The following two chapters have been added to the initial publication of The Geometry of Money (which represents less than two dozen handmade volumes thus far). Both have been included here with this next printing to clarify and expand upon important topics previously discussed in earlier chapters. They were written for readers who may not have read this book so some of the previous illustrations have been repeated for clarity.
THE SPHERE IN THE CYLINDER
Modeling Both, the Measures of Man, and of Nature
Following is an example of one of the most fundamental of all geometric transformations: A sphere may be viewed as both units of volume and units of surface-area. But for “quanta-sizing” a system of geometric form, only one of these qualities may be considered “The Unit” at any given time. For example, in the photo above, the sphere in the cylinder can be either 1.0 unit of surface (in the form of a sphere) or 1.0 unit of volume.
This “Unit” in the form of a sphere may be distorted, fused with another, or divided. If the sphere is (1.0) unit of surface, and fuses its volume with that of another identical sphere’s volume, then the result is too much surface-area for the new single larger sphere*. Geometry packages this “excess” surface area in the form of a regular tetrahedron. This tetrahedron is depicted a-top the sphere in the above photo and in the diagram below. In The Geometry of Form this is the “transitional tetrahedron”, or simply transit–tet.
Now, if in stead of combining the volumes of two of these 1.0 surface unit spheres into one, divide the surface–area of one of these spheres into two new spheres; each one now has a ½ unit surface area. This transformation results in “excess volume”, which geometry packages in two of these “transitional tetrahedrons”.
Let’s now look at the mathematics describing each of the above two transformations. We started each with “one unit of surface–area in the form of a sphere”. This sphere’s radius (found by the formula 4πr2 = surface area) is 0.282094792… Using this radius, its 0.094031597… volume is found by the formula 4πr3/3.
*This new single sphere has only a 1.58740 surface-area. The 0.4125989… quantity of excess surface-area becomes the surface of the transit–tetrahedron.
In the first example, two of these volumes were fused into one, resulting in an excess surface–area of 0.4125989… unit. Configured as the surface of a regular tetrahedron, its 0488071804… edge–length was found by a tetrahedron’s surface-area being equal to its [(edge)2 X 31/2]. Now use [(edge)3 X (21/2)] / 12 to find this transit-tetrahedron’s 0.13702032… volume.
The second example divides the sphere’s 1.0 unit of surface into two new spheres. Using the formulas above we find that the combined volume of the now two spheres is only 0.066490380… compared to the previous single sphere volume of 0.094031597… There is an excess volume of 0.027541217… unit. In The Geometry of Form, this is packaged not as one, but two tetrahedrons; each has a volume of 0.013770609… unit.
These two slightly different tetrahedrons are both constructs from transforming the same 1.0 surface unit sphere. Certainly they are related in that respect, as well as their function as to accounting for the residual quantities created in transformations. Unless fine enough calibrations are applied in measurement, these two different “transitional-tetrahedrons” in physical form would be practically indistinguishable from one another; or from the tetrahedron atop the cylinder with the sphere within, depicted in the graphic and photo above.
The Sphere and Cylinder
The cylinder in the photo is inseparable from the sphere. This is because a sphere bores a cylindrical hole through the fabric of “space-time”. The proportions of the cylinder depicted make it a “Cylinder of Maximum Volume”: given a fixed unit of surface-area, this proportioned cylinder captures the most volume using the least surface–area. The diameter of its circular end planes is equal to the “height” of its side, and to the diameter of the sphere within.
The surface-area of the cylinder’s side is exactly equal to the surface-area of the sphere within. The total surface-area of the cylinder is 3/2 times the sphere’s surface-area. And its volume is 3/2 times the sphere’s volume as well. Since this sphere’s surface is 1.0 square unit, the cylinder’s surface is 1.5 square units.
These are some of the quantities and proportions built into the very structure of transformational geometry along with their appropriate forms. These presented thus far are purely geometric in that no name (inch, foot, yard, meter, etc.) has been assigned to the Unit, nor substance assigned to the volume of the forms.
An Analogous Physical Model
Compared to a trained physicist, I know nothing about physics. But I do know something about geometric quantities that are constants of geometric structure. Some of these have been shown above.
The physicists have measured the masses of the proton, neutron, and electron and have come up with quantities describing their sizes relative to one another. They’ve found that a proton’s mass is 1836.152701 times an electron’s mass; and likewise, that a neutron’s mass is 1838.683662 times the size of an electron. Here the electron has been assigned the role of being the (1.0) Unit. Using simple subtraction, it can be determined that the difference between a neutron and a proton is 2.530961 “electron masses”.
Protons and neutrons are collectively known as “nucleons” and are, on the average, about 1837.418182… times the size of an electron. Thus an electron is about 1/1837.418182 the mass of a “nucleon”, which can also be written as .000544242… Therefore, 2.530961… of these quantities is
2.530961… X .000544242… = 0.001377455…
and is seen to be directly related to The Geometry of Form’s transit-tetrahedron’s volume quantity since it is 1/10th of its 0.0137706087 volume. From the perspective of The Geometry of Form, the transit-tetrahedron is appropriate as model basis since its role in geometry arises when accounting for surface and volume “differences” in basic geometrical transformations.
Using geometric forms to model this difference in mass between the two nucleons can be done in the following manner: First, divide the transit-tetrahedron’s volume into two separate packages, into what amounts to ½ volume “t-tets”. Each of these two volume quantities is reconfigured into the form of a star-tetrahedron (imagine the four faces of a tetrahedron being the base triangles of four other tetrahedrons; the resulting form is the “star-tetrahedron”). Any one of its five sub-tetrahedronal chambers has a volume of 0.00137706087 and is a model of this mass difference between the two different sized nucleons. Below are three views of the star-tetrahedron.
There is something else about physics that I know little about. It’s called the fine–structure constant. Its value is .007297352568…, or inversely 1/137.035999139… For what it is worth to future researchers, it should be noted that 37 of these quantities together total 0.2700020… This is 27/100 to better than .99999… fine. Also worth noting is the relationship between 27 and 37: i.e. 1/27 = .037037037… and 1/37 = .027027027…; and, 37/27 = 1.370370370…
Again, since I know nothing about physics, I’ll leave it to the trained physicists (and the reader) as to whether the above lines of inquiry have any merit. But for now, look again at the photograph at the beginning of this section. If the surface of the tetrahedron atop the sphere is the .412598947… “excess surface-area” resulting from fusing the volumes of two of the spheres (with surface-area equal to 1.0 unit) into one sphere, then its volume is .01370203…
Now, we already know that the volume of the sphere within the cylinder is .094031597…; and, that the volume of its cylindrical domain is 1.5 times the sphere’s volume making the cylinder .141047396… cubic unit. So what we are looking at, with respect to volume (in the right-hand side of the photo) is the sum of the tetrahedron’s volume and the cylinder’s volume. This combined volume is .154749428… cubic unit. The cube (to the left in the photograph) is this combined volume reconfigured.
So, what is it about this cube and the forms it represents that’s so important to your understanding of history? It is with respect to exposing the truth about where the measures of Man truly have been derived? Below is some very persuasive evidence.
Volume of cylinder = .1410473… = .9114566…
Combined volumes .1547494… 1.000000…
A.V. Ounce = 437.5grains = .9114583…
Troy Ounce 480.0grains 1.000000…
In the photo, the entire cube, or the combined tetrahedron–cylinder is the Troy ounce. The white portion of the cube, or the lone cylinder is the Avoirdupois ounce. The Troy ounce has been in use since at least 800 AD; the Avoirdupois ounce since the late 1400’s. These supposedly “arbitrary and subjective” measures of Man conform to the immutable and eternal measures of geometry to an incredible .999998… approach to perfection. Let’s see what other gems of forensic history lay buried in the simple geometry thus far presented above.
This quantity, .154749428… divided in half is .0773747…, or in thirds is .05153143… And there is .7734375 troy oz. pure silver in America’s dollar coin and .515625 troy oz. pure gold in the $10 eagle coin. The gross weight of the $10 eagle was 270 grains; which can also be written as 1/.0037037… and, .0037058… is (.154749428…)3. If .154749428… is the edge-length of a cube, the cube’s volume is 1/270. And if it is the edge-length of a cuboctahedron, then the sum of this form’s edges is 3.713986… reflecting the 371.25 grains of pure silver in the dollar coin. All of these monetary measures are better than 99.9% identical to the measures of geometry!
