Greek Parthenon…The
Athenian Acropolis Analyzed
Parthenon of Athens
Abstract: The author
reviews the efforts of Dr. Livio Stecchini to provide a metrological model of
the Greek Parthenon and provides serious mathematical proof that the Parthenon
was designed and built with great precision and likely used the meter as the
basis for the design layout. The model discussed within shows a very consistent
system of volumes, areas, dimensionless ratios and angles, not previously
discovered. Many will bolt at the idea of such technical capabilities in
ancient times, but perhaps only a very few really made it happen, and those few
may not have known exactly the total outcome we see in modern measurements.
Few
buildings have had as great of impact on human kind as the Parthenon at Athens.
The Athena Temple has long been a world center of attention. The majestic pillars have amazing dimensional
characteristics. From Dr. Levio C.
Stecchini who wrote extensively on metrology we know that researchers have long
debated whether the slight variations in the dimensions are due to error or
whether it represents extreme precision of a specific design. The general direction of the debate appears
to be centered towards the finding of which basic unit of dimension (geographic
foot, trimmed foot, Roman foot, etc) fits well and then trying to discover what
the basis for the design could have been.
Generally these approaches center on specific ratios and fractions that
seem to have been popular during the era, some 500 BC. Dr Stecchini was drawn toward what he called
the trimmed lesser foot and a basis of 48.
Fortunately,
Dr. Stecchini also printed the dimensions in meters to act as a standard when
comparing several different units of length.
I guess nobody thought to check the relationships in the metric system,
thinking perhaps that metrics came much later.
But in other work, Dr. Stecchini suggests all of our dimensional units
are related as if designed. You can
easily see most of the Dr. Stecchini work and the other research he reported upon
at
As
a structural engineer, it is absolutely clear to me that the columns were not
designed to simply hold the roof up. In
fact, it is abundantly clear that the purpose of the structure is reflected in
the placement and design criteria of the columns themselves throughout the
entire structure. Here I am going to
focus just upon the outer periphery at first.
If
we look at the Penrose data and Stecchini theory collectively, it is not hard
to find relationships between widths, lengths and heights. The length times the width in meters came out to a
standard fraction 2900 x 20 / 27. And the widths times the heights were close to
2900 / 9.
And of course, Dr. Stecchini had already noted that the dimensionless ratio
of length divided height was close to 20 / 3
for the most abundant outer columns. It
is almost obvious that if we rewrite 2900 x 20 / 27
as 2900 / 9 x 20
/ 3 one can see it is the combination of the other two. But each
uses independent data to get these results.
If
you consider there are 46 columns around the outer group and subtract 17 from
one side, there remains 29. On the inside right there is the group of 10
columns with apparent 5 across the end (counting the corners twice) to make 50
apparent columns. That multiplied by the
4 on the left end makes 100. Therefore
the 2900 is represented by (46 – 17) x 4
x 5 x 10 = 2900. And obviously 46/2 columns
plus 4 is 27 as well as 10 + 10 is 20 with 3
columns between the rows. Therefore, 20/3 and all the numbers above are represented
in the number of columns in groups. And there is redundancy such as 23 + 6 = 29.
Many
people report that there is nothing straight in the Parthenon. And dimensions reported by Dr. Stecchini and
others indicate the east side is slightly different from the west and also the
north varies from the south. So I went
back into ACAD and tried to construct the slightly crooked Parthenon. The problem in making the potential construction
is that the rectangular shape can wobble around without knowing the angles, so
a precise drawing could not be made from just that information alone.
If
one assumes, however, that the area is meant to be exactly 2900 x 20 / 27 =
2148.148148 no matter what the shape is, then the sides can be moved around
until this area results. This then
provided a bit more confidence in quantities like volume. The height appeared to be the same on three
sides with perhaps maybe only one corner raised up the 6 millimeters that Dr.
Stecchini refers to.
In
an effort to simplify the calculation of the mass of the columns I let the
taper in the model run the full height of the column instead of the shortened
version up to the capstones. Like a ton
of bricks it hit me that there are seventeen columns on the north and south
sides and here was the design criteria for the columns. The diameter of the column projected to the
top plane is the square root of 17/10 (or the area of ten column tops is 17 x
pi) and the area of the bottom of the column plus the area of the projected top
is 17 / 4 in square meters (or the area of top and bottom of four columns is 17
and there are four columns behind the six).
