ON THE MASSES OF
The most familiar of the subatomic or elementary particles are the proton and neutron (the constituents of the atomic nucleus and called collectively nucleons) and the electron. In a normal atom the nucleus is surrounded by electrons. The nucleons are much heaver than the electron. The proton is 1835.844… times the mass of the electron and the neutron slightly heavier at 1838.388… times. Thus the nucleons, again collectively, are 1837.116… times the mass of the electron.*
* These values were from The Encyclopedia Britannica, 15th Edition, Vol. 25, pg. 818, Table 1. These were the accepted quantities when I first wrote this essay in 1985. Today’s values differ very slightly with the proton at 1836.152… and the neutron at 1838.683… “collectively” these nucleons are 1837.418…
When our physical universe is viewed from the atomic plateau all matter is composed of proton, neutron, and electron assemblages. It seems that nature has divided all the “stuff” of material existence into just two different sized packets. These packets compare size-wise as the quantities 1.0 and 1837. That is to say, the proton/neutron sized packets are 1837 times the size of the electron’s packet.
There must be some reasonable explanation why nature chose, or was forced to choose this particular ratio for the packaging of energy as matter. Yet none has ever been proposed with any kind of acceptance by the scientific community. And so the mystery remains.
But could it be possible the answer is so simple and obvious that it has remained obscured by the overwhelming complexities of the world of the atomic scientists? Could the answer lay within the subtle relationships among the forms of geometry?
In the world of geometry the minimum “thing” is a point. The domain of a point can be considered to be a spherical field diminishing in strength with distance from the point. Consequently, a spherical point must be considered the model for representing a hypothetical minimum physical thing.
Now, when a sphere having a volume equal to 1.0 unit is compared to a sphere with volume 1837.116… their radii are hardly proportioned to a random ratio.
Radius Sph. Vol. 1.0 = 0.62035… = (0.006666…)^1/2
Radius Sph. Vol. 1837.116 = 7.59770…
It seems that their radii are proportioned to a power of 2/3 which is (as will be seen in what will follow shortly) a ratio underlying the very concept “the most economical packaging of volume”.*
* It should be noted that at one time “in the days when the electron was conceived more substantially than it is now (2/3)e^2/mc^2 used to be called the radius of an electron where “e” is the charge of an electron or proton, “m” the mass of an electron, and “c” the velocity of light (Eddington: The Constants of Nature).
Geometry chooses the form of a sphere as the most efficient form of packaging or containing “stuff”, when that stuff is at minimum . . . i.e., containing no smaller parts or sub-structuring. This is because the sphere is that form which contains the most volume using the least surface area.
But in the packaging of many things geometry dictates another form as the most economical. It is the cuboctahedron as Archimedes called it, or the vector-equilibrium as coined by Dr. Buckminister Fuller. This is the most economical form to accomplish the closest-packing of uni-radiused spheres. . . i.e., conglomerates of minimum things.
The cuboctahedron is a form modeling an “ideal”. A simple way to describe this form is to say it is a cube with its corners truncated to the midpoints of its edges. The eight truncated corners are equilateral triangular planes. These eight triangles each comprise one face of eight tetrahedrons which all converge at the center of the cube. The vertices of the tetrahedrons correspond to the centers of the uni-radiused spheres. All of the vectors comprising this form are of equal length.
Thus in the ideal most efficient packaging of “stuff” geometry chooses the sphere for a thing and the cuboctahedron for agglomerations of things.
The cuboctahedron scaled to unity is one of the most convincing pieces of evidence supporting the argument that the mass ratio of the nucleon and electron is rooted in the simple geometries modeling the concept “the most economical packaging of volume”. The model is constructed with eight tetrahedrons, each scaled to represent 1.0 surface-unit. They are edge-bonded with each tetrahedron having one vertex in common at the center of the cuboctahedron. Again, keep in mind that the cuboctahedron models the closest-packing of spheres (things) and that the sphere itself models the minimum thing.