The Meter Measure, the Ton, and the Element Gold
From what we’ve seen thus far, it’s safe to conclude that for some reason the “quantities” chosen to define our systems of weights (troy/avoirdupois and gram) and monetary measures mirror geometry’s choice of “quantities” with respect to spherical transformations. As we’ll see next, these measures, and the geometry from which they’ve been spawned, can also be found in the very monetary metals comprising the coins. Again, the reader should refer to the previous photo and diagram.
To start with we must quantify the tetrahedron, the cylinder, and the sphere within. Since we began with the sphere equaling 1.0 surface–unit in the form of a sphere, all we need to do now is to “name” this unit. Let’s see what happens when this one unit of surface is 1.0 square “meter”.
This makes all of the other quantities collapse into metric measures. (Yes, for readers who have not read my book, The Geometry of Money, the metric system existed eternally in geometry long before the French claimed they “invented” it in the late 1790’s.) So the “size” of the forms has been established giving them a real world, or human scale. But the “substance” of their volumes still remains unnamed, without which there is no “weight”. Pure gold seems to be that one substance commensurate to both the metric system of measurement and the base unit of both ounces, the “grain”. Have a look and decide for yourself.
One cubic meter of gold weighs 19,300 kilograms. Since the surface of the sphere is 1.0 square meter, its volume is 0.094031… cubic meter. This sphere’s cylindrical domain is 1.5 times the sphere’s volume and surface-area. To start, how much does the golden sphere within the cylinder weigh?
(0.094031…)(19,300 kilograms) = 1,814.8098…kilograms
Since there is 1000 grams in a kilogram the golden sphere weighs 1,814,809.8…grams. Given that there is 15.43235835… grains in one gram makes its weight in grains 28,006,795.5… Again, given 7000 grains in one pound makes the golden sphere within the cylinder . . .
28,006,795.5…grains / 7000 grains/pound = 4,000.9707…pounds
Made out of pure gold, with exactly 1.0 square meter of surface, this sphere weighs 4000 pounds, which is 2.0 “tons”. This means that the entire cylinder of gold, with its 1.5 square meters of surface, weighs 3.0 tons (6001.456… pounds). Both of these measures (2.0 and 3.0 “tons”) are exact measures to an accuracy of better than 99.9757%. Keep in mind that the pure gold and silver metals in American coinage are by design and legislation only necessary to be 99.9000% pure.
It seems that with respect to weight the ton is the Unit (which subdivides into pounds, ounces, and grains); whereas the meter is the Unit with respect to size. Further confirmation of this conclusion will be clearly seen in the next example.
Look at the diagram back on page five. On the right side is the transit–tetrahedron atop the cross-section of one hemisphere of the sphere equal to 1.0 surface unit (to which unit has been assigned the name 1.0 square meter). The circle equal to the height of the tetrahedron is the cross-section of the sphere having its surface equal to ½ square meter. What is the weight of the gold comprising this sphere?
Knowing its surface area we find its volume using the formulas from our previous calculations. It is .03324519… cubic meter. With gold weighing 19,300 kilograms/cubic meter this sphere weighs 641.6321… kilograms. This is 9,901,897.542… grains, which quantity is 9,900,000 grains to a .9998… degree of fine-ness. Readers of The Geometry of Money know that a quantity of 990 special cubets (described in a later section) equals a weight of 412.5 grains, which is the gross weight of the silver dollar coin.
The previously described division resulted in two of these spheres with a ½ meter surface. These together weigh 41257.9064…troy ounces. This is √2.0 tons . . . i.e., 1,414.5567… pounds! This again shows the “ton” as the preferred named unit of weight, and the meter of length, area, and volume, when gold is the named substance.
The Star-tetrahedron and the Monetary Metals
This next example references the previously introduced geometric form, the star–tetrahedron. Its relative physical dimensions (in this case) relate directly to the quantity defining the volume of the sphere within the cylinder. But in this example, this quantity is not a volume-unit but an increment of length. Here is how this works.
As we have seen, geometry clearly shows us that 1.0 surface-unit in the form of a sphere has a volume defined by a .094031… quantity. But one must look more closely to see that 1.0 surface-unit in the form of a circle has a radius measuring six times .094031…, or, .5641895… This is how one “normally” would describe this measure. The diameter of the .094031 (volume) sphere is also this measure. Many of geometry’s quantitative structural units work in this fashion.
Now a line .094031… unit long, as the side of a square, delineates an area equal to .008841941… square unit. This (.094031…)2 quantity, as the surface area of a star–tetrahedron, once again reveals in its structuring the two formative quantities defining the silver dollar coin. We see the gross weight of the coin (in grains) in its .041250847… edge-length; and the weight of its pure silver content in the two-times .37125761… sum of its edges.
I think it is relevant to mention here that because gold weighs 19,300 kilograms/cu. meter, if modeled as cube and subdivided into eight sub-unit cubes, then each of these eight cubes will have a weight of 2.0 (metric) tons + 412.5 kilograms. Each sub-cube has a surface area equal to the cylinder’s 1.5 square meters.
Let’s now make this .008841941… quantity a volume, and then name the “unit” that it is a portion of 1.0 cubic meter. The substance of this volume is again pure gold. Now let’s do the math: .008841941… X 19,300 kilograms/cubic meter = 170.6494… kilograms. This is 170,649.4… grams or equally 2,633,523.724… grains. There is 7000 grains in one pound, so 376.2176 pounds is the weight of this .008841941… cubic meter of gold.
Now remember, it is the “ton”-measure of weight that has consistently been the “unit” in all of the previous examples. What is this weight when expressed in tons? 376.2176 lbs. / 2000 lbs. = .18810 ton. But on closer inspection we also find this weight to be twice .094054… ton. What this means is that
.094031… cubic meter of gold weighs twice 1.0… ton.
(.094031…)2 cubic meter of gold weighs twice .094054… ton.
The strength of geometry’s influence upon the meter measure, the pure element gold, and the pound-based ton, can even be more clearly expressed when we re-write the above relationship:
(1/36π)… cubic meter of gold weighs twice (1/36π)1/2 … ton
In the photo of the sphere in the cylinder, the scale of the forms was established when the sphere’s surface area was set to equal 1.0 square meter, exposing the ton as the preferred weight measure when gold was the named substance. These relationships are again confirmed when the same 1.0 square meter is assigned as the surface area of a tetrahedron made of pure gold. Let’s follow the math.
The surface area of a tetrahedron is equal to its edge2 X √3. Since this surface area is 1.0 square meter its edge length equals (1.0 / √3)1/2 which is .7598356… meter. Using the edge to find the volume gives us .05170027… cubic meter. To find how much this gold weighs multiply .05170027… cu. meter by 19,300 kilogram/cu. meter: 997.8152… kilograms. At 2.20462… lbs./kilo, it weighs 2199.8059… pounds; which is 1.1 ton to better than .999911… fine.
The volume contained by 1.0 square unit in the form of a tetrahedron, by whatever name, is .05170027… “unit”. It is directly related to the Coinage Act of 1792 since the weights specified for the coins are powers of this quantity:
(.05170027…)2 = .002672918…
[3(.05170027)]2 = .02405626…
6(.05170027…)2 = .01603750…
26.72955… grams equals the silver dollar’s 412.5 grain gross weight; and, 24.05659… grams its 371.25 grains of pure silver. And .05170027…, as the edge length of a cube, makes the cube’s surface area equal to .016037508… unit, and 16.03773… grams is the 247.5 grain pure gold content of the $10 eagle coin. These coinage weights are 99.998…% in conformance with their geometric ideals.
Once again, we have clear evidence showing that the quantities structuring geometry were those quantities “chosen” for quantifying the measures of man.
Why 3.0 Tons is Special
What makes “3.0 tons” so special? In light of the above discoveries the following will reveal at least some of the reasons. Previously, it was shown that the relationship between the cylinder’s volume and the combined volume of the cylinder and tetrahedron was the same ratio as that between the avoirdupois ounce and troy ounce. This relationship is a .999998… approach to being perfect! We also saw that gold was naturally commensurate to the ton when the surface of the sphere was 1.0 square meter.
The ton is a pound and grain based measure. And the troy ounce and avoirdupois ounce are the first “grain-based” units after the grain itself. History says they were arbitrarily arrived at by what felt “right” to one or more persons. They don’t seem to be related, especially when a pound troy is only .822857… a pound avoirdupois. And despite both being grain–based, 1.0 ton is an even number of A.V. ounces (32,000) and is an awkward (29,166.666…) number of troy ounces. Again, 2.0 tons is an even number of A.V. ounces (64,000) and an irreducible number of troy ounces (58,333,333…).But 3.0 tons is special to both systems of accounting:
3.0 tons = 96,000 A.V. oz.s, or 87,500 Troy oz.s
The gallon is a unit of volume. The modern gallon of 231 cubic inches was introduced into England at about the same time as the “new” avoirdupois ounce. Using this gallon unit of measurement, 3.0 tons of pure gold equals 37.25 gallons, or 149 quarts, or 298 pints to an accuracy of better than 99.993%.
