The Simple Geometry Uniting The Silver Dollar Coin
And The Avoirdupois Pound of 7000 grains
There can be no doubt that in geometry three volume quantities are inextricably connected: 7000 units; 412.5 units; and, 371.25 units. These quantities are united via 1st dimensional equivalent geometric forms.
Start by packaging 7000 “units” (one avoirdupois pound) in the form of a perfect cube. Because of the subject matter of this manuscript, we’ll call these units “grains” and imagine them to be pure silver. Next to this cube is a 1st dimensionally equivalent tetrahedron (also in pure silver), which means that the sum of this tetrahedron’s edge lengths is equal to the sum of the cube’s edge lengths. This tetrahedron contains 6600 grains and is equal to the gross weight of 16 silver dollar coins: 16 X 412.5 grains = 6600 grains.
If instead, we start by packaging the 7000 grains of pure silver in the form of a perfect tetrahedron, and then next to it place a perfect edge-length equivalent cube of pure silver, then this cube will contain 7425 grains. This is equal to the pure silver content of 20 silver dollar coins: 20 X 371.25 grains = 7425 grains: or, 7425 grains / 30 = 247.5 grains, the pure gold content in the $10 gold “Eagle”.
How is it possible that history regards all of these quantities as having evolved independently of one another, having been arbitrarily arrived at, apparently through convenience, subjectivity, acclamation, and decree? (Note: a 6992 grain pound had previously been decreed to be the measure of the land, but in 1588 Queen Elizabeth increased the weight of the avoirdupois pound to 7000 grains. And after a 1786 team conducted a scientific survey of a thousand Spanish reales and had recommended to the Continental Congress that the new dollar contain 375.64 grains of silver, a newly elected George Washington still felt a need to conduct another survey which came up with 371.25 grains. This became the true definition of a “dollar” with the Coinage Act in 1792. Years later, in 1837 Congress reduced the gross weight of the dollar coin from 416 grains to its long since familiar standard of (412.5 grains.)
The geometry depicted above is timeless and etched into the eternal forms. This geometry, in light of the other geometry and mathematical relationships presented in this “Geometry of Money” manuscript, certainly must rule out coincidence as being responsible and that intent must have been the operative principle at work.
* All of the mathematics presented reflect the actual coin specifications to better than 99.99%. And remember, history speaks nothing about a geometric aspect to coinage, and regards all of these quantities as “arbitrary”.
Geometry Uniting the Silver Dollar
And the Avoirdupois Pound
Within the magical laboratory that is in each of our own minds, imagine the silver dollar coin perfectly divided into five equal parts. Each of these five “pie-shaped” parts is then transformed into a perfect regular tetrahedron. Next, out of the same silver alloy, construct ten cubes equal in volume to one another and having an edge-length measuring exactly the same as the edges of the five tetrahedrons. Weigh the cubes. Each is a tenth of a pound, and together they weigh one avoirdupois pound. Below is the simple mathematics supporting the assertion above:
Silver Dollar coin / one AV Pound = 412.5 grains / 7000 grains
412.5 grains / 7000 grains = 0.058928571… / 1.0
This can be re-written as _______________________________________________0.59828571… / 10
and as 5(0.117857143…) / 10
and can be illustrated by the sketch below:
The sketch depicts the actual geometric “ideals” to which the silver dollar coin’s weight and the avoirdupois pound conform to better than a 99.99% approach to perfection:
7000 grains / 7000.357131… grains = 0.99998984…
The 7000.357131… grains is the actual measure for the ten cubes given that their edge-length exactly replicates the edge of the tetrahedrons made from the dollar coin.
The Avoirdupois Ounce and The
American Gold and Silver Coins
The 437.5 grain avoirdupois ounce was chosen as “the ounce” because of its special place in the geometric hierarchy and not because of the popular usage of one ancient ounce over another, as history has us believing. No, this is a special quantity and the money architects knew all about it.
Three dimensional geometric forms known as polyhedrons, having their edge-length-sums made from the same length line are equal to one another in the first dimension of geometry. In this same sense, a cube and a tetrahedron containing the same volume are equal to each other in the third dimension of geometry, even though their edge-length-sums are different.
So when we talk about an avoirdupois ounce of 437.5 grains we are referring to its weight, its third dimensional manifestation. This quantity can be packaged within any number of different forms but geometry has preference for the cube and tetrahedron.
As a cube containing 437.5 “grains” its edge length is the cube root of this volume which is 7.59147… And since there are twelve edges on a cube its edge-length-sum is 91.09766… It can be said that this 437.5 grain cube is equal to a line of this length. The tetrahedronal equivalent of this cube in the first dimension is the tetrahedron which has its six edges equal to this same length line. This tetrahedron contains 412.47895… grains, which is the gross weight of the silver dollar coin to better than 99.99%:
412.47895… / 412.5 = .99994898…
Now watch what happens when we change the geometry of this “ounce” from its form as a cube to that of a 437.5 grain tetrahedron. Its edge length is 15.48393… making its edge-length-sum 92.90363… When we reconstruct this line into the twelve edges of a cube its edge length is 7.74196… and it contains 464.03882…grains. In 1834 congress changed the amount of pure gold in the $10 eagle from 247.5 grains to 232 grains. Two eagles contained 464 grains, which is the amount contained within this cube.
464 / 464.03882… = .99991633…
The “other” cube, considered geometrically commensurate to the avoirdupois ounce as a tetrahedron, has any one of its twelve edges equal to any one of the tetrahedron’s six edges. It is equivalent to stacking eight of the 464.03882…grain cubes together as one cube making a total of 3,712.31060…grains. This is the amount of pure silver in $10; i.e. ten one dollar coins to better than 99.99%:
3,712.31060… / 3,712.5 = .99994898…
In fact, this cube of 3,712.31060… grains models the 16:1 exchange rate between gold and silver newly set in 1834, up from the initial 15:1 ratio set in 1792. If you divide its contents by 16 you’ll get 232.01941…, the gold content of the eagle. If you divide its contents by 10 you’ll get 3,712.31060…, the silver content of the dollar.
The 1792 rate of 15:1 requiring 247.5 grains of pure gold in each coin can also be accurately modeled beginning with the 437.5 grain avoirdupois ounce. But this modeling requires an additional function, fundamental to all of mathematics: “the square root of two” (√2 or 2^1/2) . Here’s how it works.