In the details below one can then back calculate the diameter to get
1.926465 meter with infinite model precision compared to Dr. Stecchini’s theory
at 1.9267 meter. This hypothesis will
need more confirmation, and it follows.
Oversize
corners 1.9855 assume same taper
(1.9855
- tan(3.415915444) x 10.4310224 ) = 1.36287533
Area
of top plus bottom might be 29/20 x pi and base could be near 1.98555555
Image below is only used to orient the reader to the Parthenon. It is not used in any calculations. See new corner column stuff in Mathcad second figure below
The
volume of 42 of these regular (46 periphery less 4 oversized corner columns far
below), but theoretical, columns is (31 /
2 ) cubed . One can note that there
appears to be 17 columns on the north and south sides while 8 appear from the
east and west. Immediately inside there
is a row of 6 internal columns which sum to 31 and that is half of the total
number of such columns. One can then hypothesize that the outward appearance
signals the detail design criteria as further spelled out below.
Page
1 Details
(31
/2)^(3) / (42 * 17/2) = 10.4310224089636
Sqrt
1.7 * 8 =10.4307
With
this type of precision for the area and the column height it was not difficult
to find the intended volume of the entire Parthenon outer periphery which again
is a relatively simple fraction of 1100 squared divided again by 27 and 2. Using this new assumption, one can see that
the average height is calculated using the Parthenon volume divided by the area
of the base of the Parthenon to get numbers within 0.0012 millimeters of the each
other. (each are underlined in magenta)
(31/2)^(3) / 42 cubic meters x
.999839gms/cm3 x 2.7 x 100^(3) cm3/m3 / 1000 gm/kg = 239,353.4 kg
Therefore, the theoretical vision of the mass
of the Parthenon columns is:
239,353.4 kg x 25 squared =
149,595,888.9 kg.
By
changing the density from 2.7 to 2.699603137 it becomes a perfect fit.
The
first equation below comes from the product of quantum numbers in the Lyman
Series of hydrogen.
[( 1/4-1/16)* ( 1/4-1/25)* ( 1/4-1/36)*( 1/4-1/49)* ( 1/4-1/64)* (1/4-1/81)*2]^(1/2)*10^(10)=
149,597,985.58692783
The
second equation below comes from ancient Egypt where 20.625 is the value of the
Royal Cubit combined with a primary quantum number 100/96.
Of
course, the actual column has flutes which would decrease its weight by some
amount and it has a capstone larger than the imaginary top given above as the
square root of 1.7 for a diameter. If the increase in weight due to the
combination of the capstone and the entasis curvature approximately equals the
fluting loss in weight, then the decorative aspects are in balance. And it is
likely the theoretical column without decorations was the intended message.
But
then the distance between the earth and the sun varies cyclically both
relatively short range and some cycles over hundreds of thousands of years.
Therefore, a very intelligent design could let the weight vary in the columns
and essentially mimic the earth and sun distance over very long periods of
time. There is the potential for a very
complex message to be embedded in the construction of the Parthenon.
Note
these numbers agree not only in significant digits but also to the exact
decimal format. This tends to bolster the argument that the meter is
fundamental to the Parthenon design basis. Some will find it hard to believe
that the ancients knew the distance between sun and earth to that accuracy, but
there is no denying the Royal Cubit was in existence.
For
example, if one knows precisely the length of the year then the distance from
earth to the sun is also known if one is familiar with the laws developed by
Kepler and Newton. To argue that someone like a savant could not make the same
calculations of Kepler and Newton seems somewhat arrogant to me. I am guessing that the reason we haven’t
fully appreciated the Babylonian Mathematical Tablets is because they are
perhaps far different from our current methods of numerical analysis via
methods involving prime numbers and the like.
For
example, the calculation of the weight of the imaginary column above can be
modified as below:
(31/2+1/[40*365.2422])^(3) / 42 cubic meters x
.999839gms/cm3 x 2.7 x 100^(3) cm3/m3 / 1000 gm/kg = 239,356.5933 kg
This
yields the exact number for the astronomical unit. The red marked modification is not just some
random number. It is 40 tropical years. Perhaps
the reason for this type of model using the cube of 31 / 2 is to be able to
draw this particular hint from the structure.