The bushel measurement is likewise a unit of volume. It is 2150.42 cubic inches and subdivides into pecks, gallons dry, dry quarts and pints. Using this gallon dry system, 3.0 tons of pure gold equals 4 bushels, or 16 pecks, 32 gallons, 128 quarts, or 256 pints dry. These measures are accurate to better than 99.95%
Gold likes the imperial gallon as a unit of measure almost as much as the modern gallon and gallon dry: 31 imperial gallons equals 3.0 tons of pure gold. So does 124 quarts, or 248 pints. These measures have a better than 99.93% correspondence.
This next example involves a special unit of weight. It is simple to imagine. Start with a cube as the “Unit”. Its volume is comprised of an un-named substance, and thus the cube has weight; but it too is left un-named for the moment. Now assemble 27 of these cubes into one large cube with three per edge. Finally, divide this now single cube into 1000 cubets. We can now “name” the weight. To do this we go back to the original cube, the first one of the twenty-seven. Any name will do, but the choice for this one is “gram”. The 27 together as one cube weigh 27 grams. When divided into 1000 cubets each resulting little cube weighs 27 milligrams. Our entire system of weight measures is based on this 27 milligram cubet. I call it “The Chosen One”, and lay out the proof for it being this weight-measure base unit in great detail in The Geometry of Money. The picture below is from the book and illustrates what we just described above.
Now that we have the “proper” base-unit, we are ready to continue exploring this very special 3.0 ton measure. First we calculated this weight in grams knowing that there is 6000 pounds in 3.0 tons, 7000 grains/pound, and 15.43235… grains/gram. This equals 2,721,554.221… grams. Divide this by .027 (27 milligram) and we discover how many of these cubets are in 3.0 tons. The answer is 100,798,304.5.
This is 100,800,000 cubets to an accuracy of better than 99.998. . .%. When its relevant “factors” are distilled, they are found to be 480 X 480 X 437.5. Now we can apply these quantities to the white portion of the cube on the left side of the photo at the start of this chapter. Its base is 480 cubets by 480 cubets. And its height is 437.5 cubets. Obviously, here again is confirmation of the “natural affinity” between the two grain-based measures called “ounces” and, not only the ton measure, but the gram–based system of weight itself. And already we’ve seen that this white portion of the cube, reformed into the cylinder, embraces a sphere within having a surface-area equal to 1.0 square meter.
Using this 27 milligram system makes the weight of the tetrahedron atop the sphere and cylinder equal to 9,792,000 cubets. This translates to 264,384 grams or 4,080,068.63… grains. But the relevant unit quanta-sizing this tetrahedron’s weight (in gold) is the troy ounce. This is because it weighs 8500.1429… troy ounces. This is 8500 exact to a better than .99998… correlation. When this weight is added to the 87,500 troy ounces comprising the 3.0 tons of the cylinder below the tetrahedron, or below the red portion of the cube, the total is 96,000 troy ounces.
Now when we look at the photo we know that the white portion of the cube, or the white cylinder, is 96,000 avoirdupois ounces; and the entire cube, or the cylinder and tetrahedron is 96,000 troy ounces.
After examining this system of the sphere within the cylinder, it’s obvious that there are “ideal geometric quantities” or “perfect” units to which the actual substantive geometric forms only ever so closely approach. We can distill these ideals by examining the weights of the sphere and cylinder, as we already know that their size has been defined from the sphere’s surface being a perfect 1.0 square meter.
When the actual weights are grain-based units the golden sphere weighs 28,006,795.5… grains; and, the entire cylinder of gold weighs 42,010,193.3… grains. These are the “actual” quantities. The “ideal” quantities to which they have been patterned are respectively 28,000,000 and 42,000,000. The sphere’s equatorial plane is tangent to the inside surface of the cylinder and isolates the two spaces exterior of the sphere’s surface. Each of these spaces contain 7,000,000 grains of pure gold. These, together with the sphere’s four times 7,000,000 grains, comprise the cylinder’s (idealized) 42,000,000 grain total.
The conformance to this ideal modeling is once again to an accuracy of better than 99.9757%.
Some of the Structural Properties of
The Fundamental Unit
And Their Relationship to the Measures of Man
In The Geometry of Form the “fundamental unit” is a line 1.0 unit long. But unlike in Euclidian geometry, this line is configured three-dimensionally as the six edges of a tetrahedron. The Geometry of Form also recognizes that polyhedrons (in general, and of every size) have unique equivalent forms constructed from the spheres defining each of their individual vertex’s volumetric domains. In this sense, the tetrahedron’s four vertices become synonymous with the center points of four spheres.
Since this tetrahedron’s six edges total 1.0 unit in length, each edge measures .1666… unit making the radius of its formative spheres equal to .08333…, or 1/12th unit. Keep in mind that what we are seeing is that these specific quantities and ratios are literally built into, or inherent to this fundamental unit of length, regardless of what “name” humanity assigns to that unit; and so too are each of these quantities that arise from the following calculations.
First calculate the volume of this sphere: 4πr3 / 3 = the volume of a sphere. The volume of the fundamental unit of length’s structural sphere is .002424068… cubic unit. But this quantity can be equally expressed as:
1.0 / 412.5296124…
exposing the American silver dollar’s gross weight quantity of 412.5 (grains), and the other units of measure exposed in The Geometry of Money that are based on this exact same quantity, to be at the very start of geometry’s three-dimensional system of accounting. More will be said about this quantity with respect to the many measurement systems of which it is an essential quantitative ingredient and to its many “names” within those systems.
But for now, let’s compare this primal structural sphere’s volume with that of the fundamental unit’s tetrahedronal volume of:
1.0 / 1832.8200776…
Both of these ratios are describing the volumetric “size” of their respective geometric forms. And both are referencing, or scaled to, (first of all) this system’s Base Unit of volume (1.0)3. This is most easily modeled by a cube with a (1.0)1 unit edge-length. It is this cube that will hold precisely 1832.8200776… volumes equal to the fundamental unit’s tetrahedronal volume; or just as precisely, 412.5296124… volumes equal to the volume of any one of its formative spheres.
Thus far in the few paragraphs above, three distinct geometric forms have been introduced. They are (in order of volume-size starting with the smallest) a tetrahedron (1.0 / 1832.8200776…), a sphere (1.0 / 412.5296124…), and a cube (1.0). But there are two more forms inseparable from this geometry implied in the above ratios, which are far larger than the first three. They are another tetrahedron with a volume equal to 412.5296124… units, each one equal to the cube’s volume of (1.0); and it’s formative sphere with a volume equal to 1832.8200776… times this same cubical volume unit. This is because in geometry, with respect to “form”, every regular polyhedron has a “reciprocal” opposite-formed counterpart with identical quantitative units.
It’s important to keep in mind that these are the “quantities” that literally structure three-dimensional geometry at its inception. And so too must we regard the sequential geometric structures encountered in calculations associated with investigating any single form. For example, the radius of the formative spheres of the tetrahedron with a volume equal to 412.5296124… units is one-half of this tetrahedron’s 15.1835663… edge-length, which is 7.59178318… But if this same radius be expressed as (437.553729…)1/3, or the “cube root” of 437.553729… , we can clearly see that man’s 437.5 grain–based weight measure known as the avoirdupois ounce, which is our common marketplace ounce, is also mirroring what is truly a “natural” geometric quantity. And the tetrahedron’s 15.1835663… unit edge-length can also be re-written as (3500.42983…)1/3 units, or the “cube root” of 3500.42983… units, which as 3500 grains would be the ½ pound measure in this same avoirdupois system.
Another example of this can be seen in the form of a regular tetrahedron. As demonstrated in The Geometry of Money, the edge-length-sum of a cube with a 437.5 unit volume (1.0 ounce avoirdupois) is the same as a tetrahedron with a 412.5 unit volume.
This shows that the quantitative structure of “man’s” avoirdupois system of weight measures is actually borrowed, or “stolen” from some of the earliest manifestations of geometry’s own natural system of quantification. It also shows us that America’s monetary measures (412.5; and below, 371.25 and 2475) to be geometrically inseparable from the avoirdupois measures. This next example confirms this unequivocally.
Transform the cubical 3500.42983… volume unit quantity described above into a tetrahedron. A cube having its edge-lengths-sum exactly equal to this tetrahedron’s (the edge-lengths-sum of both forms equal the same length line) will have a volume equal to 8(3712.7665…) and is one of the quantitative sources for America’s preference for 371.25 grains of pure silver for its dollar coin.