Start with a cube. It contains (437.5/√2) grains. A tetrahedron, having its edge-length-sum equal to this cube will have any one of its six edges equal to 13.52646…
And like above, the “other” cube commensurate to this tetrahedron, that cube with any one of its twelve edges equal to any one of this tetrahedron’s six edges . . . it is found to contain 2474.87373…grains. This is, of course, the amount of pure gold contained in 10 original eagles to better than 99.99%:
2474.87373… / 2475 = .999948983…
To show that this relationship between the pure gold content of the 1792 eagle and “the square root of two” quantity is not just a coincidence or mathematical anomaly we need only to look at the following examples.
The “ideal” cube in the example immediately above contains exactly 2475 grains. Again, this is the amount of pure gold in 10 original 1792 eagles. A single edge of this cube measures 13.526696… and as one edge of a tetrahedron, we will find exactly (412.5/√2) grains contained within. Of course 412.5 grains is the gross weight of the silver dollar after its 1837 adjustment.
So, the 1837 adjusted gross weight of the silver dollar coin is seen to be directly, and exactly related to the weight of the original eagle’s pure gold content through the √2 quantity. And so too is that coin’s pure silver content similarly related. Here is how that works.
There is 371.25 grains of pure silver contained in every dollar coin. A cube containing 371.25/√2 grains has an edge-length-sum equivalent tetrahedron containing exactly 247.5 grains! Again, this is the amount of gold in every eagle coin until 1834.
The Geometry Underlying the Weight
Of the Original Dollar Coin
The 1792 Coinage Act set the original weight of the silver dollar coin at 416 grains. The money architects would have preferred 412.5 grains, but that would have been too noticeable a departure from that Act’s mandate: that the coin be of the same value as the then current Spanish milled dollar, or eight reale. Even at 416 grains the Spanish coins were still slightly heavier being around 423.9… grains. And the purity of the reale was 0.9305… compared to the 0.8924… of the US coin.
This means that the original Coinage Act not only violated the very mandate within the Act itself, by specifying a coin clearly not of the same value as the Spanish milled dollar, but also (knowingly) violated Gresham’s Law, which states that bad money will drive good money out of circulation. They couldn’t make the coin too much heavier than their 412.5 grains target because the greater the weight the more difficult it would be in the future to make the adjustment back down to the geometric ideal.
It is clear that the architects added 3.5 grains to the ideal 412.5 grains target to arrive at 416 grains. Why not 3 grains or 4 grains? Why 3.5? The answer is found in the geometry surrounding these quantitative relationships:
416 / 412.5 = 1.00848484… / 1.0
The equation above means that if the 412.5 grain coin be regarded as the unit and quantified as volume = 1.0, then by comparison the 416 grain coin equals that unit (1.0) plus an additional 0.00848484… of that unit. Now this portion describing the 3.5 grains quantity can also be written as:
1.0 / 117.8571…
and is a quantity readily modeled by two different geometric solids; again, the cube and the tetrahedron which share a common edge-length. The sketch below at left depicts these relationships.
The quantity___________________ (1.0 / 117.8511…)
is modeled in the simple sketch above at right. A single cube is the numerator above the line. Its edge length is equal to 1.0 unit. Below the line, as denominator, are 1000 tetrahedrons each with its edge length equal to 1.0 unit. This is the ideal geometric model from which the 3.5 grains quantity was derived. This becomes clear when the models are scaled to the coinage:
Again, the sketch above and to the right, depicts a cube scaled to the volume of the 412.5 grain silver dollar coin. Each of the 1000 tetrahedrons below the line, with the same edge length as this cube, has a volume of 48.6135… grains. Translating this geometrically modeled quantity to its numerical equivalent:
412.5 grains / 48,613.5911… grains = 0.00848528…
0.00848528… portion of the ideal targeted unit of 412.5 grains is 3.5 grains. When combined they produce the 416 grains of the original 1792 dollar coin.
412.5 grains (X) 0.00848528… = 3.50017857… grains
It should be noted that the coinage weights correspond to their geometric ideals to better than 0.99994898… and is a correspondence which will be seen to be repeated time and again throughout the evolution of the monetary system right up to the present day. Similarly, the cube and the tetrahedron, and one or another relationship regarding an equality in their edge lengths will be seen to guide each and every change in the weights and proportions of the individual monetary units.
As will be demonstrated in later chapters this geometry extends even into the proportions of the paper notes issued by the Federal Reserve and their predecessors.
The Geometric Lineage from the Troy Ounce
to the Gross Weight of the 1792 Gold “Eagle”
The 480 grain troy ounce, packaged in the form of a regular tetrahedron, leads directly to the 270 grain gross weight of the original gold “eagle”. This is accomplished by three simple transformations.
First, find its one–dimensional cubical counterpart, that specific cube having its edge-length-sum equal to the sum of the tetrahedron’s edges (its volume or weight is 509.11688… grains). Second, take exactly half this quantity and package it again as a regular tetrahedron. And third, find this tetrahedron’s edge-length-sum equivalent cube. This cube will be found to contain exactly 270 grains, which is the gross weight of the 1792 ten dollar gold coin.
The Geometric Lineage from the Avoirdupois Ounce
To the Pure Gold Content of the 1792 Gold “Eagle”
The 437.5 grain avoirdupois ounce, packaged in the form of a regular tetrahedron, leads directly to the pure gold contents of all “eagle” coins. This is accomplished by the same three simple transformations as the troy ounce above.
This time the one-dimensional cubical counterpart to this tetrahedron contains 464.0388…grains. This is two times 232.0194… grains. In 1834 congress adjusted the “eagle’s” gold content down from 247.5 grains to 232 grains. It was raised to 232.2 grains just three years later in 1837.
Like with the troy ounce above, this cube (equal in edge-length-sum to theavoirdupois ounce as a tetrahedron) is divided in half, and this half volume is repackaged again as a tetrahedron containing 232.0194… grains (the 1834 gold content). And its edge-length-sum cubical counterpart is found to contain 246.0937…grains, which is a 99% accurate approach to the original coins 247.5 grains pure gold.
The Geometric Lineage from Silver Dollar’s Gross Weight
To the Pure Gold Content of the “Eagle” Coin
In 1837 the gross weight of the American silver dollar coin was reduced from 416 grains to 412.5 grains. Since the silver content remained the same, the purity of the coin was increased slightly to an even .900. This same act mandated the gold coins to likewise have a .900 fineness raising their pure gold content ever so slightly to 232.2 grains.