If
we take the weight at 239,356.5933 kg and divide by the area of the base from a
theoretical round column of 1.926465 meter, then the pressure is 82117.02155. That number divided by 10000 and taken to the
sixth root is 1.420383215 which is similar to the hydrogen hyperfine frequency
of 1.420405.
In
the second page of details below, one can now make the assumption that the area
of the east or west face is meant to be 2900 / 9 square meters. Therefore, if the width of one side is
slightly smaller, then the height must be slightly higher and vice versa. These
are average numbers for perfect geometric rectangular shapes and represent the
starting points that were then distorted (not errors) to include
additional information.
Width
x height at that width = 2900 / 9
(2900
/ 9) / ((31 /2)^(3) /(42*17/2)) = 30.89076
Area
of columns = 2900 / 27 * 20
Average
length is area divided by average width
(2900
/ 27 * 20) / (2900 / 9) / ((31 /2)^(3) / (42 * 17 /2)) = 69.540149
Atan
[ (((17 -1.7*pi)/pi)^(1/2) –sqrt 1.7) /
{ (31 /2)^(2) / (42 * 17 / 2)}] * 180/pi
=3.415954
Precision
It
is obvious that the designer and the builders had the capability to design,
fabricate and install the columns with great precision. In more modern times we have had major
problems simply measuring the columns. It is clear that there was intended some
very important message in the column geometry.
The consistent repetition of certain design elements such as the 42
columns proves beyond any doubt that precision was widely available and used
routinely.
If
there is then a change from say the length of the east face compared to the
west face and a difference between the north and south, it may feel like the
dimensions are a little random or contain minor errors. But if the east and south sum to the exact
same number as the sum of the north and west, then it feels a lot less like errors. It seems likely that the perimeter has some
major message yet to be discovered. One should note that the half perimeter of
100.431 less 90 leaves the height at 10.431, repeating the use of 9
and 10 as model elements.
If
one wants to leave a message in a structure such as a rectangular shape,
leaving a perfect rectangle with 90 degree corners does not leave much information
behind. It seems strange to me that
people like Kepler and Newton were deeply disappointed that planetary orbits
were not exactly circular and thereby, for them, confirming the creation of the
solar system by God. Yet perfect
circles, squares or any geometric shape conveys very little information. If I were creating any important shape, I
would make it so that it could be solved inherently within a few of the
measurements.
Constructing
the Model of the Parthenon Periphery in AutoCAD
Using
just the Stecchini measurements as an example for the four sides and the
assumption of an exact area of 2900 / 27 x 20 square meters, two separate
solutions are possible. The area when
say the north side and east side make a perfect 90 degree angle is just a bit too
high at 2148.176 sq. meters. Therefore,
the west side can be shifted left or right to distort the rectangular shape and
shrink the area a small amount. One can
draw a radius for the south side shown in red and draw the azimuth at some
trial and error value. From the end of
that line one can draw the radius in green to represent the length of the west
side. Then one can draw in blue the
radius for the north side and where the radii intersect completes the figure
with Stecchini numbers. Then ACAD allows
you to measure the area easily and with great precision. If the area is off, then another trial can be
easily drawn and the area measured.
Of
course, one might wonder how anyone even in modern times using a crane can set
the base column blocks weighing in excess of 9 tons to such precision. And how would we make them precisely even in
modern times?
Part
II
The
Parthenon Design has been the center of serious debate for centuries,
particularly the last two. A man who
played a major role in arriving at our current state of affairs is Nikolaus
Balanos who worked on the restoration from the late 1800s until shortly before
his death during the Second World War. Dr. Stecchini included some of the
Balanos Data in his efforts, available online with special thanks to:
http://www.metrum.org/key/athens/dimensions.htm.
Dr.
Stecchini, a major scholar, was a bit testy towards Balanos’s efforts because
he was primarily a curator and not a scholar.