And if we were to now take the 412.5296124… unit volume tetrahedron’s formative sphere and divide this sphere’s 1832.8200776… unit volume into two spheres, and then the two into four spherical volumes, and so on infinitely, all the while reforming them back into a single closest packed unit, the volume will grow from the original’s volume until a limit is reached. In The Geometry of Form this limit is known as a sphere’s “Maximum Volume Potential”. It is a property of every sphere. This sphere’s MVP is 2475.17766… and is the source of another of America’s monetary measures in that the 1792 Coinage Act called for precisely 247.5 grains of pure gold to be contained in its $10 eagle coin.
Now look at the following pairs of measures:
3712.5 / 3712.7665… = .9999282…
412.5 / 412.5296124… = .9999282…
2475 / 2475.17766… = .9999282…
On the left in red are America’s monetary measures for her gold and silver coinage. To the right in red are the most primal formative measures distilled directly from The Geometry of Form’s fundamental unit of length. As the strings of “9s” clearly demonstrate, America’s (behind the scenes) monetary architects managed to create a system enshrining this timeless geometry to a degree of perfection beyond the ten-thousandths number place! And many hundreds of years earlier in England, when King Richard IV declared 437.5 grains to be the new ounce of the realm, and that 16 of those ounces containing 7000 grains (2 X 3500) would be the new pound, it was this same geometry back then being enshrined into this secretly coalescing global system of weights and measures.
An equivalent form to the fundamental unit of length configured as the sum of a tetrahedron’s edges is this same line (1.0) reconfigured as the edge-length-sum of a cube. It too comes with its own unique set of eight formative spheres. Since a cube has 12 edges, each measures 1/12th unit in length, making the radius of its formative sphere 1/24th unit.
A sphere with a 1/24th unit radius has a volume of .000303009… But this quantity can be equally expressed as 1/3300.236…, or just as equally:
[(1.0 / 412.5296124…) / 8.0]
showing once again that this volume is a power of the now familiar 412.5296124… quantity. It is also showing us that the volumes of the cube’s eight formative spheres altogether equal the volume of just one of the tetrahedron’s formative spheres.
After the 1873 Coinage Act, the gross weight of the previously debased (in 1853) fractional dollar coins was adjusted slightly upward from exactly 384 grains to 385.80895… This new, rather awkward, number of grains was now simply 25 grams exact. This was the “targeted” weight sought after ever since the first coinage act in 1792 even though America has never officially adopted the metric system to this day! Nonetheless, it is “the weight” of 25 grams (by whatever name) that was needed to finally conform to the geometry. Here’s the reason why.
America’s standard coinage silver is an alloy consisting of 9 parts pure silver and 1 part copper. In the 25 grams comprising one dollar in any combination of fractional coins (half-dollars; quarters; and dimes) there is 22.5 grams of pure silver. This is .723391798… of one troy ounce, which is the ounce comprising the base-unit of weight measurement for gold and silver, and for coinage systems throughout the world.
When one calculates the volume for the cube with an edge-length-sum equal to geometry’s fundamental unit of length (1.0) we find it to be (1/12)3. This is also 1/1728 exact, which means that a larger cube equal to 1.0 unit of volume will hold exactly 1728 of these cubes. Moreover, these fit perfectly within the larger cube with twelve cubets per edge. This cube’s volume quantity can be expressed many different ways, but as 8(.000072337963…) it is clearly a power of the quantity of pure silver contained in the coins comprising one fractional dollar. These are the same measures to the exact same tolerance as the Illuminati’s entire clandestine system of weight measures based on the 27 milligram cubet, where 2.4 cubets equal one grain.
.000072337963… / .723391798… = .0000999983179…
In the beginning of my book The Geometry ofMoney, I show how 1.0 troy ounce of silver (or any substance for that matter) in the form of a perfect sphere has the exact same “face value”, i.e. the same surface area as a cube of silver having a volume of .72360125… the volume of the one troy ounce. This cube weighs 22.506514… grams.
Look carefully at the image to the left, which appears on page 136 in The Geometry of Money. There, these two forms are modeling the grain (the cuboid in the foreground) and the gram (the ten cubes in back). Both measures are constructed from identical cubets, each weighing 1/10,000th gram. The cuboid representing the grain contains 648 cubets compared to the 10,000 in the gram assemblage. These forms model the actual measures to better than .99998318… fine.
Now use these same models but change their common cubet sub-unit of measure from a 1/10,000th “unit”, or .0001, to .000072337963… “unit”. As demonstrated above, this is the 1/8th sub-divisional-unit quanta-sizing the unique internal spatial domain of geometry’s fundamental unit of length when configured into the edges of a cube. Using this scale makes the ten-cube assemblage equal to .72337963… “unit”; and the cuboid in the foreground, with the 648 cubets, equal to .046875000… of the same “unit”.
To apply this natural system of geometry to an earthly human system of measurements requires that the unit be “named”. It seems clear that the Troy Ounce of 480 grains, or 31.10347680… grams was the chosen base unit. Have a look.
.72337963… X 480 grains = 347.22222… grains
.72337963… X 31.10347680… grams = 22.49962152… grams
.046875000… X 480 grains = 22.5 grains (exact!)
The amount of pure silver in the fractional dollar coins is 347.2280629… grains, which is also 22.5 grams (exact!):
347.22222… / 347.2280629… = .999983179…
22.49962152… / 22.5 = .999983179…
This shows that man’s measures imitate geometry’s measures to better than a .99998 approach to perfection! Now, the ten cubes in the background represent the pure silver content in any combination of coins comprising one fractional American dollar. But what then does the cuboid represent, with its 648 sub-unit cubets, shown in the foreground of the image that, in this system, equals exactly 22.5 grains?
It is the Eagle, America’s original ten dollar gold coin. The gold in these coins is alloyed with copper in a proportion of 11 parts pure gold with 1 part copper. If the assemblage of ten cubes of silver in the background of the image is seen as the pure silver content of one fractional dollar, then the cuboid in the foreground contains the exact pure copper content in one ten dollar gold coin. The pure gold content of the coin is 11 X 22.5 grains, or 247.5 grains; 12 X 22.5 grains is 270 grains and is the coin’s gross weight. And in the Illuminati’s 27mg system of weight measurement 648 27mg cubets model the Eagles 270 grains.
There is something else very special about this 648 unit quantity regardless of what substance it is comprised or what system of measurement in which it is referenced. The gross weight quantity of the American silver dollar coin (which was just shown to mimic the internal structuring of the fundamental unit of length) and the mathematical constant π, combine to “create” this special quantity:
π (412.5296124…) = 2(648)
Now what happens if we change the name and instead of the troy ounce being the reference base-unit (a weight modeled by a geometric volume) assign the name “inch” (a measure of length) to these same geometrically derived quantities? For example,
648” = 54’ or 18 yds
648 sq” = 4.5 sq’ (one-half square yard)
2(648 sq”) = 1296 sq” (one square yard)
also, remember the equation above but now it is “inch” based
π”(412.5296124…)” = 2(648 sq”)
Look at the above equations. Notice first of all that 648 square inches equals ½ square “yard” exactly. Thus a “square yard” is naturally subdivided into units of this fundamental geometric quantity. But the equations above also clearly demonstrate that man’s ultimately arbitrary and subjectively derived measure known as the “yard” just happens to be coincidentally the product of two primal geometric constants when denominated in “inches”. Overtly, one square yard is simply 36” X 36”. How can we attribute to “coincidence” the fact that one yard is exactly π, or 3.1415926… inches multiplied by 412.5296124… inches? Covertly, meaning “secretly”, is this really what a “yard” is equal to “geometrically”? This is just one mathematical/geometrical construct exposing its true underlying derivation in geometry.
Since we now know that the square yard is a creature of pure geometry so too must be the acre measure, which itself is based on whole unit quantities of yards or feet. There is 43,560 square feet, or 4,840 square yards in one acre. There is also exactly 6,272,640 square inches in one acre. If these are disassembled into individual square inches, and then placed edge to edge forming a single line, this newly configured acre will measure exactly 1.0 inch by 99.0 miles! Don’t believe it? Do the math, and then realize we’ve all been lied to. None of our measures are arbitrary and all of them conform to the same clandestine system of geometry.
Now, let’s see what happens when the “foot” is named this system’s fundamental unit of length. Remember! This is still geometry’s fundamental “unit”; we are just using a different name for “unit” and seeing to where it leads us.