But even with these new quantities, the geometric lineage, as illustrated in the above examples, remains consistent. Start again with a regular tetrahedron. This time have it contain the 412.5 grains, the new weight chosen for the silver dollar. As we have seen in other sections, the edge-length-sum equivalent cube for this tetrahedron contains 437.5223… grains and is the avoirdupois ounce to better than 99.99%. Half of this cube packaged as a tetrahedrons volume will have an edge-length-sum equivalent cube containing 232.0312… grains. Once again, this is the pure gold content found in the eagle between 1834 and 1837.
“SQUARE ROOT OF TWO”
And The Pure Metal Contents Of The Gold And Silver Coins
The pure metal contents of the 1792 gold “eagle” and the silver “dollar” are directly related both to one another as well as to the avoirdupois pound. This can be clearly seen in the relationships of the geometric forms modeling these quantities. In this particular modeling it is again the 1st dimensional “lineal” component, or the edges of the respective forms that is being regarded as equal.
Start with the 371.25 grains of pure silver molded into the form of a perfect cube. Imagine next to it a tetrahedron (also in pure silver) so scaled that the sum of its edges equals the sum of the silver cube’s edges. The weight of this tetrahedron is 350.0178565… grains which is 1/20th of avoirdupois pound (to better than a 99.99% approach to perfection).
When the 247.5 grains of pure gold is similarly modeled as a cube and its corresponding edge-length-equivalent tetrahedron (also in pure gold) is weighed, it will be found to contain 233.3452374… grains, which is 1/30th of an avoirdupois pound (again, to the exact same better than a 99.99% approach to perfection).
The above relationships to the avoirdupois pound can be expressed another way: as a 2/60th and a 3/60th portion. The tetrahedrons are literally successive “sixtieths” of an avoirdupois pound.
These very specific quantities of pure gold and silver can again be seen united to one another; this time through the √2 (square root of two) proportion. Start with the gold’s weight:
√2 (247.5 grains) = 350.0178565… grains
“the square root of two” times 247.5 grains equals 350 grains, or 1/20th of an avoirdupois pound. And 350 grains in the form of a tetrahedron was just shown to have an edge-length-sum equal to that of a cube weighing 371.25 grains (the pure silver content of the dollar coin).
Here is another way of looking at the above data:
√2 / 247.5 grains = 1 AV. LB / 20
This shows that a √2’s portion in relation to “one unit” is the same as 1/20th an avoirdupois pound’s relation to the 247.5 grains defining the eagle.
Now, to again travel full circle, divide the 371.25 grains by √2 proportion:
(371.25 grains) / √2 = 262.51339… grains
If you model these 262.51339… grains as a cube, its edge-length-sum equivalent tetrahedron contains 247.5 grains.
And, like with the avoirdupois pound above:
√2 / 262.51339…grains = 371.25 grains
This shows that a √2’s portion in relation to 1.0 unit is the same as the pure silver content of the American dollar coin’s relation to the cube equal in edge-length-sum to the tetrahedron containing the 247.5 grain weight of the gold eagle coin.
For one last example, take a tetrahedron with a volume of (412.5 / √2) grains and look at the cube with the same edge length. The volume of this cube is 2475 grains, or the pure gold content in 10 gold eagles.
All of the above is further evidence that the founders recipes for the new American monetary system was anything but arbitrary or subjective as our history presently has us believing.
The Geometric Properties of the “Gram”
And 371.25 Grains of Pure Silver
Let’s model a unit of mass as a cube of pure silver. We will call this unit 1.0 “gram”. It has several unique properties in this form that it imparts to the greater system of weights and measures in which it is the fundamental unit
1 – edge = 1.0 unit of length
1 – face = 1.0 square unit of surface area
1 – volume = 1.0 unit of mass
The “kilogram” is the next order of magnitude in this system. Whereas a single cube models the gram, a cube comprised of 1000 cubic grams models the kilogram unit. Its edge measures 10 “gram-length” units; and each of its six faces, 100 surface units.
Fifty of these gram-length surface units can be reconfigured into the surface of a single cube. The amount of silver contained within this cube is 24.056261… grams. This is equal to 371.2448437… grains of pure silver which by definition is one American silver dollar coin to a better than a 99.99% accuracy.
371.2448437 / 371.25 = 0.999986111…
One face, of a 1.0 cubic gram unit, reconfigured into the surface of another cube contains .0680413… gram. One cubic kilogram has 600 “square” gram-length units comprising its surface. If each surface unit is individually reconfigured into a new cube’s surface, then the total amount of volume contained within the 600 cubes as one system is 600 X .06804… gram = 40.82482906… grams, which is also 630.02339… grains.
The kilogram’s 600 gram–length surface units can also be sub-divided into 12 units of 50 each. As we’ve already seen above, 50 surface units, reconfigured as the surface of a cube, contains 371.2448437 grains.
These transformations can be illustrated once again by the cube and tetrahedron scaled to the same edge–length. Imagine in your mind’s laboratory a quantity of 371.2448437 grains divided evenly between 5.0 tetrahedrons. The edge-length of these 5.0 tetrahedrons is 8.572724979… , and as the edge of a cube encompasses a volume of 630.02339… grains, exactly.
What is important for the reader to see is that this geometry clearly reveals a quantity amounting to 371.2448437 grains to be intimately related to the metric system’s kilogram unit. Its astounding to think that after eons of human development mankind just “coincidentally” came up with two, supposedly unrelated systems of measurement, that mirror the relationship among the eternal forms of geometry with a correspondence to better than 99.99%!
371.25 Grains Was
A CHOICE ROOTED IN GEOMETRY
And Concealed From Humanity
We know from historical records that the avoirdupois ounce has been the standard weight unit in England and her colonies at least since 1588 when it was decreed that 7000 grains replace the previous 6992 grains per 16 ounce pound. The troy ounce had developed much earlier and remained the preferred ounce for coinage, precious metals, and gunpowder while the avoirdupois ounce became the standard for just about everything else.