Balanos could have stated more detail about his methods of measurement
and actual equipment used. Therefore, many have been quick to claim that
Balanos could have distorted the data to fit his scheme of how the Parthenon
was designed. Dr. Stecchini was particularly critical that Balanos measured the
column spacing on the straight instead of on the curvature. However, Balanos may have the last word on it
all if the analysis below stands up to scholarly scrutiny.
If
Balanos did, as most assume, then his measurements should sum to a lower number
because he is measuring the chord and not the very slight curvature. Therefore, if we simply make the assumption
that this may have been true, we could simply multiply each measurement by the
angle of curvature.
Dr.
Stecchini, a scholar and not an engineer, was unable to decipher what Balanos
did on the north and south sides since the totals of the numbers in the columns
do not come close to 69,521. With computers and spreadsheets it was easy to
determine that the third number is actually the same for 14 different
measurements. So the miscopied 4263 is actually 4290 and the south side is
actually 4293 instead of 4294 if we believe in the totals. But since he was
doing everything by hand, we should forgive this error of omission in
describing his method.
The
data below is simply the Balanos data put into a spreadsheet and some
calculations made on it. This
spreadsheet proves the Parthenon precision and the Balanos measurements almost
heavenly inspired. The patterns that
prove this are far beyond even the wildest imagination of anybody at the time
period. A model will show the Balanos numbers to be nearly perfectly done.
This
spreadsheet needs to be studied for hours to get the whole picture.
But
one should first take some time and look closely at column E which is simply
the differences between column B and A.
One can see that the negative numbers in green progress downward in
sequence -5, -4, -3. This is the first
of strong evidence that the measurements represent a design and are not random
errors. Then notice that the positive
numbers in blue descend in the sequence of 6951 which is the exact sequence in
the sum of the north column of 6951x. Would the normal reader have picked up on
that?
If
one looks at the five inner columns shown in rows 5 thru 9, they appear to be
just what one might expect with some random errors when setting 25 ton
columns. At the bottom highlighted in
red letters we see that the difference of the averages is 0.4, nothing to shout
about yet. But then when we calculate to
the right the differences of the averages of the 14 inner north and south columns,
the difference is the reciprocal of 0.4 at 2.5.
One
can see that the row 3 and row 11 of column A and B are identical. Strangely the collective sum of column A plus
B without row 3 is 59,711 which is 60,000 subtracting 17 squared. This is beginning to again look like a very
serious mathematical puzzle.
But
now we can do something that seems it just could not have been contemplated by
anyone but someone with a very high level of mathematical capability and puzzle
mentality. One can see that columns I
and J attempt to use color coding to show how column H comes into
existence. You can see via the
background colors in column I and J that they are echoed in column G which
indicates how column H is generated. This essentially makes east and west spacing
a reverse image effect. One can see in the first four numbers of col A and col
B compared to the last four numbers before the totals that it is very nearly a
reverse image except that 3662 is slightly different from 3668. It is the
puzzle mentality that demonstrates a system and then varies one element to make
the explorer dig deeper. But what is the chance of seven numbers being in
reverse order? Perhaps the probability
is one in ten million. It demonstrates that the columns can be set to 1mm
precision if they wanted to.
Of
course, the Mathcad model assumption makes the difference between the sums of
column I and J exactly 2.89 or the square of 1.7. This is shown here just for a check on the
typing and number insertion accuracy.
The
bottom of column H shows that the negative sum 15.827888 can be easily
converted via 2500 and the square root of ten to 3.15568808 (see green bottom
right above). This is a particularly
interesting number in that the earth period around the sun is 31,556,926
seconds and one can see that the similarity is good to seven digits, a very
remarkable event for such low precision input numbers. This parallels the mass
of the column converted to the earth sun distance given above in the beginning
section.
The
first thing I noticed was that the 3696 measurement repeated itself from the
second position on the east end to the opposite position on the west end. Therefore, what might happen if we looked at
the comparison of measurements at opposite locations? In column F we see the results of subtracting
column B from column C but in the inverse order. We see that 3668 in C4 has
3662 in B10 subtracted to get 6 in column F, etc.