We’ve just been looking anew at the acre so let’s take up with the foot measure where we last left off with the square inch. Let’s disassemble the acre into its individual 43,560 foot squares, and like with the inch measures re-assemble them into a single line. This acre now measures 1.0 foot by (two units of) 4.125 miles.
The equation below is showing one way the quantity 648, as “feet”, relates to the mile measure. The equivalent right side of this equation exposes the mile blended with powers of America’s afore-mentioned monetary measures of 270 (grains) and 412.5 (grains).
648 feet = [1.0 mile (in # feet)(2.70)3] / [16(4.125)]
The Number of God
The volume of the formative sphere associated with the fundamental unit in its 1st dimension as a line (configured into the six edges of a tetrahedron) is 1/412.52961… Now a quantity of 412.52961…, when multiplied π times, or 3.1415926…, results in a remarkable product: 1296. This perfectly even quantity is a surprise, given the two irreconcilable factors from which it is produced:
π (412.52961…) = 1296 exact
And here again is an example of where the value of a “name” becomes all important. If 412.52961… is a quantity of “inches”, then 1296” is also 108 “feet” or 36 “yards”. And if the quantity π is also a number of inches then it becomes 1296 “square” inches which is 1.0 square yard or 9.0 square feet.
Now a length or area of π inches, or π square inches, divided into 412.52961… parts (the reciprocal relationship to the above equation) makes each part equal to .007615435… inch or square inch. This measure is, of course, special to the geometry of form being a creature derived from the geometry of the fundamental unit and the mathematical and geometric constant π. Confirming this special-ness is the fact that:
13,000(.007615435…inch) = 99.00066142…inches
and that 99 inches has previously been shown to be a “natural” sub-unit of the surveyor’s chain of 66 feet; i.e., 8 times 99 inches equals 66 feet. In this case,
8(99.00066142…inches) equals 66.00044095… feet
which is .99999… the actual measure of 66 feet.
If instead of 99.00066142… “inches” in the above equations we name the measure “square inches”, then the visual geometry resulting from the equation can be seen as a line consisting of 99 one inch “squares”. Each of these squares consist of a bit more than 131 sub-units of area with each measuring 1.0 inch on one side and .007615435… inch on the other. It is 13,000 of these units that together create this 99.00066142… measure. Repeat this 98 more times and one will arrive at a larger square measuring 99 inches per side (or, two times 4.125027559… feet). This square contains 9,801.065481… “square inches”. Just as 99 inches is the “survey chain’s” designed occulted sub-unit, the square of this measure is the sub-unit of 1.0 “square chain” in that there are 8 of these per side, 64 all together. 10 of these “square chain” measures constitute 1.0 acre measuring 43,560 square feet. And 640 acres constitute 1.0 square mile. Once again, here they are, the western world’s (pre-metric) measures of land.
All of the above quantities were derived directly from agglomerations of a more fundamental geometric measure: .007615435. . . In this example, we’ve “named” it either inch, or square inch. The square described above, with the two times 4.125027559… foot edge-length (99.00066142… “inches”), contains exactly 1,287,000 units each measuring .007615435. . . square inch. Another way of describing this is that each of these .007615435. . . square inch units is 1/1,287,000 the area of the greater (99” X 99”) square of which it is a part. Each of these units is a
portion of the entire 99.00066142… inch square.
By its structural appearance alone this is a very interesting looking number. That it is a repetition of the quantity 777 gives it a particular mystic, since throughout history the triple seven has had an inseparable spiritual connotation. For what it is worth, to many religious sects 777 is The Number of God.
There is something else too, along these lines, with respect to this special unit of area derived from the properties of the fundamental unit. In the example above this area was modeled by the 1.0” X .007615435…” rectangle; but when reconfigured into a square, once again its divine property shines forth. Have a look.
(.007615435. . .)1/2 = .0872664. . . and
30(.0872664. . .) = 2.617993878. . . and
2.617993878. . . = (φ)2 , or (φ + 1.0)
The first equation above reads: the square root of the .007615435. . . area is a length measure equal to .0872664. . . The middle equation then shows that every 30 of these measures equal a length of 2.617993878. . .units; and the third equation equates this quantity to the mathematically essential phi proportion . . . which is more popularly known as The Divine Proportion, as well as the Fibonacci ratio and is found throughout all of nature’s constructs. Specifically, 2.617993878. . .* is the second power, or “square” of φ, which as we see is also (φ + 1.0).
*Note: φ is equal to [(5)1/2 + 1.0] / 2, which equals 1.618033989…; and (φ)2 = 2.618033989… The quantity 2.617993878. . . derived from the above geometry, compares to the implied ideal as:
2.617993878. . . / 2.618033989… = .9999846
And lastly, and again for what its worth, in the cosmology which parallels “The Geometry of Form”, the inherent potential of Unity is the release of seven others identical to itself. Together, the eight of them combine to form the next power of the “Original Unit”. In the beginning of geometry, 7 “others” is the potential of 1.
We’ve already seen that when geometry’s fundamental unit of length is re-configured into the edges of a cube, each edge equals .0833333… and encompassed a volume of .000578707… We also know this volume as 1/1728 cubic unit, and that it requires exactly 1728 of these cubes’ volumes to perfectly fill a cube with a 1.0 “unit” edge-length; 12 cubes per edge. Since we’ve given this unit the name “foot”, its “cube” equals 1.0 cubic foot; and each of its 1728 natural sub-unit cubets (each one with edge-length-sums totaling 1.0 lineal unit, 1.0 foot) long ago we’ve come to call them cubic “inches”. There are twelve per edge; and we call this cube’s edge-length a foot. Each face of this larger cube measures 1.0 square foot.
Now, let’s make the inch the base unit of length. Again, look at the larger tetrahedron inspected earlier with the volume of 412.5296124… cubic “inches”. Transform this volume into a perfect cube and it will have an edge-length equal to 7.444205890 inches. Now this is a very interesting
cube. Agglomerations consisting of eight of these cubes…together form one larger cube and create the perfect size cube for embracing within the confines of its face-planes a very special sphere having a volume of exactly 1728 cubic inches . . . that’s 1.0 cubic foot exactly! This sphere’s perfect volume of 1728 cubic inches can then be re-package as a single cube; its edges are divided “naturally” into 12 units, each unit 1.0 inch in length.
In the light of the above described relationships this 1.0 cubic foot unit has been exposed as a primal construct of geometry; by whatever “name” you want to call it. It is inherent to, and derives from, The Geometry of Form’s fundamental unit of length (1.0 unit) in its forms as both the six edges of a tetrahedron and the 12 edges of a cube. It is this cubical volume of 1.0 cubic foot and any one of its individual face’s surface areas (of 1.0 square foot) that has patterned all of humanity’s customary units of length, area, and volume.
Measures of Length and Area
Measures of length and area are synonymous with measures associated with land here on Earth. Appropriately, man’s measures of land have been derived from the geometric relationship between a sphere’s volume (like earth) and its surface area. To prove this, we must return to the cube equal to 1.0 cubic foot.
Any one square face-plane of this cube measures 1.0 square foot. Re-formed into a sphere’s surface it will encompass a volume of .094031597… cubic foot. This quantity is the maximum volume that 1.0 surface unit is capable of enclosing. The relationship between this sphere’s surface and volume (1.0 / .0940…) is equally expressed as 10.63472310… / 1.0. Specifically,
1.0 / .0940… = 10.63472310… / 1.0
This means that for every unit of volume there is ten-plus units of surface. Specifically, there are 10 “square surface units” with each square measuring 1.06347231 square “feet”. This square became man’s fundamental unit of land measurement. Let’s see why?
The edge-length of this square is 1.031247938… “foot”. This is 1.03125 foot (or, 33/32 foot) to a .99999800… degree of perfection. And [(2)9 X 1.03125 foot = 5280 feet], which is one “mile”. In “inches”, 1.031247938… foot is 12.37497526*… inches; and (2)9 of these units measure 5279.989444… feet. Again, this is a “mile” to the same .99999800… degree of perfection.
* 123.7497526… X 3 = 371.2492578… ; and, 123.7497526… / 3 = 41.24991753…both of which are a .99999800… degree of congruence to America’s 371.25 grains and 412.5 grains monetary measures.
But America’s prime coinage measures of weight make the behind the scenes manipulation even more blatant since as portions of a foot, show the same geometry at work: (.2475′ + .37125′ + .4125′ = 1.03125′). The Geometry of Money (the predecessor of this book) proved beyond any doubt that this is by design.