So by 1792, these two different, seemingly unrelated “ounce” units of measure had become deeply inculcated into the American states, as they had been throughout much of the world by that time. When Congress was being persuaded as to precisely what amount of silver would define the new American “dollar” there was never a mention of (occult) geometry. They argued over how many grains of silver were in the Spanish reales, the coins they’d used most in the colonial era. And allegedly it was settled after careful scientific studies were conducted on a thousand coins taken from current circulation. This supposedly was done twice: first in 1786 when they concluded the coins contained 375.64 grains; and later after Washington became president, and they adopted a new 371.25 grains as the standard.
In the paragraph above I used the words “allegedly” and “supposedly” because I have serious doubts that any real scientific study was conducted. Why? Because 371.25 as a volume quantity is built into geometry itself and occurs in transformations along with other quantitative standards: 412.5; 437.5; 480; 270; 247.5; 258; 232; and others. These of course also just happen to be the monetary standards chosen by the founders of this nation.
Look carefully at the very simple equation below:
(437.5 / 1600) + 0.5 = (371.25 / 480)
This is illustrating an inseparable relationship among these specific numerical quantities etched into a single fabric through mathematics. Now, let’s add some substance to the abstraction above:
(1 AV.OZ. / 1600) + 0.5 grain = (371.25 grains / 1 TROY OZ.) grain
What this shows is the troy ounce and avoirdupois ounce directly and exactly related through three other simple quantities: 1600, which is a magnitude of 16, the number of ounces in the avoirdupois pound; 0.5 of one unit; and lastly, the portion amounting to 371.25, mirroring the pure silver content chosen by the Congress to be the foundation of their new monetary system.
These same constants arise together by way of a slightly different modeling. Imagine a perfect tetrahedron scaled so that its volume is equal to the formula below.
(1 AV.OZ. X 16) + 0.5 TROY OZ. + (371.25 / 480) grain / 16
Notice how closely it mirrors the equation above it: instead of the AV.OZ. being divided by 1600, its multiplied by 16; instead of adding 0.5 grain, it is a 0.5 TROY O.Z. that is added; and the “=” sign in the equation above becomes a “+” sign for the numerator.
This formula resolves itself into a quantity amounting to exactly 452.5483399 grains. Since we know this is the volume of our tetrahedron, a cube can now be constructed and scaled so that the sum of its edges is equal to the sum of this tetrahedron’s edges. In geometry, this tetrahedron, and this cube, are “first-dimensional equivalent” forms. And what is the volume of this cube? It is
exactly 479.9999996 grains.
Of course, this is one TROY O.Z., or 480 grains by any practical standard. This proves once again that in geometry a 371.25 unit volume quantity identifies and unites in one system, through proportional shape and form, the specific volumes 437.5 and 480. In light of all of this, how these measures came to be mankind’s two different “ounce” units must rank among humanity’s greatest mysteries. Perhaps this manuscript will help shed some light on the answer.
EQUAL “FACE VALUES”
When speaking about money the term “face value” arises. Face value is marked on every paper “note” or coin. It denotes the amount or worth of the specific instrument of exchange. For example, with the American silver coinage, the total face value of four quarters was the same as two half dollars or ten dimes. The face value of a one-dollar Federal Reserve Note is the same as a 1935 Peace Silver Dollar, or any of the others that preceded it. But the “true value” of these instruments of exchange often proved to be deceptive and misleading.
As early as 1853, when the fractional coinage was first debased (by reducing its silver content), the true value of two half-dollars, four quarters, or ten dimes was no longer equal to the true value of a single one-dollar coin. There was another minor adjustment in 1873, but for reasons of benefit only to themselves, the money lobbies purposely made the monetary system victim to Gresham’s Law. To somebody who knew what was going on, it was obvious that the “face values” and thus the fractional coins themselves were deceptive. After 1873, twenty American silver one-dollar coins contain almost exactly one troy ounce of pure silver more than does twenty dollars in halves, quarters, and dimes (precisely 1.000914038… troy ounce). Clever money-interests could, and would put this knowledge to use for their own benefit while the general public hadn’t a clue.
There is a geometric analogy to this monetary concept of “face value” that mirrors its inherent implicit ambiguity and was probably well known to the actual monetary architects behind the scenes in 1792. They probably knew that a troy ounce of pure silver could be packaged in the most economical manner (geometrically speaking, that means using the least surface area) in the form of a perfect sphere. Now for purposes of an honest deception known only amongst the “inner circle”, they made a cube of pure silver scaled so that its surface area was exactly equal to the surface area of the spherical troy ounce of pure silver. Now the two different forms, nonetheless, have the same “face values”. They are second dimensional equivalent forms.
Like the actual coins in their debased version, where the “face values” of the factional coins were less than the “true value” of a whole dollar-coin, so too is the silver content of the cube proportionally less than one troy ounce. And this difference models the coinage difference.
Here’s how it works; the math is simple. Start with a perfect sphere containing 480 grains. Calculate its surface area by first finding its radius from its volume:
4(π)r3 / 3 = 480 cubic grains; therefore r = 4.857180125…
Using the radius of the sphere (and the formula 4r2 = surface area) it’s surface area is found to be 296.4683134… surface units. This is the surface area of the cube of silver as well. But to find the cube’s volume its edge length must be found first. Since the cube consists of six square faces each must measure 296.4683… / 6, or 49.41138557… square units. The edge-length of this square, and this cube, is 7.029323265… Therefore the volume of this cube is:
(7.029323265…)3 = 347.3286021… grains
To bring America’s coinage into line with the rest of the worlds’ metric accounting in “grams” rather than “grains” the 1873 adjustment to the 1853 debasement mandated that one dollar in any combination of fractional silver coins have a gross weight of precisely 25.0 grams. Ninety percent of this weight is pure silver, and .90 X 25.0 = 22.5 grams. There are15.43235836… grains in a gram; therefore:
22.5 X 15.43235836… grains = 347.2280631… grains
This is very nearly the exact same number of grains as are in the cube of pure silver described above, which is equal in “face value” to the spherical troy ounce of pure silver. The “purity” of this correspondence is to better than a .999 fineness:
347.2280631… / 347.3286021… = .999710536…
A GEOMETRIC “ARTIFACT”
Even though one will need to look a little deeper into the geometry, we can see the silver dollar coin’s pure silver content of 371.25 grains clearly imprinted in the geometric modeling of the “troy ounce” of 480 grains. It is important to keep in mind that this modeling is of “geometric quantities” and that any particular named substance assigned to the forms is unimportant. This is because the resulting data will be the same for any name assigned to the contents of the forms.