It
seemed peculiar that the negative numbers are -5, -4, and -3 in sequence. But note that the positive numbers 6-9-5-1
are the same digits that are in the sum of the north measurements. Note that
the digits 6 + 9 + 5 + 1 = 21 and
-5 + -4 + -3 = -12 and 21 and 12
are the digital reciprocal of each other and included in the trailing digits of
69512 and 69521. All these considerations would not occur
in any other unit of measure except metric millimeters. The probability of the 6951 digits occurring
accidentally is one in ten thousand.
Combined with the 5-4-3 sequence, the probability goes even lower. But
that is only the tip of the iceberg.
In
the right column G we see that the normal difference between column B and
column C is simply multiplied by column F (3696-3668) x 6 = 28 x 6 = 168. The sum of column G is then 170 which is
double the total number of columns at 85 and is ten times the 17 columns on the
north and south sides. Further, the sum
of the numbers in column G down to the midpoint is 140 and the sum of the numbers
in column G up from the bottom to the midpoint is 14 and there are 14 column
spaces of 429X on the north and south rows.
These are hardly numbers that we would expect to happen randomly from
errors and would not occur in any other units of measure than the millimeter.
And there is a major amount of other data that indicates the design criteria
uses 17 as the basis.
This
comparison uses all of the east and west column measurements collectively and
indicates that each measurement is accurately recorded in millimeters. The precision of the stone would have to be a
small fraction of a millimeter or less for Balanos to get these results
reliably over the entire system of measurements.
If
one examines the lower portion of the graphic above, you can see a comparison
of the inner column spacing which all begin with the digits 429X. You can see the difference of the east and
west average is 0.4 while the difference of the north and south average is
2.5…these numbers are the reciprocal of each other. Again not something we would expect from
differences due to errors or random numbers.
It
appears then that each of the Balanos numbers is exact to the nearest
millimeter. This could not happen unless
the actual tolerance was much smaller.
Even in modern times we cannot measure something like these huge columns
and get the measurement consistent if the construction tolerance isn’t much
less than half of the smallest measurement unit (the mm in this case). It then
becomes a worry whether anything of this size could even be measured that
precisely in modern times.
There
seems to be only one alternative as to how all this could happen. The Parthenon is designed, fabricated and
installed with mind boggling precision.
This
brief paper is just the tip of the iceberg.
There is a whole bunch of supporting evidence to follow when I get
around to it.
Balanos
Mathcad Model
There are a lot of rules one has to follow in the Mathcad procedures and not get off in wonderland or freeze up your computer. The model calculation below is amazingly simple. No hint is given to suggest the Balanos numbers. There are just four primary equations entered into the “Given” command with hopes to solve for four unknowns.
This
might seem a strange number to be considering but the distance from the earth
to the sun in kilometers is considered to be 149,597,870.6 via the astronomical
almanac and the astronomical unit defined years ago. Modern measurements suggest that number
should be a tad higher via a factor of 1.0000010178 but since the astronomical
unit is deeply embedded in computer programs and other publications, it has
remained the older value.
(20625)^(1/2)
/ 96 ) x 100,000,000 = 149,597,985.58692783
Obviously
these can be shown to be the same number. This makes the earth/sun distance
appear to be far from random.
Entasis
curvature is 1.91 cm overall
Page
2 details
Displacements
If
width is less, height is greater
Flutes
may be 1/tan(3.4159154444)/320 mm deep
This
was done repeatedly and the solution above was finally attained with the
assumption of the southeast corner at 91.0165249 degrees. I could have taken it
to greater precision but thought this adequate to make the point. My hope was that the resulting angles would
somehow signal that they were intentional and not accidents.
In
the grey area above the main drawing, one can see the angle dimensionless
ratios lead to the same year namely 365.246 and 365.248. With more precision they might converge to a
single value.
Exact
Mathematical Model
One
then might surmise that there is an exact mathematical solution using
trigonometric identities such as the law of sines, law of cosines and area and
perimeter considerations. This effort is
mathematically intense and the details not included in this preliminary paper.
There
is one more peculiarity which Dr. Stecchini appears to have overlooked. He was providing argument that the curved
lengths and widths would be 69639.7 mm and 30935.6 mm respectively. If he had multiplied these together and
realized that in ancient times they often “squared the circle” and also used area
to project volume, he would have noted the following.