This square base unit of land measurement aggregates, once again, like the classic chessboard: 8.0 per edge, 64 in all. Using the modeled ideal measure of 1.03125 feet/edge for this base unit, makes the edge-length of the next (now a composite) square measure 8.25 feet; which is also both 2(4.125) feet or, 99 inches. Again, 8.0 of these 99-inch squares form each edge comprising the next composite square unit of area. This square measures 792 inches per edge, which is also 66 feet. This is the length of the surveyor’s chain, which was the measuring rod in use from the early 1600’s right up into the 1960’s. One square chain contains 4356 square feet, which is a tenth of an acre: thus 10 square chains equal one acre. A square, 80 chains per edge, is a square mile, and contains 6400 square chains; which is also 640 acres.
It’s clear to see that regardless of what “name” man assigns to the “unit” in the geometry unfolded above, to geometry it is simply the “unit”. The evolving geometric forms are always the same, and they will always have the same internal sub-structuring. There will always be a sphere, perfectly quanta-sized by a 1728 unit volume, resulting from a radius derived from the edge of a cube having a 412.5296124… cubic unit volume. And the cubical form of this 1728 unit volume will always be sub-structured with a 12-unit edge-length with each individual “unit” equal to the fundamental unit of length in its form as a tetrahedron. So regardless of the name, the geometry is the same.
The previous geometric units of length, area, and volume descended from the properties of a line equal to 1.0 lineal “unit”. Man’s customary units of volume (such as our various gallons and their derivatives) are based on the cubic inch . . . and the same occult geometry.
For example, the Roman gallon contains 216 cubic inches (6x6x6) and is overtly an eighth of a cubic foot. But its occulted genealogy is
(1/6)π(412.5296124…) cubic inches
The modern gallon of 231 cubic inches overtly is the Roman gallon with an additional 15 inches. But again, covertly, the modern gallon is:
(1/6)π(412.5296124…) + (1.0/.0666…) cubic inches
Clearly, the formative structural quantities of the fundamental unit of length combine along with π to create the Roman and modern gallons. And so too can the dry gallon (or gallon dry) measure (268.8025 cubic inches) be derived from a relationship between the two specific geometric forms noted above. These are the sphere with a 1.0 unit surface area, and the sphere with a 1.0 unit volume.
In The Geometry of Money, we saw man’s measures of land fittingly spawned from the geometric relationship between geometry’s fundamental unit of surface area and the spherical volume within its confines. Traditionally, produce from the land has been measured in bushels. Beginning with two dry pints to one quart dry; then four quarts dry to one dry gallon; four gallons dry to one peck; and finally two pecks to a bushel. Thus like the squares of land measures above, the bushel essentially consists of 64 pint dry sub-unit measures.
The dry pint volume unit is a portion of the volume unit contained by a (1.0) unit surface area in the form of a sphere’s surface. This volume of .0940315… unit is then divided by the spherical surface area quantity containing (1.0) volume unit: 4.835975… units. The resulting portion is the pint dry measure.
.0940315… / 4.835975… = .019444183…
Since this.0940315… volume measure is a portion of a cubic “foot”, this .019444183… quantity also is scaled to the cubic foot. Therefore,
.019444183… X 1728 cubic inches = 33.59954… cubic inches
Man’s pint dry measurement is 33.6 cubic inches; geometry’s measurement is 33.59954… cubic “units”. They are the same measure to better than a .99998… correspondence.
An Interesting Cube Indeed!
GEOMETRY clearly shows that the American monetary measures, along with the grain/gram conversion quantity, to have been “borrowed” or “stolen” from the fundamental constructs of geometry! This is simple to prove.
The gross weight of the silver dollar coin is 412.5 grains. In this case this “number” is first and foremost a “quantity” of weight-standards; but to geometry, it is simply a “quantity”, devoid of substance leaving only volume. This quantity is most easily modeled geometrically by a simple cube with a volume equal to 412.5 units.
The American gold coin in circulation at the same time was the eagle. It was the equivalent in value to ten of these silver dollar coins. Therefore the gold eagle was worth the same as the 4125 grains of coinage silver comprising the ten silver dollars. This too is most easily modeled geometrically by a simple cube with a volume equal to 4125 units. This is a very special cube. We can see why I say this first by examining its remaining parameters:
Edge-length = 16.03767165…
Surface area = 1543.241472…
The quantity describing the length of this cube’s edge is the same quantity describing the pure gold content in the ten dollar eagle coin. But this quantity (of grains of pure gold, 247.5) is here denominated in grams:
16.03767165… X 15.432358… grains/gram = 247.499096… grains
247.4990960… / 247.5 = .999996…
Moreover, and of great significance, is the commensuration between the surface area quantity of this cube and the standard quantity for converting grains to grams, or grams to grains:
15.43235835… / 1543.241472… = .00999996…
Another example of this can be seen in the form of a regular tetrahedron. When it is scaled having an edge-length equal to 15.43235835… (once again, the grain/gram conversion quantity) then its surface-area will be 412.501209…
This is really quite remarkable since “history” tells us that these human designed weight quantities were arrived at subjectively and arbitrarily; and that the gram was a derivative of the meter measure, which was simply a ten-millionth part of the earth’s quadrant running through Paris France from the north pole to the equator. History does note, however, that the birth of the metric system amidst the French revolution in the 1790’s, and the Coinage Act of 1792 in the USA, both occurred contemporaneously.
Prior to this point in time, the “quantities” 15.432…, 16.037…, 412.5, and 247.5 existed only in the eternal relationships among the forms of geometry.
The Winchester Bushel
A “Natural” Unit of Account
If one investigates the bushel measure (2150.42 cubic inches) it too will be found, contrary to the historical derivation, to be firmly embedded in the hierarchy of geometric form. Whereas we just saw the pint dry measure coming from the surface unit’s volume in the form of a “sphere’s” surface, in the case of the bushel measure, among other means, it can be derived directly from the primal geometric forms of both the “cube” and “tetrahedron”. Here’s how that works.
Start with 1.0 unit of surface in the form of a cube. In The Geometry of Form this is the “Unit” in its second-dimensional form (in contradistinction to Euclid’s two-dimensional square). If this cubical surface-unit is magnified by a factor of one-thousand then we have a cube with a 1000 unit surface area. The resulting cube will have a volume of 2151.6574…, which is the bushel measure above to better than .9994… fine. And a tetrahedron with a 1200 unit surface area has a volume of 2149.13986… which, like the cube, is the volume of the bushel to better than .9994… fine. When we find the average volume between these two very unique geometric forms having essentially equal volumes it is 2150.39863… This is .9999900…the bushel measure.
As mentioned earlier there are eight dry gallons in this bushel measure with each gallon equal to 268.8025 cubic inches. Not only is this “quantity” embedded in the geometric hierarchy, but it appears to be a quantity “natural” to our physical world relating directly to at least two of the elements: silver and copper, both prime monetary metals. Watch what happens when we “name” geometry’s fundamental volume unit (in the form of a cube) this time as “1.0 dry gallon”.
Since there are 268.8025 cubic inches within this cube its edge-length is the cube-root of this volume which is 6.45373476… inches; and, the area of any one of its six square facial planes is (6.45373476…)2 making its total surface area 249.90414…square inches. This is 250 square inches to .9996… fine. So in a very real sense, the “ideal” geometric forms on which the macro-modeling of the bushel measure is based views the 1000 square inch surface cube divided into eight gallon sub-cubes with each having a 250 square inch surface area. The original 1000 square inch surface becomes 2000 when the bushel divides into separate gallons. But here is what is most remarkable:
Take the edge-length of this cubical “Unit”, 6.45373476… inches, and reconfigure it into the twelve edges of a new cube. The volume of this new cube is .155557002… cubic inch. If this cube is filled with pure silver it will weigh exactly 412.665… grains. This is the gross weight of 1.0 American silver dollar to .999598… fine!
Now, make another cube with a surface area equal to any one of the six square faces of the gallon dry in its form as a cube. . . i.e., equal to the square of this Unit’s 6.45373476… inch edge-length. The edge-length of this new cube is 2.6347261…* inches; its volume is 18.28969351… cubic inches. If this cube is filled with pure copper it will weigh exactly 41257.6732 grains. Since there are 41.25 grains of pure copper in each silver dollar coin, this cube of pure copper (derived from the geometry of the Winchester dry gallon) will make alloy for exactly 1000 of these coins to a precision of .9998… fine.
*Note: This quantity of length is directly related to 1.0 surface unit in the form of a sphere in that its .6347261… portion is also that amount over 10 units in the simple equation describing the surface/volume relationship in a sphere equal to 1.0 unit of surface area:
1.0 / .094031… = 10.6347231… / 1.0
This can be translated to read “The ratio between 1.0 surface unit and the maximum amount of volume it can possibly contain, is the same ratio as that between 10.6347231… surface units and 1.0 unit of volume. Again, this is just another piece of blatant evidence revealing both the nature of geometric structure and its measurement basis, and the bushel measure’s geometric pedigree as well since this portion has been directly distilled from the cube made from one face of the Winchester bushel’s gallon dry when it is itself modeled as a cube.