This cuboctahedron can both encompass a sphere within its triangular surface planes and itself be encompassed within another sphere. The surface of the sphere within is tangent to the cuboctahedron’s eight triangular faces; and the surface of the encompassing sphere is tangent to each of its fourteen vertices. Together, these two spheres are an equivalent model of this cuboctahedron (which again is the form modeling the most economical packaging of things).
The volumes of these two spheres compare as:
1.837585… / 1.000255… = 1.837116… / 1.ooo…
Within the proportions of these two spheres is a precise reflection of the numbers patterning the mass distribution between electron and average nucleon.*
* A sphere with a volume of 1.837… units produces the same number patterning as a sphere of 1837… units except that the radius is 1/10th , the surface area 1/100th , and the volume 1/1000th that of the larger sphere. They are intimately related. They are powers of one another.
All of this is clear evidence that at least in the realm of geometry the otherwise undistinguished number pattern 1 8 3 7 is fundamental to modeling the concept “the most economical packaging of things”. This congruence between the geometric ideals and the composition of physical reality could probably be argued as “coincidence” if the above mentioned model was the only geometric model based in unity indicating such congruence. However, as what follows here will clearly show this is but one of many models illustrating the most economical packaging concept. And like the model just presented, they all reveal the number pattern 1 8 3 7 in their underlying proportional structuring.
Before introducing some of the other models, something further should be pointed out regarding the tetrahedronal form. It is the minimum three dimensional (3-d) form, being composed of four points. This is clearly the case since three points defines the minimum plane (2-d), two points the minimum line (1-d), and one point (again) the minimum thing (0-d). The tetrahedronal form is also the one polyhedronal form which contains the least volume with the most surface area. It is the opposite of the sphere which is the form containing the most volume with the least surface area.
Previously it was noted that the eight 1.0 surface-unit tetrahedrons combined to form the cuboctahedron having a direct proportional relationship to the masses of the nucleons and electron. A tetrahedron is also
the minimum non-nucleated system while the cuboctahedron is the minimum nucleated system (with its at minimum 12 spheres surrounding a centrally located 13th sphere).
In the cuboctahedron described above, the prime vectors of the 1.0 surface-unit tetrahedron (its edge-length and height measure), each as a sphere’s radius, was shown to be related to the mass ratios of the primary elementary particles. But this tetrahedron has one other measure mid-way between its height and edge. This is its perpendicular from any edge to opposite vertex. This measure, also as a sphere’s radius, creates a sphere with a 1.1935471… volume. . . a quantity equally expressed as
1 / (1.8378387… − 1)
In the same way, this sphere’s surface area of 5.4413980… expressed as
1 / 0.1837762…
again shows just how deeply this numeric 1837 recipe pervades this primal form of geometry scaled to one unit of surface*.
* This same one surface unit in the form of a cube’s surface creates a 0.0680… volume. Subdivided into 125 sub-cubes makes the volume of each these small cubes 1 / 1837.11… . This is of course the size of an electron with respect to the nucleon. I didn’t know this at the time I first wrote this article in 1986.
Another ideal form of geometry that reveals this same 1 8 3 7 influence while at the same time modeling the concept of packaging volume in the most economical manner (i.e., using the least surface area possible) is the cone of maximum volume. To make this cone from a circle of area, a pie-shaped wedge measuring 1 − (.666…)^1/2 of its 360 degrees must be removed. Stated another way, 0.18350341… of 360 degrees. If we look at this cone’s cross-section and quanta-size it with a 1.0^1/3 unit radius, then this cone’s apothem measurement (distance from tip to base circumference) is
and again reflects this very special portion.
So in essence, the edge of a cube with volume 1.0 and the edge of a cube with volume 1.83711… if properly oriented, delineates the cross-sectional profile of this ideal cone of maximum volume (see illustration, page 132, in chapter titled The Ideal Cones Of Maximum Volume).
