to what we have now. This insight leads to the conclusion that the Stone-Age site of La Marche had been tampered with,
or even entirely staged fourteen millenia ago, in order to provide medium for camouflaged
science-art.
I believe that the Frame holds the universal patent on this type
of communication regarding the two most fundamental mathematical
ratios - Pi, and Phi. It also holds the global patent on Equinoctial Precession riddles. No other set of thirteen whole
numbers can rival
the Frame in usefulness, because the Frame seizes
the best available opportunities. Could this be by chance?
How many possible combinations are there of thirteen whole
numbers
in
the range from about 10 to about 180, as long as their total
falls somewhere between 1,000 and 1,300? Each
number can appear more than once, and the order matters as well. In other words, the odds against are stupendous. The Frame is the best solution for the above stated objectives among octillions of competing combinations possible.
The Game Rules, and Gamepieces
We
can scramble the Frame into about 4,000 unique combinations of
segments, but we can break it up into only 156 unique pieces
(a piece is a segment, or a sequence of neighbouring segments). That's where to look for rational meaning. In addition, we have something called 'the Strong
Connection'
between points B and G. This connection proves of great importance
in a number of ways. Hence it connects a few more segments across
the Frame in an equivalent of a direct connection. Sequences of directly connected segments become gamepieces, or logical
objects, or modules, when they make sense in the context with other gamepieces, when subject to mutual addition, subtraction,
multiplication, division, and even rounding. The main rule is that the gamepiece
for the next move must be either a part of the first gamepiece, or
immediately adjacent to it. Segments connected across the Frame by the
points B and G are also considered immediately adjacent. ( as if B and G were connected through subspace )
The Frame plays at least ten major games
Game 1 - 360 degrees of order symbolizing Pi
Game 2 - Quoting Pi to eighteen decimals
Game 3 - Quoting Phi to ten decimals
Game 4 - The Strong Connection (all combinations work, plus, a Pi approximation good to six decimals)
Game 5 - Frame arranged by segment size (Zodiac and Osiris numbers)
Game 6 - Frame ordered by unique segment values (Zodiac and Osiris numbers)
Game 7 - Equinoctial Precession value quotes on three levels of accuracy - (Zodiac and Osiris numbers)
Game 8 - Cyclical fractions and the 'Wheel of 113' (the cycle as a pie-chart of moduli)
The Frame also presents geometry games
Game 9 - The Hex-machine (a grand design of three generations of 6-pointed stars
Game 10 - A system of two 5-pointed stars
Within these major games we find plays, which could be seen as complete games by
themselves. For instance, the Frame as eleven unique values can be
divided into two neighbouring groups, one all Osiris numbers, the other
all non-Osiris numbers, whose underlying order is all Osiris values,
however.
Game One - Opening Moves (beginner level)
The Frame serves as the doorway into the picture, but there must be something to entice us into its game, some
attention-getter, a sign above the door, and a handle. Sure enough,
there are at once several opening moves, all in one big section.
The
Section of Regular Proportions
The section of five segments spanning the points H to
M exhibits a variety of regular proportions. The relative distance
intervals are: 1-2-4-3-1, or
(27-54-108-81-27).
It
is a certainty that these whole proportions, as well as the actual
numbers look deliberate. That makes the Frame legitimately interesting,
so we may feel justified in studying it some more.
The Frame lends itself to transformation into a
pie-chart
Pi & Phi
The number 16 is of interest in part because it could represent the first two digits of Phi, the acclaimed Golden Ratio, and it is the base of the hexadecimal
system.
A check of the next door neighbors of 16, reveals that
they total 314. Of course, those are the first three digits of Pi,
the best known ratio of all.
We have here Phi embraced by Pi. Is this a sign of things to come?
Without these three segments, the entire rest of the Frame forms a regular pattern, and this pattern again symbolizes the Pi!
Twice 340 (680), next to twice 108 (216) The arrangement is shown below.
What
is the ratio between the groups?
1
Pi (3.14.....)
!!!
680/216
= 3.14...
The first three
digits are those of Pi.
2
Pi (3.14.....)
340/108
= 3.14...
These two shorter sections nest within the larger order. Things
are now interesting.. We have at once two approximations of Pi
(680/216, and 340/108).
3
The second 340 section (clockwise) is next to the segment of 80. Their total of 420, is a composite number expandable into 3 x 140,
or 314
without the multiplication symbol.