I should point out that those who have read The Geometry of Money may remember that on page 247 we see that a perfect 1.0 cubic foot of pure silver is actually comprised of 800 units with each one containing “371.25 grams” (which same quantity as “371.25 grains” is the pure silver content in the dollar coin). This metal embodies this geometry and mathematics to a .9999 degree of perfection.
One can further reference page 169 in The Geometry of Money where I showed that today’s standard gallon of 231 cubic inches is .85936700 the capacity of the still in use 268.8025 cubic inch gallon dry. And, that this compared to the 412.5 grains comprising the silver dollar coin being .859375 the weight of 1.000 troy ounce. These measures are in the same proportions to .999990 fine! The much older Roman gallon of 216 cubic inches is only .803563… the capacity of the Bushel’s gallon dry, but this too mirrors the much later 25 gram “fractional silver dollar’s” .803768… portion of (again) 1.000 troy ounce. These are the same quantities to .999745… fine.
Referring back to the two cubes of metals derived from the edge and face measures of the Winchester bushel there is certainly no coincidence at work here. No other metals filling the voids of these two specific cubes (extracted from the geometric properties of the Winchester Bushel) will be base-ten-denominated in whole numbers of units comprising 412.5 grains each.
Here is further evidence that this geometry unites the natural properties of mass and volume for the element silver, with first the supposed “arbitrary” monetary measures devised by the men back in 1792; and second, in relation to other measures of weight and volume that became firmly established before and after that time. For we see that the same cubical volume–measure containing exactly 412.5 grains of pure silver is also:
.0000160273549… barrel oil; and, 16.037730… grams of pure gold are in each $10 eagle coin (different powers of the same quantity to .9993… fine).
.0000723121… bushel dry; and there is .723391… of 1.0 troy ounce of pure silver in America’s fractional dollar coins (.9996… fine).
.00000562379… cord foot; and .5625 troy ounce is the gross weight of the $10 dollar eagle coin (.99978… fine).
.00008998690… cubic foot, which is a 1/10,000th part of .900 cubic foot to .99985… fine: and, American coinage silver is .900 pure silver.
.155496… cubic inch; and a pence (a unit of weight used in the 1792 Coinage Act) is 1.55517… gram (.99986… fine).
Maybe silver is special with respect to geometry and man’s systems of weights and measures. In the following section, the reader will be introduced to a new quantity and its powers and various forms of its expression. This quantity structures one of the most important conceptual relationships among the geometric forms. One of its powers is .00037412297…, and as a portion of 1.0 barrel dry this measure of volume filled with pure silver weighs 1.0004231… pound. This is 1.0 pound exact to a precision greater than .9995… fine. . . finer than the purity of America’s gold and silver coinage. And if we have a complete 1.0 barrel dry of pure silver it will weigh 2674.04899… pounds. This quantity as grams is the gross weight of 100 silver dollars (26.729550… X 100) again to a precision greater than .9995… fine. Why are these measures so special to geometry, to nature, and to man?
Balance And The American Silver Dollar
Modeling the Concept And the Quantity 374.1229…
The act of weighing something originally involved the use of a “balance”. A balance is the simplest of all scales. Geometry recognizes the importance of “balance” and has special forms and scales (as in sizes) which embody “equality”. The now familiar Roman gallon of 216 cubic inches is one such real-world example when modeled as a cube. This is because a cube with a 6 unit edge-length has both its volume and surface area equal: 216 “units”. These two opposing qualities (volume being expansive and dispersive, surface being contractive and tensive) are in harmony, they are equal. Every outward pressing volume unit has one surface unit containing it. They are balanced, but only in this specific size cube.
Similarly a tetrahedron with an edge-length equal to “the square-root of” 216 also has its volume and surface in balance one-to-one. Each measures 374.1229746… units, and like the previously discussed cube, only this specific tetrahedron embodies the concept of “balance”. The height of this tetrahedron is twice the edge-length of the cube: 12 units.
Another way of expressing this unique 374.1229746… quantity is 1 / .002672918… We saw using the example of the cube above that only by naming the cube a Roman gallon were “cubic” and “square” inches made the units of measure. Similarly, if we call the 374.1229746… units in this tetra-volume “grams”, then this .002672918… quantity represents a 1 / 10,000th part of 26.72918… grams, which is the gross weight of the American silver dollar coin!
26.72918… grams / silver dollar’s 26.7295503…grams = .999986…
This gross weight “quantity” (26.72918… grams) as the edge-length of another tetrahedron generates a volume quantity of 2250.653…grams. This is the weight of pure silver in 100 “fractional dollars” (22.5 grams/fractional dollar) to .9997… fine. The surface area of this tetrahedron is 1,237.496372… a “quantity” which, as a measure of length, can be 12.37496372… inches, or 33/32 of one foot. This is recognized as the edge-length of the square comprising the fundamental unit of land which we saw back in the chapter on Measures of Land on page 180.
A weight measure of 374.12297… grains is also 24.242760… grams which can be equally expressed as 1 / .041249427… Again, this is a quantitative equivalent to the grain weight of the silver dollar coin.
At its inception, each of two most fundamental constructs embodying the first two dimensions of the Unit, i.e., . . . the line configured as the six edges of a tetrahedron; and the unit of area configured as the surface of a cube, enclose a specific volume quantity. The 12 unit of surface, as a cube’s surface, holds within exactly 124.707658…volumes contained by the 11 unit of length when configured as the sum of a tetrahedrons edges. Three of these 1.0 surface unit cubes together contain 374.1229746…of these units of volume, which as was just shown above, can be equally expressed by the quantity 1 / .002672918… and can represent a 1 / 10,000th part of 26.72918… grams, or equally 412.5 grains. This is the gross weight of the Silver Dollar, to .999986… fine.
The following exercise will show just how deeply embedded in geometry are some of the weight measures “chosen” in the 1790’s for both the American silver dollar coin and again the “new” metric system in France, in which its new gram based-unit is equal to the weight of 15.43235835… grains. Just above we saw how three one-surface unit quantities (configured as cubes) together hold a volume equal to 374.1229746… “units” (of the volume contained by the line of 1.0 unit as the sum of a tetrahedron’s edges). If these three 1.0 unit surfaces are reconfigured into a single two-dimensional square, then the edge-length of this square is the square-root-of-three, or (3)1/2. How does this relate?
Look at the first two equations below. Note the “quantities” involved: the square-root-of-three, or (3)1/2 times the exact grain/gram conversion ratio ( 15.43235835 ); and, what is arguably the weight of the American silver dollar in grams in the first equation, and grains in the second!
(3)1/2(15.43235835…) = 26.729628…
(3)1/2(15.43235835…)2 = 412.50120…
(3)1/2(.037412297…) = .0648000000040…
Now look at the third equation immediately above. It too is dependent on “the square root of three units”. But most importantly, it literally unites the two different measures of the same weight:
412.50120… X .0648000000040 = 26.729628…
One ten-thousandth part of the equilibrious-tetrahedron’s volume is .037412297… This equation again shows that the proportions defining the grain/gram quantitative relationship is built into the system of geometry and was NOT a creation of the French scientists. One “gram” divided into .0648 unit quantities contains 15.43209876… such quantities. Of course we now call these quantities “grains”.
The following example also demonstrates just how important the name is that is assigned to any particular unit. This equation makes no mathematical sense without the quantities being assigned the names (first) grains, and then grams:
[(3)1/2 / 3]104 grains = 374.1166815… grams =
1 / .002672918…grams
Another example showing just how deeply rooted in geometry is this quantity 374.12297… can be seen when modeled in the form of a cube’s volume quantity. The edge-length of this cube is 7.20562173… which is also the surface area of 1.0 unit of volume when modeled in the form of a tetrahedron.
Certainly, the previous examples firmly establish that this 374.12297… quantity is an essential part of the very fabric of geometry itself. Now, take a look at its impact on some of the other measures of man.
A few pages back we saw how the Winchester Bushel and its subdivisions are actually measures built into the structure of geometry. They are very special quantities not just to geometry, but as we also saw, to the volume size packaging of the 412.5 grain units of pure silver and copper. The fact that these physical elements themselves conform to the geometry of the gallon dry measure is a signpost that there is some kind of “special-ness” associated with this unit of volume. Some of this becomes apparent when this measure is used with another measure just shown to be special as well. This is the quantity 374.12297… which, as a measure of gallons dry, reveals some quite astounding correlations among some other common units of volume measurement.