More evidence as to the suitability of this number 1837.11… as the parameter in the ideal (most economical) packaging of volume is the fact that as a tetrahedron’s volume, the height of the tetrahedron is 20.396… If this is used as a model for the mass of an average nucleon, then a tetrahedron of volume 1.0 models the electron’s mass. The fact that the edge-length of this electron equivalent tetrahedron measures 2.0396… (which is recognized as a “power” of the nucleon’s height measure) shows that these two particular sized quantities are intimately and geometrically related to one another.* The fact that nature has packaged all of her material “stuff” in two packets equivalent to these special volume ratios, seems to indicate that the eternal and immutable forms of geometry served as her guiding influence.
* And if the tetrahedron’s volume is 1.83711…, its height becomes 2.0396… or the same as the edge-length of the volume 1.0 tetrahedron.
And Its Role
IN THE GEOMETRY OF FORM
Builders learn quickly that measurement, though essential to their craft, nonetheless can never be exact. No matter how much care is taken, (in practice) it is impossible to cut anything precisely, or perfectly, in half. No tool can ever be devised that would enable us to cut one thing into two pieces such that each halves’ measurement could accurately be described by the same number. For if measurement could be precise to enough decimal points, sooner or later a discrepancy in size would be detected.
For framing a quality house in wood this may require all of the pieces to be within 1/16 of an inch (of designed measurement) for the resulting building to come out straight and smooth. For a cylinder to operate properly in an internal combustion engine the pieces must conform to within 1/10,000 of an inch. More sophisticated devises require even greater precision; yet, none expect nor require exactitude.
This same constraint must be observed by nature as well. For example, Earth may be considered a sphere but only to a certain degree of tolerance. On close inspection we find it’s not truly spherical. And more refined measurements would disclose mountains and valleys showing that it is not even smooth. Yet if Earth were reduced to the size of a one inch ball bearing its surface would be as smooth as any one inch ball bearing we can presently construct. The measurable discontinuities on the large scale become inconsequential on a different, much smaller scale.
Again using Earth as an example, where does one draw the line between it and the rest of universe? The “at-most-sphere” doesn’t end with a sharp line of demarcation for it gradually thins with distance from Earth’s surface. But the “at-most-sphere” is not “nothing”. It’s composed of lighter elements and can be weighed. So in a very real sense it must be considered part of this sphere called Earth. Yet rarely is it ever considered part of the sphere in measuring its radius simply because its density is well below our normal considerations for concern (unless you are a re-entering space craft). Nonetheless, in truth airplanes fly through Earth, not above Earth very much in the same way that submarines travel through their respective medium.
Even in Geometry, with its inherent system of formal relationships, tolerance plays an essential role. For example, a 1.0 volume-unit in the form of a sphere has a radius described by the number:
And a 1.0 surface-unit in the form of a tetrahedron has its height described by the number:
These certainly are different numbers, but their difference is very slight. Their congruence extends through the first four number places (0.6203505…/0.6204031 = 0.99991…) which means these numbers are the same quantity until after the fourth decimal place. Thus the Unit, in its form as a spherical volume; and the Unit, in its form as a tetrahedron’s surface are (within a certain tolerance) commensurate forms.
As illustrated here, these two easily modeled three dimensional forms are exquisitely related within the bounds of the same focal plateau. Expressed graphically, this concept might resemble the convergence of the two lines below.
Thus the idea of a focal point draws an analogy with this idea of number congruence, or near congruence. If nature in some manner is using the Unit-volume sphere, and the Unit-surface tetrahedron as available building materials for cosmic constructions, the (near) congruence of these two measures could be artfully employed within the bounds of their parallel tolerance.
Another good example illustrating these built-in commensurate bounds of geometry can be seen by way of a further introduction to what I have come to call the “Transitional-tetrahedron” (a.k.a., “Transit-tet”, or “T-tet” for convenience). This form arises under the following conditions.
Imagine 1.0 surface-unit in the form of a sphere. If two of these identical surface units combine their volumes into one spherical unit, then excess surface area needs to be ejected. This excess surface area is modeled by the surface area of a regular tetrahedron. . . i.e., the “Transitional-tetrahedron. Its volume is:
Now if instead, one of those same spheres divides its 1.0 surface-unit into the surfaces of two new spherical units, then excess volume must be ejected in the transformation. This excess volume quantity is packaged in the form of two identical “Transitional-tetrahedrons”. The volume of this form of the Transit-tetrahedron is:
So beginning with the same 1.0 surface-unit in the form of a sphere, we see two completely opposite transformations use the same geometric form to account for the resulting surface and volume inequalities which arise as a result of geometry’s “packaging economies”.