This modeling scheme begins with the troy ounce in the form of a cube, and its’ corresponding equal edge-length-sum tetrahedron, which contains 452.548339… grains. The edge-length-sum of each of these forms is equal to the same length line. Also, any one of this tetrahedron’s six edges is equal to the edge of another cube that contains eight troy ounces, or ½ Dutch pound (there being 16 troy ounces in one Dutch pound).
Now, a .773432 portion will appear in this model after the tetrahedrons’ volume of 452.548339… grains is multiplied by 16.
16 X 452.548339… = 7240.773432
This sum, exactly 7240.773432 grains, is a startling discovery. When sorted out, what we clearly see is this:
7000 grains = (one avoirdupois pound), plus
240 grains = (one troy ounce / 2), plus
0.773432 grain = (one 371.25 / 480 grain)
It is an unpredictable surprise to find an exact avoirdupois pound, and an exact troy ounce divided by two comprising the bulk of this quantity. But it is the “artifact”, this .773432 portion of one grain that is of most interest since it is the same proportional amount of one troy ounce of pure silver that is contained in the American silver dollar coin. This geometric modeling shows that the actual silver dollar coin conforms to these geometric “ideals” to better than 99.999% since:
371.25 grains / 480 grains = .7734375 EXACT
.773432 / .7734375 = 0.999992889…
What the above modeling reveals is that a .77343… quantity is naturally occurring when in conjunction with a quantity of 480. This relationship is timeless, and certainly predates America’s coinage system. The following example shows this same “artifact” appearing in an even more basic construct that is itself fundamental to what I have come to call “The Geometry of Form”.
The Geometry of Form is a dynamic geometric transformational system. In this system, the first two dimensions of the “fundamental unit” assume different forms than those of classical Euclidian geometry. Euclid’s one-dimensional line 1.0 “unit” in length (with no width or height) is reconfigured as the 6 edges of a tetrahedron; each edge measures 1/6 unit. Euclid’s two-dimensional surface unit (with no thickness) is Euclid’s line moved through space the length of this line delineating a “unit” square; one surface unit. But in The Geometry of Form the area of this square is reconfigured as the surface of a cube. Each of this cube’s six faces is 1/6 unit of surface area.
Geometry loves expressing its unit’s three different manifestations, its three dimensions in this manner. We know this because when we stack three tetrahedrons of this size, one atop another, their combined height is the exact height of the 1.0 surface unit cube! And when this geometric fractal-like arrangement is reproduced on a larger scale, having a new tetrahedron’s edge being equal to the edge of the 1.0 surface unit cube, we find a stack of three of these tetrahedrons precisely equal to the height of a cube having a volume equal to 1.0 unit. So regarding geometry’s third dimensional form of the unit, both Euclid and The Geometry of Form arrive at this cube of volume 1.0.
These are the three dimensions of the fundamental unit in The Geometry of Form. They are illustrated in the image below:
The large yellow square is one face of a cube equaling 1.0 unit of volume. The small red square is one face of a cube equaling 1.0 unit of surface. The three small two-tone blue triangles represent three tetrahedrons viewed head-on. Each tetrahedron represents 1.0 unit of length (as the sum of each tetrahedron’s edges). The remaining three larger blue and green tetrahedrons provide the transition between the second and third dimensions of the unit (and are known as the spin-quanta of the “transitional-tetrahedrons” in the larger Geometry of Form system).
One of the key attributes of the unit being manifest in this manner (as opposed to Euclidian) is that its “forms” as a line and a surface each have both three dimensional spatial extension and specific volume’s. And it is in the volumetric relationships among these primitive forms where we will again find a quantity based on this 371.25 quantity.
All volumes are portions of, or multiples of the “unit of volume” represented by the yellow square. One surface unit in the form of a cube has a volume of .0680… and approximates a “sixty-eight one-thousandths” portion of the standard volume unit. This can be equally expressed as about 1/14.7; thus it requires almost 15 red cube volumes to fill the yellow cube.
And one lineal unit in the form of a tetrahedron has a volume of .00054560…, or 1/1832.820… Which means about 1833 small blue tetrahedron volumes will fit into the yellow cube.
When we explore the relationships among and between the volumes of the three distinct dimensional manifestations of “the fundamental unit” we discover the following:
(.068041…)(.0005456…) = .0000371238… / 1.0
(.068041…) / 1832.82… = .0000371238… / 1.0
(.0005456…) / 14.696… = .0000371238… / 1.0
The essence of this discovery is that a quantity exists absolutely inherent to the geometric organization of the “fundamental unit” in a three dimensional space frame that is a power of the 371.25 grain unit defining the American Silver dollar. In fact, it is a 1/10,000,000th (one ten-millionth) portion of this quantity to better than 99.99% perfection:
.0000371238… / .0000371250… = .999969…
Now look at the following geometric arrangement. Keep in mind the previous image of the three forms defining the three dimensions of the fundamental unit, especially their volume relationships and the stacks of three tetrahedrons so essential to this geometry.
Again, like in the previous diagram we are looking head-on at a stack of three two-tone-blue tetrahedrons. They are stacked in front of a single red tetrahedron. We know from the previous illustration that each one of the blue tetrahedrons is made from a single line of “unit” length (the sum of its six edges equals 1.0). Geometry also shows us that three unit-length lines forming the edges of a single tetrahedron will reach the same height as this stack of three. The same holds true for four lines comprising a stack of four tetrahedrons, or four lines comprising a single tetrahedron. The same holds true for five lines, six and so on. . .
This means that in The Geometry of Form, with the large red tetrahedron reaching the same height as the stack of three, they are equivalent linear models. Either one equals the height of the red cube, suitably showing the commensurability between the first two dimensions modeled in this manner.
So we see among these forms a .0000371238… quantity unifying the proportional relations between them, and thus must be considered inherent to the fundamental unit of measure itself. But we still must be surprise when we discover that in the illustration immediately above are all of the essential quantities defining the American silver dollar.
Without specific scale these are just four tetrahedrons with indefinite size. Assigning scale to one allows us to calculate the size of the others. Let’s see what happens when the large red tetrahedron has a volume of 371.25 grains. Each of the blue tetrahedrons must then contain 13.75 grains. The four tetrahedrons altogether contain 412.5 grains. These are exact measures, and likewise are an exact recipe for the silver dollar coin:
gross weight = 412.5 grains
pure silver weight = 371.25 grains
copper alloy weight = 13.75 grains X 3
In all probability the money architects behind the scenes knew of this geometry when they ushered this coin through the halls of an oblivious congress.