30.9356
x 69.6397 = 2154.346 and the cube root of 10^(10) is 2154.435
Therefore,
his estimate of the curved length and width may have been within 1/10th
millimeter of error even with that very difficult type of measurement and
theoretical endeavor. And what does this
say about the potential precision of the design and build effort? Even the curvature issue seems like it could
have a basis in the meter as a measure.
Conclusion
My
original intention was to essentially prove Dr. Stecchini correct in his
assumptions of precision in ancient times.
It never occurred to me in the beginning that the meter as a basic
measure might have been involved, but it certainly looks like it now. It doesn’t prove Stecchini wrong since it
uses his same numbers in some cases but he never draws the entire puzzle
together like this model does in such an organized fashion and using prime
number fractions like were typically used from ancient Babylonian periods
through Egyptian times and well into the modern era. And perhaps most importantly, prime numbers
have often been the center of focus for savant calculations. Whether the apparent orderly relationship
with the meter is accident, influenced by God or the result of the efforts of a
savant type mentality, it seems worthy of professional analysis.
There
seems little doubt now that the temple was created with miraculous
precision. It would seem imperative that
any restoration program include a new precise measuring of the entire
site. It seems likely to me that there
is an encyclopedia of information in the other dimensions.
Precision
in General
Most
readers will have no idea how precise the Parthenon was or how precise people
such as Balanos and Penrose could measure the huge stone columns. Perhaps if we consider a simple exercise in
modern times, we can make the point better.
Assume
you have a very large wooden table capable of holding the footprint of one
column roughly 2 meters in diameter or a little over six feet square. On this table assume there is a nice flat
piece of quality drawing paper well secured to the table. The simple objective
is to draw a circle 2 meters in diameter and then cover up any trace of the
center with another smaller piece of paper.
Next see if someone else on the team can find the center again under
almost ideal conditions inside a building on a drawing table with air
conditioning and uniform lighting.
The
first problem you have is to find something that will draw a circle accurately
that is 1 meter in radius. In fairly
modern times before the computer era, this was routinely done in drafting rooms
with a metal bar with a sharp pin at one end and a graphite holder that was
adjustable to about ½ mm. Typically the
pin was located at the chosen center and a tick mark made with the graphite and
then more precise scales were used to measure the line from the center. Appropriate adjustments could be made until
the line for the circle was probably within 0.2 mm of the chosen radius. You
might think this is all easy, but just tipping the arm a little will change the
circle radius slightly.
But
the graphite was often over 1 mm in diameter.
It was sharpened obliquely to a fine edge but on one side like a wedge. It was very important as to whether this
sharp edge was located towards the center or away or something askew. Because the graphite wore down significantly
in making a line some 19 feet long, the use of the line in subsequent measurements
required one to know how the graphite was positioned. If the graphite was turned so the sharpened
side was towards the center, then the extreme inside of the line would always
remain the same distance from center no matter how far the graphite wore
down. If the graphite was 180 degrees
from the center, then the outside of the line remained similarly constant.
Now
to find the center of the circle, one needs to draw a couple chords of any
length and plot the perpendicular lines from the midpoint of each chord. Where these lines intersect is the re-captured
center. But just finding the right place
on the circle is harder than you might think.
And striking arcs to find the perpendicular to the chord induces more
error. The intersection of these lines
could throw off the center over a millimeter by itself, particularly if the
chords are not selected at the optimal locations. If the angle between chord
perpendiculars is small, the apparent intersection is quite lengthy and finding
the center questionable.
Chances
are that this effort done by amateurs would find a major discrepancy in
centers, some probably different by 2 mm or more. This effort done by professionals would sometimes
still be off 1mm or more.
As
a project engineer and project manager of multi-million dollar projects, I wouldn’t
even think about bidding a job where the owner thought he wanted 1 mm of
precision even using computers, cranes and precision optical equipment.
So
how are we to interpret the measurements of Balanos and Penrose outside on a
rock floor in the wind and dust? To measure within a millimeter seems almost
ridiculous. Yet, it appears that it was
done and repeated on several occasions.
The primary concern of the times was whether one should measure straight
across an area or on the curvature of the columns in that particular
section. In the Cella Dr. Stecchini is
all over Balanos for measuring across at the assumed location of the Goddess
Athena. In that effort Balanos came up
with 21731 mm.