The cross-section of the 1.0 surface-unit sphere is a circle. The largest equilateral triangle that fits perfectly within this circle’s circumference is the face of a regular tetrahedron. This tetrahedron is the ideal form of the Transitional-tetrahedron. Its volume is:
There is one other Transitional-tetrahedron that has appeared with great significance in The Geometry of Form. This arises with the transformation of a seven point system into an eight point system (i.e., the Decahedron, modeling Unity’s system of seven points of potential, transforms into the actualized eight points of the Star-tetrahedron). Much more will be said about these characters and their transformations throughout this text, but for purposes here it is sufficient to know that the difference in volume between these two systems is:
and modeled in the form of a regular tetrahedron it becomes another form of the “T-tet”.
Clearly, these four tetrahedrons are the same tetrahedron within the bounds of the ten-thousandths number place. This boundary might be termed “the commensurate bounds of The Geometry of Form”. This is to say that geometry itself is imperfect when scrutinized beyond its built-in commensurate limits. But within the bounds of this focal point, this tolerance, the forms of geometry are commensurate.
The drawing above provides us with further insights into the workings of number congruence within the idealized system of geometric forms. The circle represents the cross-section of the spherical 1.0 surface-unit. Its radius measure is 0.2820947… The equilateral triangle within the circle appears to be tangent to the circle’s circumference when in fact the lineal measure from the triangles center to its respective corners is 0.2817994… This particular triangle represents one of the four faces of the Transit-tetrahedron which perfectly models the excess surface ejected when two of the 1.0 surface-unit spheres fuse their volumes into one single sphere.
There is no disputing that this particular tetrahedron is intimately related to this specific sphere since its very existence was born from the need to balance the surface differential arising from fusing two of these spherical volumes. But even without this knowledge, from the seemingly obvious congruence alone, one could deduce that the two forms were related. Not until the drawing is enlarged sufficiently in scale could we determine visually their miniscule difference. And this difference would grow more and more apparent as the scale is increased relative to us, the observers.
The square in the drawing is another example of this tolerance built into the structuring of geometry. It represents one face of 1.0 surface-unit (this time) in the form of a cube. Like the tetrahedron’s face, this cube’s face also appears to be congruent with the circle’s circumference. Both of these forms apparent visual congruence, and the fact that they are both scaled to the same 1.0 surface-unit, would lead one to conclude that these idealized forms of geometry are united within one system of quanta-sization. But again, it is only when the specific measurements are scrutinized beyond the designed tolerance do we see that the cube’s face radial length of 0.2886751… is slightly greater than the sphere’s radius. This difference is practically indiscernible on the scale of this drawing.
In this next drawing these forms have been rendered three-dimensionally. The figure on the left shows the 1.0 surface-unit cube and the sphere intermeshed. The cube fits nicely with the sphere, with its edge mid-points appearing to just touch the sphere. The figure on the right shows the cube delineated by its diagonal planes, this time sitting atop the cross-section of the 1.0 surface-unit sphere. The tetrahedron set on the sphere’s cross-section and intermeshed with the planes of the cube is the ideal form of the Transitional-tetrahedron. It appears to just touch the upper face of the cube. In fact, the cube’s height measures 0.4082482… while the height of the tetrahedron is 0.3989422… (which is also the radius of a 2.0 surface-unit sphere; meaning that this tetrahedron fits perfectly within a hemisphere equal to 1.0 surface-unit).