For example, 374.12297… gallons dry is 100,565.1908… cubic inches. But this measure of volume is also 58,000.3741… imperial fluid ounces; that’s exactly 58,000 to a precision of better than .99999… fine! (Note: the “zero” separation creates and clearly shows a well defined “discrete” quantity.) This means that, to the same precision, this measure is also 11,600 imperial gills; 2900 imperial pints; and, 1450 imperial quarts. These two supposedly unrelated systems (gallon dry and gallon imperial) become correlated or commensurate when this special measure embodying harmony and balance comes into play uniting the two.
“Proof” that the relationships above are NOT coincidence can be found when we look at the reciprocal expression of this special measure: 1.0 / 374.12297… which equals .002672918… Again, as a portion of a gallon dry, it is at the same time .003110343… the standard American gallon which was England’s wine gallon for centuries before; .0248814… its pint measure; and, .0124413… the standard American quart. Now, look at these measured quantities in light of the following:
31.1034768… = number of grams in 1.0 troy ounce
24.8827814… = gross weight in grams of 1st debasing
of the fractional silver dollar (1853-1873)
12.4413907… = gross weight in grams of 1st debasing
of the fractional ½ silver dollar (1853-1873)
The measure corresponding to 1.0 troy ounce is the same measure gleaned from pure geometry to a precision of .999998495 fine! So too do the others correspond to at least .99999+ fine.
There is something else even more remarkable about this 374.12297… measure of gallons dry. It is also 1.647968284… cubic meter. This translates to 1.000 cubic meter plus .647968284… of one cubic meter and may be visualized in the form of two perfect cubes. The larger cube contains exactly 1000 liters, 10 per edge. The smaller cube contains 648 liters to a precision of .99995… which may be reassemble as individual liters into a perfect cuboid 9 liters by 9 liters by 8 liters.
Now refer back to the photo on page 4. The cuboid in the foreground models the most efficient packaging of 648 cubets. Any one of the ten cubes in the background models 1000 cubets. Without naming the “unit” these 10 cubes and 1.0 cuboid are size-less and without substance. Nonetheless, they display the same ratio or proportioning as the 1000 liter and 648 liter measures derived from dividing the 374.12297… gallon dry measure. This is
1000 / 648 = 1.543209877…
By now the reader should recognize 1.543209877… as a power of 15.43235835…, which is once again, the grain/gram conversion ratio. These are powers of the same quantity to better than .99998 fine.
(Earlier) the reader was shown that the relationship between the square-root of three and the quantity defining the number of grains in a gram was directly related to the gross weight of the silver dollar coin in measures of both grains and grams. The equations proving this have been restated below.
(3)1/2(15.43235835…) = 26.729628…
(3)1/2(15.43235835…)2 = 412.50120…
Below (at left) is a drawing of a rectilinear geometric solid which has been derived from these equations. This “cuboid” is an equivalent manifestation of this data. Its volume is 412.50120… cubic units; there is 26.729628… areal units in each of its four rectangular faces; and the edge-length of its two square faces is 15.43235835… lineal units. As presented (without any “named” unit) it is a size-less geometric form void of any substance. Its two lineal components are in a 1.0 / 8.9098762… ratio and are illustrated in the “cuboid” on the right.
In The Geometry of Form there is a similar ratio: 1.0 / .89089871… Though they differ by a power of 10, these quantities are the same measure to .999900216. . . fine. This ratio describes the difference in lineal measures which arise when “the unit”, modeled as any geometric solid, divides from one-into-two (forms of the original). But, which quality of the solid-form unit is being divided? Is it one-unit of surface in the form of the solid; or is it one-unit of volume? This ratio 1.0 / .89089871… illustrates this difference.
A simple way to understand this principle is by using a cube with a 1.0 unit edge-length for the solid geometric form. This cube is one-“unit” of volume. After dividing this volume, each new cube’s volume will be .5 unit. Its edge-length is (.5)1/3, which is .793700526… On the other hand, if this same cube’s surface area (of six units) divides one-into-two instead of its volume, each new cube will have a 3 unit surface area; each face measures .5 square unit. This makes its edge-length (.5)1/2, which is .70710678…
The resulting lineal measures of these two different choices for dividing this cube create the ratios seen below:
.70710678… / .793700526… = .89089871… / 1.0
Now, there can be no question that this ratio is not only fundamental to geometry, but as such, must also be a significant quantity to other geometrical constructs. For example, .89089871… as the surface area of a cubeoctahedron (a cube with its eight vertices truncated to the mid-points of its edges) results in its volume becoming .0680749… which is equal to the volume of a cube with a 1.0 unit surface area (to an accuracy of .9995… fine). Readers of The Geometry of Money may recall that in the chapter titled “The Great Metric Hoax” the cuboctahedron is shown to be the geometric model for the grain, and for the 480 grain troy ounce as well. So we shouldn’t be too surprised when .89089871… as the sum of the cuboctahedron’s edge-lengths results in any single edge measuring .037120780… unit. Again, this is a power of the same 371.25 quantity defining the number of grains of pure silver in the silver dollar coin to better than .99988… fine.
And if a power of this 8.9098762… quantity becomes 890.98762… units of the “illuminati’s” 27mg weight standard, then altogether they weigh 24.0566659. . . grams; or, 371.2510…grains. And again, referencing the 27 milligram system, 891 cubets assemble together as one perfectly complete cuboid (9 x 9 x 11).
Another example may be found in the “star-tetrahedron”. This form is comprised of five smaller tetrahedra: one in the center completely surrounded by the four others. The significance of this form to all of geometry cannot be over stated and is explained in my previous book Some Thoughts on Universe, An Introduction to The Geometry of Form. The star-tetrahedron is the form of Unity’s “potential” actualized. If the edge-length of the star-tetrahedron is .890898718… then its surface area is 4.124188… and the sum of its edges (or edge-length sum) is 16.03617… Again, these are both “monetary measures” directly from the geometry of form:
4.124188… / 412.5(grains in silver dollar) = .009998032…
16.03617… / 16.03773…(# grams of pure gold in eagle) = .999903…
The original form of Unity’s potential is called the “decahedron”. In its form as a geometric solid it has ten equilateral triangles. They combine into two five-sided pyramids which are in turn base-bonded into one form with seven vertices. If .890898718… is a decahedron’s surface area then .4535903… will be its edge measure quantity. This quantity is certainly a power of 453.592370…, or the number of grams in one pound avoirdupois
.4535903… / 453.592370… = .000999996…
This same decahedron will have a volume equal to .05627466… which quantity is also a power of the 270 grain gross weight of the ten dollar eagle coin with respect to the troy ounce:
.05627466… X 480 grains = 27.0118… grains
Multiples of this special .890898718… quantity also show these same “monetary measures” in gram-based units to an accuracy of .9999+:
18X = 16.03617… (pure gold content in eagle)
27X = 24.05426… (pure silver content in dollar)
30X = 26.72696… (gross weight of dollar coin)
What the reader must come to understand is that this is “forensic historical evidence”. It is based upon an irrefutable level of mathematical certainty from which is exposed a magnificent hoax. This outright “lie” has been perpetrated upon all of humanity by some “others” amongst us who remain hidden from view. This group, by what ever “name”, has been uncloaked by their own works . . . something they never imagined would ever happen.
Silver weighs 10,490 kilograms per cubic meter; this is 10,490,000 grams. The 412.5 grains of pure silver is also 26.72955… grams; which is .00000254809… cubic meter:
26.72955 grams / 10,490,000 grams/m3 = .00000254809…m3
There are 39.37007874 inches in a meter. Therefore a cubic meter contains
(39.37007874”)3 = 61,023.74409… cubic inches
Since 412.5 grains of pure silver as a cube measures .00000254809… cubic meter, in cubic “inches” it is:
(.00000254809…)61,023.74409… cubic inches = .1554939921 cubic inch
The edge-length of 412.5 grains of pure silver in the form of a perfect cube is the “cube root” of .155493… cubic inch, which is .537738591… inch. This means that the edge-length-sum of its 12 edges is 6.452863087… inches. If this edge-length-sum becomes the edge of a larger cube (in the same way that the edge-length-sum of 1.0 cubic inch became the edge of a 1.0 cubic foot cube) then this larger cube, by man’s own measures, has the name “1.0 gallon dry” and is an eighth part of the 268.8 cubic inch Winchester Bushel:
(6.452863087… inches)3 = 268.6936184… cubic inches
268.6936184… cubic inches / 268.8 cubic inches = .9996…