Dr.
Stecchini and Penrose came into fairly good agreement on the length of the
outside. On one model we used Penrose
numbers of 69541.3 and 69537.3 for the south and north lengths indicating a
difference of only 4mm. If we take the
average of those two numbers and multiply by 5 (five columns across at Athena)
and divide by 2 x 8 columns on the ends of the lengths, we get 21731.03125 mm. If we do the same procedure using Dr.
Stecchini theoretical numbers then we get 21730.8 mm. It looks to me like Balanos is winning this
one. Of course, it doesn’t mean that the
curvature numbers are wrong and do not signify something important too. Note that the issue of units of measure is
avoided here. No matter what units you
employ, the ratio suggests that the designer/builder had the capability of
great precision.
How
can we be sure that this precision actually occurred? Penrose (an architect who thought the angles should have been 90 degrees) made his measurements
prior to the publishing of his main book in 1888 while Balanos (a civil
engineer) did most of his work in the early 1900s. Both men had the training and equipment of
their times to make very precise measurements.
One should not expect two erroneous numbers to be a
perfect ratio of each other. And as far
as I know, there was never the suggestion by anyone that the width of the Cella
was in any way related to the length of the Stybolate. Yet there it is… the perfect ratio of
5/16. There are 5 columns across the
Cella and two rows of eight columns on the east and west end which define the
lengths of the north and south sides.
There can be no question that the design was intentional and no question
that both Penrose and Balanos made very precise measurements.
And
finally if we explore some other combinations we see the following:
8 x 17 x 100,000,000 / ( 9 x 21731) =
69537.118
This
says the outer appearance of 8 columns on the east and west multiplied by 17 on
the north and south is tied thru the number of columns (23 total = 9 south+ 9
north + 5 east) in the Cella length (9)
times the Cella width to get one specific length on the distorted
shape. All this indicates that the
design was precise and the design was executed with precision.
2^(36)/10^(6)/10^(1/2)
= 21731.0066
How
about throwing another wrench in the pudding? There are 46 columns in the
circumference and from a distance there are 17 columns on the side and 8 on the
end and 6 inside on one end. Therefore, there are (46 + 17 + 8 + 6 = 77 columns
and then multiply by 100 followed by dividing that 7700 by (17 – 8) =
855.5555555 (similar to a conjugate ratio approach). Now if we call that number inches and
multiply by 25.4 to convert to millimeters = 21731.11111 mm. If that were the
criteria, Balanos was within 0.1111 mm and of course means the execution was
also within 0.1111 mm, totally mind boggling.
It is likely these redundant scenarios were purposefully designed to
enhance the probability of being discovered.
But
what are we to think about the measurements in general? How can you measure something that is 21.7
meters or 69.5 meters and get it right within a millimeter while measuring out
in the open in the hot sun, wind and dust.
While you can simply reach across the table and draw chords, how do you
draw chords on a solid marble object? Can you really ever find the center of
the column? Are you simply making assumptions regarding the regularity of the
flutes? Maybe one could make a big caliper and measure the diameter if the
flutes are opposite each other exactly.
This would say something about those two flutes but not much else.
Col
A B C D E G H I J K
The
first thing I noticed was that the sum of the east and south very nearly equals
the sum of the west and north at 100392 or 100391
mm. The proof that this was likely not a
Balanos objective but more likely the original design lies in the variability
of the component numbers particularly in the columns A and B (east and west). One can see from the highlighted numbers in
yellow (3696) and light green (3662) that there is
some hint that possibly these measurements are mirror images of each other and
that is addressed in the right side analysis and discussed further below.
This
very data is used by many archeologists and associates to “prove the Parthenon
was not built with precision”. The analysis I have seen seems to emphasize how
the east and west columns are not perfectly aligned and that column spacing
varies even within a given end such as the 5 mm difference between B6 and B5
(4295-4290 = 5). In examination of the ten inner spaces, one can see that even
by modern construction standards, getting four exactly at the average of 4295.0 would be outstanding quality (see numbers
in red type). The total of these spaces for the east is 21474 and similarly
21476 for the west. The sum of the total widths (30870 + 30880) times the sum
of the total lengths (69512 + 69521) and then divided by 2 E6 is the number
4292.643875 (approx. double the area in mm).