Now when I make the statement that geometry itself is imperfect, what I mean by that is this: If geometry was perfect we would expect that the transitional-tetrahedron (0.0137706087…) accounting for the volume discrepancy when dividing the surface of the 1.0 surface-unit sphere to be the same transitional-tetrahedron (0.01370203224…) that accounts for the surface discrepancy when fusing two such volumes. They certainly are not the same. Yet within a certain tolerance defining the commensurate bounds of The Geometry of Form, they are the same. Again, the same logic applies to the previous modeling of the 1.0 surface-unit cube and sphere. If geometry was perfect we would expect the sphere’s radius (0.2820947…) to be described by precisely the same number as half the cube’s face diagonal (0.2817994…); and the T-tet’s height (0.3989422…) to be the same as the height of the cube. They are not. But within the focal point inherent to this system of geometric forms, they can be regarded as the same.
Now out of this chaos of apparent non-commensuration there is an underlying order to the variance among the forms. We find that the cube and sphere vary from their implied ideals:
(0.28209…/0.28867… = 0.97720…)
in the proportion as that found between the height of the tetrahedron and cube:
(0.39894…/0.40824… = 0.97720…)
Moreover, this same ratio (0.97720…) describes the difference in volume between the aforementioned, Transitional-tetrahedron generating, Decahedron/Star-tetrahedron transformation. This number sequence, divided in half (0.97720…/2 = 0.4886…) results in the edge length of the ideal Transitional-Tetrahedron.
There are many other examples which will appear in other contexts throughout this text which show in every instance this focal plateau. It is as if the system were inherently imperfect if scrutinized too closely, yet completely commensurate within the focal range.
THE ACTUAL AND THE IDEAL
The simple diagrams above illustrate how the “actual” formative relationships in geometry are often times based on some higher, more perfect “ideals”. This schematic depicts the “actual” relationship between both a 1.0 volume-unit sphere and cube; and, a 1.0 surface-unit sphere and cube.
The smaller light blue square represents one face of the 1.0 surface-unit cube set upon the cross-section of the 1.0 surface-unit sphere (depicted by the turquoise circle). The larger orange square depicts one face of the 1.0 volume-unit cube with the same circle this time representing the cross-section of the 1.0 volume-unit sphere. In both relationships the forms of the respective sphere and cube pairs are seen to be nearly commensurate, but not perfectly. This is because tiny tips of the small and large squares protrude very slightly past the circles circumference.
The diagram on the right is illustrating this idea of perfect commensuration. We could say that this arrangement is the “ideal” that is patterning the “actual” relationships depicted on the left. In this case, the vertices of the cubes share common points on the circumference of their surface-unit cube be adjusted so that its face does fit perfectly on the cross-section of the 1.0 surface-unit sphere, it is no longer 1.0 surface-unit (it now being 0.9549296…). But this respective spheres. Again, this is the “ideal” pattern to which the “actual” relationships can only hope to approach.
If the size of the 1.0 surface-unit cube be adjusted so that its face does fit perfectly on the cross-section of the 1.0 surface-unit sphere, it is no longer 1.0 surface-unit (it now being 0.9549296…). But this “adjusted”, slightly smaller cube is found to be perfectly commensurate to the sphere equal to ½ surface-unit since this is the largest sphere that can be held within this cube.
The simple diagram above clearly illustrates this “actual” relationship between the 1.0, and ½, surface-unit spheres. It seems totally unpredictable that a square, and the two circles natural to that square, would model such a transformation. This is but one simple insight into how geometry is structured.
The following section briefly reviews this degree of variance between the “actual” unit-forms and their “ideal” patterns, and their relationship to some of the formative constants inherent to the Geometry of Form.
OF THE VARIANCE BETWEEN
The radial length emanating from the center of the 1.0 surface-unit cube’s face varies from what would be the “ideal” radial length (if it was in fact equal to the 1.0 surface-unit sphere’s radius) as 1.0 unit varies from 0.97720502…(i.e., 0.2820947…/0.2886751…). This could be considered one measure of the imperfect commensuration between these two 1.0 unit-surface forms. And this same number patterning likewise separates the 1.0 volume-unit sphere from its cubical counterpart. We can see this since the volume of the cube corresponding to that depicted in the “ideal” schematic (with the sphere being equal to 1.0 unit) is 0.9778479… rather than the “actual” volume of 1.0.