The Constructive Geometric Properties of
THE 412.5 TETRAHEDRON
Remember some of the associative geometry, exposed in previous sections, that is inherent to a tetrahedron having a volume equal to 412.5 (grains)? A good example is that the sum of its edges as the edges of a cube produced a cube equal to 437.5 (grains), or one avoirdupois ounce; and that the cube equal to one-half avoirdupois pound has its edge-length equal to that of the 412.5 (grain) tetrahedron. But some of the most revealing associative geometry is found in what might be considered the “constructive geometry” inherent to any particular form. Here’s how it works.
Imagine a perfect regular tetrahedron with its four vertices. Each vertex has its own “spherical domain” with the vertices corresponding to the centers of the four spheres. The diameters of the spheres are equal to the edge-length of the tetrahedron. Every different sized tetrahedron has its own specific corresponding sphere system. These two geometric models, the tetrahedron, and its corresponding system of four spheres are equivalent: one gives rise to the existence of the other.
Regarding the 412.5 grain tetrahedron, the weight chosen for the American silver dollar coin, the four-sphere system to which it corresponds demands each sphere to have a weight of 1832.689212… grains. This is significant since it was shown in the previous section regarding an “artifact” and the weight quantity of 371.25 (gains), that a weight quantity of 1832.820776… was seen to be literally built into the proportions uniting the three dimensional geometric representations of the fundamental unit in The Geometry of Form.
In fact, this is such an important concept to grasp it deserves review and further scrutiny. The fundamental unit of length is a line equal to 1.0. In The Geometry of Form it is configured as the sum of the six edges of a tetrahedron, and has a volume equal to 1.0/1832.820776… “The Unit’s” 2nd dimensional manifestation is configured as the surface of a cube. Its surface area equal to 1.0, contains a .0680413… volume, which if divided into 1832.820776… separate portions gives rise to each being .0000371238… The dollar coin’s pure silver content of 371.25 grains enshrines this fundamental geometric quantity*. And now we know that four spheres, each with a volume of 1832.689212… also comprise a tetrahedron of 412.5 grains!
1832.689212… / 1832.820776… = .99992821…
* It is worth noting that these two measures differ only in magnitude: the one is 10 million times the other. At the very same time these coins were introduced in America in the 1790’s, French scientist are said to have measured the earth’s quadrant from north pole to the equator passing through Paris. It is alleged that they divided this measurement into 10 million parts to create the “meter’”; and from it, the metric system.
A SINGLE LINE
SILVER DOLLAR COIN
In geometry, the simplest dimension of the unit (aside from the “point”) is the “line” of unit length. From this single line, the gross weight of the American silver dollar coin, its’ pure silver content weight, and the weight of its’ copper alloy can all be modeled after a few simple transformations.
First, we divide the line into two equal halves. We will model the weight of the pure silver with one half; and with the other half, the weight of the copper alloy. Each of these two halves will be further sub-divided in the following manner.
The first half is divided into six equal segments and assembled into one geometric form as the six edges of a regular tetrahedron. The remaining half is divided first into three segments, with each segment further sub-divided into sixths. Each of these clusters of six line segments assembles together as the six edges of three regular tetrahedrons. Each of these three tetrahedrons is exactly 1/3 the height of the single larger tetrahedron, and each is 1/27th its’ volume.
Thus far our model consists of three small tetrahedrons and one larger tetrahedron. But they are without size until we impart some kind of scale to the forms. Since I already know the answer, I will get to the essence of this modeling by assigning a weight of 371.25 grains (the weight of pure silver in the dollar coin) to the larger tetrahedron. Now, simple calculations show the weight of each of the smaller tetrahedrons to be 13.75 grains; together the three weigh 41.25 grains (the weight of the copper alloy in the dollar coin). Thus the four tetrahedrons together weigh 412.5 grains (the gross weight of the dollar coin).
Again, this model is simple and elegant with a 100% approach to perfection. It shows unequivocally that there exists an inseparable geometric bond between a weight measuring 412.5 “units” and one measuring 371.25 “units”.
THE GEOMETRY OF 1965
The Geometry behind the 1965 Debasement
Of the “Fractional Silver Dollar”
The year 1964 marks the last time the U.S. Mint issued the standard fractional silver coins (half-dollars, quarters, and dimes) it had routinely minted ever since 1873. In 1965 the comparatively worthless cupronickel-clad-copper-core imitation “silver” coins were foisted on the monetarily naive American public. Most people never gave it a second thought. After all, they almost looked identical and they were shiny, just like their predecessors. But unlike their predecessors, their intrinsic value was far less than their face value to the point of being mere tokens.
Even less known to the public, was the occult geometry to which the monetary architects had designed their new coinage to conform. It seems obvious now. The same secret traditions that first manifested in the monetary geometry of 1792 had been handed down through the generations (of illuminati?) and again reappeared in 1965 to dictate the weight change in the newly debased coins.
To understand what they did, we must begin where they began, and that was with the true definition of a “dollar”. They knew, and know to this day, that ever since 1792 a “dollar” is, and always has been 371.25 grains of pure silver. So being the honest and honorable men that they certainly must have seen themselves, they created “for us” (in their imitation silver slugs) a fractional dollar system of coins that is equal to a true “dollar” in a very real intrinsic way. Only a very few people were, and are aware of what you are about to read.
Following the time honored recipe book handed to them by their illuminated ancestors, they again began with a cube of pure silver weighing 371.25 grains. This is the essence of one “dollar”, its true definition, in the form of a cube. Next to it, again in pure silver, they created a tetrahedron so scaled that the sum of its edges equals the sum of the cube’s edges.
This tetrahedron of pure silver is equal to the cube of pure silver. But the equality of these two forms holds true only for geometry’s 1st dimensional system of measurement accounting. It is not true in the 3 dimensional system accounting for volume or weight, since the silver tetrahedron weighs only 350.0178… grains. And 350.0058… grains is the gross weight of the new fractional dollar coins ($.50, $.25, $.10) ever since the change over began. So the new fractional dollar is seen to be a geometric equal to America’s original 1792 silver dollar with respect to its 371.25 grain pure silver content.
But it is also directly related (through the same geometric forms) to the 412.5 grain gross weight of the dollar coin as well. Here’s how that works.