Therefore, the 429X type dimension is not some random selection.
All
the prior curiosity building observations caused me to spend the extra time and
try to create a model using the Balanos data above. In spite of how easy what
follows looks, it was not that easy.
There are a lot of rules one has to follow in the Mathcad procedures and not get off in wonderland or freeze up your computer. The model calculation below is amazingly simple. No hint is given to suggest the Balanos numbers. There are just four primary equations entered into the “Given” command with hopes to solve for four unknowns.
It
didn’t make sense to me that the sum of the north and west compared to the
south and east should be off by one millimeter (100392 vs 100391). Since 100391 is a large prime number and also
the square root of the primary hydrogen atomic weight (1.007825^(1/2) =
1.003905) the two sums of sides were set equal to that prime number. That accounts for two of the equations. The
difference between the south and north was finally set at 10 mm. Increasing or decreasing that difference
single-handily throws significant more error into the resulting set of numbers.
It is amazing how important that difference is.
So this is a serious demonstration of why you don’t want to try to convey information with perfect rectangles and geometric figures or 90 degree angles.
All
in all this seems to indicate a design plan very likely far outside of the
Balanos capabilities or interest. And
quite likely this whole scenario is far outside of the capability of most
readers. After studying many megalithic
structures it gets easier to find these models.
Knowhow at ctcweb dot net
Copyright 2008-2018 All Rights Reserved James D Branson
It
didn’t make sense to me that the sum of the north and west compared to the
south and east should be off by one millimeter (100392 vs 100391). Since 100391 is a large prime number and also
the square root of the primary hydrogen atomic weight (1.007825^(1/2) =
1.003905) the two sums of sides were set equal to that prime number. That accounts for two of the equations. The
difference between the south and north was finally set at 10 mm. Increasing or decreasing that difference
single-handily throws significant more error into the resulting set of numbers.
Using anything smaller than 10 distorts the
results making the smaller numbers larger and the larger numbers smaller and
odd numbers either direction create 0.5 decimal while even numbers retain the
integer values.
Using a
difference of 8 makes north 69512 but the other three are changed one mm, as in
69520, 30871 and 30879. The fit of this
model with the Balanos data is mind boggling. One can see the 38641 difference
contains the prime numbers of 17 and 2273.
The north contains 13 as a prime and the south has 19. The prime 19 is seen in the columns as 46 – 27
= 19. The 13 is seen as (46 – 20 ) / 2 = 13 where 46 is the total number of
peripheral columns.
Given
Eq.
1
shf2 – nhf2 = 10
Eq. 2
nhf2 + whf2 = 100391
Eq. 3
shf2 + ehf2 = 100391
Eq. 4
nhf2 – ehf2 = 38641
Mathcad Balanos Data
69511 69512
69521 69521
Find(nhf2,shf2,ehf2,whf2)à 30870 30870
30880 30880
Where
nhf2 is north, shf2 is south, etc
100391
and 10039 are prime numbers
38641
factors to 17, 2273
69511
factors to 13, 5347
69521
factors to 19, 3659
30870
factors to 2, 9, 5 and 7 cubed
30880
factors to 2^(5),5,193
The
secondary beauty of this model is that it could have been discovered even if
the measurements were off several millimeters at all locations. The key is to make the sum of north and west
equal to the sum of south and east.
So this is a serious demonstration of why you don’t want to try to convey information with perfect rectangles and geometric figures or 90 degree angles.
While
Balanos may have been measuring something a little different from what Dr.
Stecchini and Penrose were measuring, it is clear that Balanos connected with
the designers and their objectives.
Readers
who are not computer programmers might think it easy to program a computer to
come up with these puzzles. But in fact a
programmer must know the puzzle ahead of time.
If you don’t know how to make the inter-linking then it will require a
lot of trials to find something that you like.
And to get things to interlock repeatedly would take millions of trials
and error removal. In modern day computer programming you simply need to have
the objective in mind before you start programming. I think we have a long way to go to reach the
competence level of the Parthenon.
Knowhow at ctcweb dot net
Updated
3-18-2018
Copyright 2008-2018 All Rights Reserved James D Branson
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