The number sequence 0.977… defines some fundamental relationships at the very heart of The Geometry of Form. For example, the height of the “ideal” form of the Transitional-tetrahedron varies from the height of 1.0 surface-unit in the form of a cube (again) as 1.0 unit varies from 0.977204704…(i.e.,0.39894215…/0.40824829…).
Moreover, geometry’s minimum surface unit, the equilateral triangle, when imparted with the motion of spin exposes a natural quanta-sization for the transformation with the 0.977… unit. This is because spinning an equilateral triangle into a circle creates additional surface area. If the circle’s area be quanta-sized as 1.0 surface-unit, then the area of the triangle at rest is 0.41349667…, giving each of the three edges of the triangle a 0.977205024… length measure. This triangle, folded into a tetrahedron’s surface, is the “ideal” Transit-tetrahedron.
And as a last example, one of the most primal, if not the most primal transformation in The Geometry of Form, the actualization of Unity’s potential, when the Decahedron transforms into the Star-tetrahedron releasing Unity’s inherent potential of seven “others”, a volume ratio of 0.9771975… is seen to pattern the transformation.
as Design Function
It seemed conspicuous to me that when number describes the size (in terns of volume) of “one unit of surface” in the shapes of geometry’s three most basic forms the inclusion of a zero separation is common to each. I first noticed this characteristic from a builder’s point of view rather than one practicing simple mathematics.
Look at the volumes of “one unit of surface” in the forms of a:
Sphere = 0.0940315…
Cube = 0.0680413…
Tetrahedron = 0.0517002…
A builder could utilize this zero separation to facilitate keeping within the bounds of a given tolerance. There is a clear line of demarcation separating the primary quantity from its residual, which is so small after the zero that it can be ignored or discarded. In this light, many of the forms of geometry which comprise the characters in our transformational scenario exhibit this same anomaly. This leads to the speculation that herein lies some greater significance.
Listed below are some of these other quantities:
Transit-tet (vol. transformations) = 0.01370…
Transit-tet (S.A. transformations) = 0.013770…
Transit-tet (Deca/Star-tet) = 0.0137500…
Sinus Tetrahedron vol. = 0.01600…
Decahedron vol. = 0.60300…
S.A. Volume 1.0 tetrahedron = 1.49000…
DERIVATION OF THE
The minimum geometric expression of “surface area” is the equilateral triangle. From this triangle flows all the essential proportioning and surface-volume functionality of the “Transitional Tetrahedron”, or “Transit-tetrahedron”, or simply “Transit-tet” for short. This is the source for the ideals to which the Geometry of Form is patterned.
Spun into a circle, the equilateral triangle increases its surface area. The change in surface area due to imparting “spin” to the triangle is described by the ratio:
Fold the triangle into a tetrahedron and volume is captured within the tetrahedron’s surface planes. Spin the tetrahedron on its’ base triangle and it becomes a spinning cone with a volume now greater than the tetrahedron at rest. The difference in volume between the tetrahedron and cone is also described by the ratio:
The two-dimensional triangle spins into the two-dimensional circle; the three dimensional tetrahedron (made from the triangle) spins into a three-dimensional cone. And they both increase their respective qualities by the same measure. The tetrahedron’s base triangle spins into the circular base of the cone and aligns with the circumference of the cone’s base in the same way as the original two-dimensional triangle aligned with its’ circle. This circle formed from the tetrahedron’s base is the largest circle that can be drawn in the original triangle. And if it be considered the cross-section of a sphere, then this sphere’s surface-area is equal to the area of the circle made from spinning the original equilateral triangle (i.e. 1.0 surface-unit).
Now if this sphere divides its surface-area into two new spheres together equal in area to the original sphere’s surface, excess volume must be ejected to keep the surface-volume accounting in balance. The amount of volume ejected is modeled by two of these tetrahedrons. The diameters of these two new spheres are equal to the heights of these tetrahedronal “excess volume” packets. And if two of these spheres were to instead combine their volumes into one spherical package, excess surface-area must be ejected . . . this time modeled in the form and amount of one of these tetrahedron’s surfaces.