Between 1837 and the first debasing of the silver content of the fractional coins in 1853, a “dollar” in fractional silver coins weighed the same as the silver “dollar” coin. So we could divide the whole dollar coin into ten parts and each one of these parts would weigh the same as an 1837-1853 silver dime. This is where the money architects did their magic. They converted the dimes into perfect tetrahedrons and then created a single cube that is equal in edge-length to the edge-length of the tetrahedrons. With this cube’s substance being the same .900 silver as the tetrahedrons made from the dimes, it will weigh 350.0178… grains and is identified as the weight of our current imitation fractional silver dollar coins ever since 1965. Of course we could forget the dimes altogether and just divide any silver dollar after 1837 into ten parts as tetrahedrons and the same geometry holds true.
So coming in from two different quantities that are synonymous with a real “honest” dollar, geometry leads us directly to the “token” dollar coins of 1965.
This geometry corresponds to the actual coinage to better than a 99.99% correlation. This is regardless of which choice of refined constants I am about to present to the reader. So except for somebody knowing this geometry was at play at the time, which the Congress approving this Act (maybe with a few insider exceptions) certainly didn’t, one surely would have reason to ask: why were these people persuaded to vote for this specific quantity as opposed to at least two other more logical choices?
The answer to this question involves the first constant to be refined. Just what is this specific quantity that was approved by those legislators? The answer is in The Coinage Act of 1965, TITLE I, Section 1. (c), “setting the weight . . . of the quarter dollar (at) five and sixty-seven one-hundredths grams, and of the dime two and two hundred and sixty-eight one-thousandths.
$.25 = 5.67 grams; $.10 = 2.268 grams
The 1965 Act eased the transition over the next few years to slug-status for the $.50 coin because of the Kennedy assassination and subsequent 1964 half-dollar silver coin in his honor. But by 1971 they too had become mere tokens and took on the proscribed weight in line with the other denominations. The fifty-cent piece now weighed 11.34 grams making two of them the equivalent of 22.68 grams. This is the exact weight mandated by the Congress for the new fractional silver coinage, and it is measured in “grams” not “grains”.
So why in the beginning of this chapter was I using grains as the unit-system base rather than grams? Well I obtained the figures from the U.S. Mint, which is in grams. But when I converted them into grains it appeared to me that grains could just as easily have been the actual base unit:
22.68 grams = 350.0058874 . . . grains
Clearly, 350 is a pretty “round” number compared to 22.68; especially if one is aware that 350 grains is exactly 1/20 avoirdupois pound of 7000 grains, This is the pound Americans have used every single day since before 1792 through today (2015)! And think about this: If any combination of these coins equals $1.00, you have a 350 grain weight. But more importantly, $1.25 of these coins equals 437.5 grains, and is exactly one avoirdupois ounce. This means that twenty bucks in change weighs 16 ounces, exactly one pound. Now that is pretty neat!
If we convert 350 grains into grams we will get 22.67961850… grams, which is extremely close to what Congress had in mind:
22.67961850… / 22.68 = .999983…
In fact, its doubtful that any coins coming from the U.S. Mint conform to proscribed standards any where close to a 99.998% working tolerance, so in practice both 350 grains and 22.68 grams will produce the same coin. But why choose “grams” rather than “grains” as a base unit especially when there is no apparent correlation to greater magnitudes of whole unit aggregations such as with the 350 grains and its natural affinity with the ounce and pound. And it’s obvious that these coins, by whatever name given to their weight unit, are themselves near perfect weight standards for our grain-based system of ounce and pound. So again, why did they cloud that natural utility by using a gram-based standard?
Let’s look at these weights in light of the history preceding them. In 1873 the gross weight of the fractional (silver) dollar was set at exactly 25 grams. It remained this weight until The Coinage Act of 1965. The banking interests in the late eighteen-hundreds were lobbying Congress to bring the American fractional coinage into line with the European metric units. They wanted to raise the fractional dollar from its then 384 grains (24.88278144… grams) to an even 25 grams (385.8089588… grains), which they did in 1873. This brought a fundamental change to the coinage system by removing it from the avoirdupois system of grains, ounces, and pounds and placed it instead firmly on the metric system of grams and kilograms.
And those fractional coins from 1873 through 1964 can also be used for little weight standards in the same way we have just seen with our present fractional coins, except that they will aggregate in metric units. Four dollars in fractional silver coins weighs exactly 100 grams (4 X 25 grams). Forty dollars weighs 1000 grams, or one kilogram; $400 in these fractional silver coins weighs 10 kilos; etc.
It seems obvious now, that after 1965 the fractional coins reverted to correlating with the avoirdupois grain, ounce and pound despite their measuring rod being scaled in grams and kilograms. This is a blatant inconsistency, and is out of character for these money masters regarding past performances unless there is some deeper, more compelling reason for their specific choice. And there is, of course.
For the answer we must look more closely at the quantities cited above, magnified through the power of geometry and simple calculation. And remember, the weight of our present day silvery–clad fractional coinage in the form of a tetrahedron, has an edge-length-sum equal to the same edge-length-sum as a cube containing the pure silver content of one whole (real) silver dollar coin.
We started with this cube perfectly equal to 371.25 grains. In “grain-length units” its edge is 7.18712…, and its total surface area 309.929007… ; the corresponding equal edge-length-sum tetrahedron has a volume of 350.0178… grains, and an edge-length of 14.37425… But an altogether different picture emerges if, looking at the same two forms, instead of grains we scale them in grams.
The same cube (in “gram length units”) is far more revealing. The cube’s volume is now 24.05659534… cubic grams with a 2.886764… unit edge, and total surface area of 50.00046298… “square gram-length units”. Wow! A near perfect 50.000 unit surface area? And the tetrahedron’s volume is 22.6807756… cubic grams. Its edge measures 5.773529… and it has a 57.735561… surface area. Additionally, the measure bi-secting any face plane is 5.0000231… (gram-length-units). Again, this is an almost perfect 5.000 units.
If we focus even more closely on the edge and surface measures of this tetrahedron (in gram length units) we see that the edge can be re-written in a couple of different ways:
(10√3) / 3 = 5.7735026…
This compares to the actual measure as:
5.7735026… / 5.773529… = .9999954…
This shows that the two quantities correspond to better than 99.999% perfect.
The surface area can be expressed as:
50.000 units plus 7.735… unit,
in the same way as 5.0000 units plus .7735… unit describes the edge.