4
Pi (3.14)
So, 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, totallling 314, the first three digits of
P.
This 314 expresses
the ratio found twice in the ten segment section, but
on the
scale of 1:100. Allowing decimal
shifts of scale is a necessary part of this method. Otherwise, 3.14 units
in lifesize would be too
miniature ( .14 of half a millimeter equals .003 of an
inch.
5
360 degrees of order
Pi (3.1416)
!!!
Embedded
in 314 is 16,
the last segment left in the circuit.
We have
314 &
16, which symbolizes 31416
- Pi rounded to five digits. Onee circuit
around
the Frame devoted to Pi is complete! This game is a win.
6&7
Game Two - Quoting Pi to eighteen decimals -
Opening Moves
Pi (3.14..) !!!
& Pi (3.1416)
!!!
Let's shift our focus one segment to the left in
the counter-clockwise direction.
The
millimeter values 27 and 139 round to centimeters as
3 and 14.
This
is symbolization of Pi by a two segment sequence. but
once we acknowledge, we cannot ignore the 16, which
is next to it : 3-14-16..
Pi rounded to four decimals.
8&9
Pi (3.14159..)
!!!
The above three segments with the addition of 175 make
sense as two pairs.
The first pair says 3_14.
A subtraction operation in the second pair, 175 - 16 gives 159.
So, 3
14 159 - the first 6
digits of Pi.
We also get to the same result with just three of the four segments.
314
(175+139)
159
(175-16)
Note that 175 is used twice.
Game Three - Introduction to Composite Numbers and
Phi !!!
The Game Two is not over yet, but already a new game gets underway. Subtracting
from 175 on the left helped
us with the Pi's first six digits. Now, doing the same on the
right of 175 leaves 62 - the fraction of Phi rounded to two decimals.
Does Phi find additional support here? The two
segments
to the left of 16 make a pretty play on Pi. Could the two segments to
the right of 16 possibly make a pretty play on Phi?
175 +
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 attention to
composite numbers. Although 288 breaks down into various multiples of
whole numbers, the presence of 16 before 288 indicates that if we
look at 288 as a composite number, it should be in the following
manner:
16 * 18 Without the multiplication sign:
1618,
the
first four digits of Phi!
So, Phi does find additional support, here.
Note:
If the 175
(one-half millimeters) was originally a hair over, it
would round up to 18 cm. Then we could read the location
as 16-18 yet again! (I can't remember
just now, will check it later)
Yet
another indication seems to be that we should not view composite
numbers as such unless there is some clue in the context to do so.
We shall keep all this advice in mind.
10&11
Back to Game Two
Pi
= 3.1415926..
The first eight digits of Pi
3
14
159
(175-16)
260
(113+147)
314
(175+139)
159
(175-16)
260
(113+147)
The two combinations
making for 3.14 159 are next to a segment pair, which adds
up to 260
millimeters, or 26
centimeters.
To sum it up, a
section of six segments (three segment pairs), and a section
comprising five of
the six segments, are both directly readable as Pi in eight
correct
digits.
12
Pi
= 3.1415926535..
To get up to ten digits of the Pi fraction,
next, we would like 535.
What's next-door to
260
(113+147)? It is 175 on the left, and 80 on
the right.
First, 175:
In fact, 175 was so far proving very versatile in operations yielding
control values of Pi and Phi. It worked with
virtually everything around it:
175 + 139 = 314
175 + 113 = 288 =
16 x 18 (1618, Phi in four digits)
175 - 113 = 62
(0.62 is Phi
minor rounded to two decimals) instance c of Phi
175 - 16 = 159
(a group of three digits from the fraction of Pi)
175 / 108
= 1.620..
(Phi rounded to two decimals, 81+27=108) instance d of Phi
The number 175 does everything here. Since it also works as part of the
composite number 288, we may check if it might do the job for us as a composite
number all by itself. It does do that.
175
= 5 x 35 (535)
Pi = 3.1415926535..
The
section of six segments reading out Pi values is
shown in color in the pie-chart above. It covers just over
half the circuit.
13
Next-door to the right of 260 is
80mms, or 8 cms. This was simple.
Pi =
3.14159265358..
After 8, the next three Pi digits are 979.
14
The next nine segments
average seventy-nine
units each !!!