All of the relationships cited above derive from the proportions of an equilateral triangle. These relationships are inherent to the forms themselves regardless of how they are quanta-sized. But with these modeled transformations quanta-sized in this manner, this specific size tetrahedron is recognized as The Geometry of Form’s idealized “Transit-tetrahedron”.
The Sphere and Cylinder
of Maximum Volume
When a spherical 1.0 volume-unit distorts its volume into the form of an ideal Cylinder of Maximum Volume (which is the most efficient cylindrical proportioning for packaging volume out of the range of possible cylinder shapes) more surface area is needed to balance the equation. How much more?
The surface area increases from the sphere’s 4.835975… to the cylinder’s 5.535810… “creating” an additional 0.6998345… surface quantum. Geometry models this needed surface area in the form of a sphere. This is probably a correct supposition since the volume of this sphere (0.0550512…) and the combined volume of four Transit-tetrahedrons (0.0550824…) is the same to well within the commensurate bounds of The Geometry of Form.
This is also consistent to what has been observed in other similar geometric transformations. For example, when the spherical 1.0 surface-unit transforms into an ideal Cone of Maximum Volume a quantum of excess volume is ejected in the process. Two transit-tetrahedrons model this volume difference.
HOW THE GEOMETRY OF FORM
ACCOUNTS FOR SURFACE
WHEN DIVIDING THE VOLUME Of “UNIT” SURFACE AND
“UNIT” VOLUME SPHERES
When a “Unit” in the form of a sphere’s surface, or a “Unit” in the form of a sphere’s volume, divides its spherically packaged volume into two new spheres equal to the original Unit’s volume, an additional amount of surface area is needed in both transformations.
In the case of 1.0 unit in the form of a sphere’s surface dividing its 0.0940… volume into two new equal spherical packets, the amount of surface area needed to balance the transformation is 0.2599210… This surface area is found to be that of a tetrahedron measuring exactly ½ the volume of the (0.01370…) Transit-tetrahedron. The height of this tetrahedron is:
When 1.0 unit as volume is packaged in the form of a sphere and then this volume is divided into two spherical packets the amount of surface area need to balance the transformation is 1.25697… The Geometry Of Form models this needed surface area in the form of a sphere’s surface. The radius of this sphere is:
Clearly the radius of the sphere is scaled to the height of the tetrahedron. And just as clearly, both forms provide the same function to their respective “Unit” sphere’s transformation of volume into two equal spherical packages.
THE “ONE UNIT” SPHERE
And The Transit-tetrahedron
What do we typically think of when we refer to a “one unit” sphere? Generally, it is either a sphere with a volume equal to “one unit”, or a sphere having a “one unit” surface area. In some cases, it is possible reference is being made to a sphere with a diameter or radius equal to “one unit”. But, there is at least one other “one unit” sphere that geometry seems to favor. It may be thoroughly hypothetical, but nonetheless seems to exist prominently in the primal hierarchy of ideal geometric forms. Coming to recognize and know this special sphere provides yet another glimpse into the secrets that structure the inherent workings of the geometric world.
This unique “one unit” sphere has its two primary qualities (i.e. surface area and volume) totaling exactly 1.000… unit. This sphere has the following dimensions:
surface area = 0.917377…
+ volume = 0.082622..
radius = 0.270189…
There is a direct link between this unique sphere and geometry’s surface-accounting transit-tetrahedron. For starters, we find that six times the volume of this transitional tetrahedron equals the volume of this special “one unit” sphere.
6 X 0.013770608… = 0.08262(3648…)
This relationship is commensurate through the one-hundred-thousandth decimal place. (Its interesting to note that from this discovery it was found that 6 times any regular tetrahedron’s volume produces the maximum volume potential of any one of its four vertices spherical domains.
It is important to note that the number 6 is seen to be the governing agent whenever the concept of equating surface and volume is concerned (resulting in the creation of three dimensional equilibrious forms). For example, a sphere with a 6 unit diameter has the same number describing both its’ surface area and volume. For each unit of volume there is a unit of surface. This holds true for a cube having a 6 unit edge-length, or a tetrahedron measuring two times 6 units in height. Even the cylinder of maximum volume with a 6 unit diameter results in an equilibrious form.