And since the cube’s edge is half the length of the tetrahedron’s it too can be re-written as:
(5.0000 units plus .7735… unit) / 2
This is pretty uncanny, since the cube’s pure silver content by any measure is still:
.7734375 troy ounce;
It’s clear to see that when scaling these two forms in the gram measure, many more historic geometric ratios and mathematical constants manifest, and are literally built into these forms.
Refining the 1965 Debasement Model
The previous geometric model is based on the cube being a “perfect” 371.25 grains, or its equivalent 24.056595… grams of pure silver. This created an equal-edge-length tetrahedron of 350.0178… grains, or 22.680775… grams.
What happens to the model if it starts with the tetrahedron exactly equal to 350 grains? Its volume in grams is 22.6796185… (which is 99.998…% of the mandated 22.68 grams exact). But its edge-length is 5.000 units plus .773431… unit. This .773431… portion of a unit, another “artifact”, is an even closer approach to the .7734375 portion of the troy ounce that defines the silver dollar coin, and the cubes’ volume in the first drawing.
.773431… / .7734375 = .99999190…
It’s almost as if we’re being compelled to look at the tetrahedron having its edge-length being exactly the sum of 5.000 units plus the ideal .7734375 portion. Take a look.
This specific scaling shows that the resulting tetrahedron is 350 grains to better than 99.999%; and at the same time, its 22.679692… grams is the mandated 22.68 grams to better than 99.99% perfect. The corresponding cube having its edge-length-sum equal to the tetrahedron’s edge-length-sum (meaning its edge is exactly 5.7734375 / 2) now contains .773400 troy ounce of pure silver. Its 371.2322… grains is still better than a 99.99% approach to “perfection”. These forms, in this scaling, truly embody the proportional ideals at the heart of the true definition of a “dollar” going all the way back to its inception in 1792.
How does this compare to the mandated 22.68 grams (exact)? The tetrahedron’s edge length becomes 5.77346360… and its volume is 350.00588… grains; and the cube’s edge is half the tetrahedron’s edge and its volume is 371.23730… grains. Once again, both forms approach their modeled ideals to better than 99.99%; the 1/20th avoirdupois pound of 350 grains for the tetrahedron, and the pure silver content in the dollar coin of 371.25 grains.
There can be no doubt that the money architects behind the scenes in 1965 employed the very same geometric dance between the cube and tetrahedron as their predecessors had performed going all the way back to 1792.
The Cube with a Surface Equal to 50.00
A Special Case Example
The cube containing the 371.25 grains of pure silver, when scaled or “quanta-sized” in gram-base units was seen to have a surface area very close to 50.00 square units (50.000462…). It is 50.00 to a .999990… degree of perfection. The drawing below depicts the tetrahedron-cube relationship when the cube has a surface exactly equal to 50.00.
Once again, both the cube and tetrahedron correspond to within 99.99% of the ideals. But there is something more implied by this 50.00 unit surface area. Remember, in this model all “units” are gram-based. This means that the “measuring ruler”, the fundamental unit quanta-sizing this model is 1.0 gram in the form of a cube. Of course, in our physical world the size of this cube will depend on its constituent substance. The volume of the cube in the sketch above is 24.056261… grams, or 371.24484… grains. There is something notable about both expressions describing this quantity, an amount that became the dollar’s measure of pure silver.
First we’ll look at the 24.056261… grams. This quantity can be exactly re-written as [125 / (27)1/2] grams; or, as [48.1125224… / 2] grams. Now, the “27” in the first expression points directly at the illuminati’s system of weight measures based on the 27 milligram cube, which is the subject of a later chapter. But the second expression, 48.1125224… / 2 grams, is even more revealing as to the ultimate source from which these measures have been distilled. Here is the view now coming into focus.
A few pages back, we saw how, in The Geometry of Form, Euclid’s 1.0 unit “line” is re-configured into the six edges of a regular tetrahedron. Each edge of this tetrahedron is 1/6th lineal unit; and, there is now a 1/1832.82… volume captured within the confines of the line. We also saw how this volume quantity divided the .068041… volume of the 1.0 surface unit cube creating a .0000371238… quantity. What was not mentioned is this tetrahedron’s .0481125224… surface area. Multiply this quantity by 500.0, and the product becomes 24.056261… and is recognized as the same quantity defining the number of grams of pure silver in the dollar coin (This is using the cube with a perfect 50.00 unit surface).
In the geometry of form, this .0481125224… quantity is also an important volume since it is the volume contained by 2.0 cubes with each equal to .5000 unit of surface. Thus, for every 2.0 of these cubes there is 1.0 unit of surface area containing a .0481125224… cubic unit volume. As the reader can plainly see, geometry has very high regards for this quantity. So too did the money-masters and their technocratic architects who established the parameters for America’s monetary system. Let’s have another look at the silver dollar coin’s 24.056261… cubic grams of pure silver, in the form of the cube with a surface area equal to exactly 50.00 square gram units.
The sketch above shows that the edge-length of this cube is (3001/2 / 6). But this can also be written (10 X 31/2) / 6, which translates to 2.886751… gram length units. This means that this cube is really comprised of 10 smaller versions of itself per edge, or 1000 in all. Their surface areas are each .5000 square gram unit; and each volume .024056261… cubic gram. This volume is also equally .37124484… cubic grain.
Broken apart into the 1000 individual cubes, with their .5000 square gram unit surfaces, unleashes 500.0 square gram units. In terms of surface area, this is equal to 5.000 squares with each comprised of 100 square gram units. These 5.000 squares are equivalent to those exposed to view if 1.0 kilogram in the form of a cube was sitting atop a flat surface.
The above data shows that the silver dollar’s 371.25 quantity of grains (as described in the 1792 Coinage Act) in the language of “(48.112522… / 2) grams“, is simply a “power” of the quantity measuring the surface area of the tetrahedron with edge-length-sum equal to 1.0 unit; or equally, it is a” power” of the volume captured by 1.0 unit of surface area in the form of 2.0 cubes surfaces.
Most certainly, all of this is known to those amongst the illuminati’s coven consisting of those members entrusted to the perpetuation of this hidden knowledge regarding, not only humanity’s weights and measures, but ALL things scientific: for instance, “parallel universes” for mental paraplegics . . . I’ll just leave it with that one example. It’s worth another whole book in itself.