113 + 146 + 27 + 54 + 108 + 81 + 27 +139 + 16 = 711 = 9 x 79 Those are the three digits we
are looking for. Besides this, the composite 711 also translates as 3 x 237.
Pi
= 3.14159265358979..
Time to note that in reading
the Pi-value out to fifteen digits, we have covered more
than the
entire Frame, once again.
15
At this point, I had to go back to the web to load up on some more Pi
decimals
:) To extend the Pi sequence by three more digits, we need a 323.
Is it not amazing that 711 translates as either 9x79, or 3x237, and nothing else? That's six digits of Pi in a row, 979 323, but the seventh digit, the 7 is off by one. So, is there a pure way to get the digits 323?
We've stopped on the segment of 16, at the point B, where we've been before. This is also where the Strong Connection cuts acrosss
the Frame to G. For more on the
Strong Connection use this anchor, or just take my word that it exists, and that it is very important.
Note, how in the image below, the B-G line travels through the
origin point of the purple square, which is also central to quite a lot of other
systems. In this way the Frame acquires another loop, and 16 becomes directly connected to both 113, and
146, and likewise to the sequence 80, 113, 146, whose total is 339.
This sequence is exceptionally important.
What a coincidence then that in the pie-chart 16 balances beautifully over the 339
section.
The symmetry between the two is
the best possible (count the intervening spaces, 436, and 435).
339
- 16 = 323 !!!
By the way, 339+16 also plays a role, seen later in this study.
Pi =
3.14159265358979323....
!!!
16
After this, the Pi digits are 846
264 33.. Of course, there is 80, as part of
the section of 339. We get another sought after digit, but the Game Two comes to an
end here. I just don't see a way to continue by the same method.
My final comment on this game - It is simply
beautiful - It reaches eighteen Pi decimals, and certainly
represents the Stone-Age mark to beat! On a different note, the
Frame is
now a mnemonic helper as well to remembering the first eighteen
decimals of Pi,
besides other things.
Pi
= 3.141592653589793238....
!!!
Game Three Resumes:
Around the Frame in Phi
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).
e
113
& Phi
Does 113 have anything to do with Phi?
Phi = 1.6180
339 887..
where we find 113 in
339
= 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 twice 113 equals 1,000 units even. Of interest again, the
average segment length in the next three sequences of three
segments is 113.
f
1.618
034
Phi rounded to six decimals
First, C to F is 340, and D to G is 340 again.
Obviously there is some emphasis on 340, here. Is it
because Phi rounded to six decimals is
1.618 034
?
g
Phi!
1.6180339..
There is no
way to express a zero all by itself (1.618
0 3), but, we could replace the 8
with 80. We see this value in the very next sequence of three
segments 80_113_146.
It is 80 being first in a short
sequence totalling 339 !!!
This sequence works for multiple purposes. Being important, it has a
name - Tri-balance..
h
Phi!
1.61 80 339 887.. (Phi in ten decimals)
What is left of the Frame after
we remove the Tri-balance (339) from it? Not 887?
It is 887! The Frame
(1226) divides into sections of 339 and 887.
This marks the third time, we have absolved the entire
circuit of the Frame. Game Three is over, and everybody won again ..
Visual
Confirmation!
There is an obvious inner
divide in the image, as the line subtending
both the 339 and 887 sections visibly serves this purpose. From H to E,
the image has a point on this line eight consecutive times out of nine
possible, and the miss is by only a little bit. The
girl now
seems to be under a yellow umbrella :)
Follow
the Arrow
Look
at the shaft of the umbrella, a perfect arrow striking through the
bull's eye
- as the >B and G points
connect across the image by a line
straight through the red
crosshairs - the center of the Square, and the
engraving's
geometrical system. The line divides the 120 degree
angle FGH
down
the middle into two 60 degree angles, thus creating
a component of a regular geometrical figure - the
hexagon.
The pivotal 0,0 point in the
center creates an equilateral
triangle with the points E, F, and G.
How nice, the head rests on a triangle's basis, and
is about equidistant to
the other two. This
triangle
simply begs experimental completion into a hexagon centering in the 0,0
point, the center of the Square.
That hexagon then fits the figure
from head to toe. Note the containment of the figure within the envelope. The
fact that
this hexagon is the offspring of two other
hexagons also merits a mention!
see: The Frame - Game 9 - The
HexMachine |