There is further evidence linking this “one unit” sphere to the hierarchy of primal geometric forms. When the surface accounting transit-tetrahedron is viewed as a construct of four spheres (i.e. the four vertices of the tetrahedron are the centers of four spherical domains) each of these spheres will have a volume of 0.0611812… If a sphere this size be pulverized by repeatedly dividing its volume into a multitude of ever smaller and smaller packages and then reconstituted together into one new spherical form (representing what is known as its maximum volume potential) then this sphere will have attained a volume of 0.08262… and is recognized as the sphere equal to “one unit”.
Another indicator of this sphere being special comes from the Decahedron which gives birth to the transit-tetrahedron as an ejected volume particle en route to actualizing as the Star-tetrahedron (refer here to appropriate chapter). The Decahedron’s size relationship to the “one unit” sphere, which was shown above to be the maximum volume potential of the transit-tet spheres, is described by the ratio below:
0.082622…/0.603005… = 0.13701…/1
This shows that the volume of the unique “one unit” sphere compares to the volume of the Decahedron in the same way that ten transitional-tetrahedron volumes compare to a sphere with volume equal to 1.000…
DIVIDING 2 UNITS INTO 3
Two 1.0 surface-units, each in the form of a sphere’s surface, can combine their volumes together and then redistribute it among three new forms. And, if apportioned just right, it is possible to end up with three 1.0 surface-units; one each, on three separate forms.
As the illustration shows, two 1.0 surface-unit spheres have the potential to become two 1.0 surface-unit cubes plus one 1.0 surface-unit tetrahedron. The exact proportions are as follows:
The combined volume of each of the two 1.0 surface-unit spheres is 0.188063195… This volume is divided into two portions: 0.068142895… for each of the two cubes; and, 0.051777404… for the tetrahedron. The surface area on each of the three forms is the same: 1.000(9943…)
Note: The precise volume of 1.0 surface-unit in the form of a cube is
The precise volume of 1.0 surface-unit in the form of a tetrahedron is
THE SPHERE AS MODEL OF
“LEAST ENERGETIC STATE”
The sphere represents the minimum “energetic” state of a volume unit. Volume in the form of a sphere requires the minimum amount of surface area for its containment. Any deviation from the perfect sphere will increase surface area. On the other extreme is the tetrahedron. As the form of the same volume unit it represents the greatest amount of surface that a single volume unit can achieve. Specifically, the tetrahedron is the maximum omni-symmetrical distortion of the spherical volume unit.
Volume can be analogous to “energy”. It can be modeled any number of ways. One way is to imagine a very large number of tiny particles all trying to flee the center of the sphere. Volume is disbursent by nature. Surface counters this outward radial pressure. It operates at an ideal ninety degrees to its direction. The volume’s outward pressure is just balanced by the surface’s inward containment. Surface is tensive.
“ We can set up a model in a vacant three-dimensional space-frame. Visualize a single sphere with both volume and surface qualities. Liken the volume to millions of radii emanating outward from the center of the ‘
sphere and ending at the inside surface; and liken the surface to a very thin, hard shell. Now imagine for a moment that this volume is a unit of energy with the potential for a vibrancy dependent on its character or mood.
For example, in its least vibratory state it is spherical. Its energy of volume (as seen within the sphere) evenly emanates radially outward from the center impinging on the inside of the sphere’s surface everywhere the same. But if the volume quantity concentrates its energy by focusing it more and more intensely on to fewer and fewer inner surface points, then the form of the volume unit moves away from the sphere. Running through its range of potential distortion, we would see the sphere move from a high frequency geodesic through less and less frequencies until arriving at the tetrahedronal form. All the “energy of volume” is concentrated into just four points. This is the maximum limit of surface potential resulting from the omni-symmetrical distortion of a spherical volume unit.
Since there is more surface now on the re-formed volume unit something will have had to happen to the original quantity of surface along with each distortion. This can manifest one of two ways. First, it could be cracked and fissured with a net-like quality exposing the volume it is still containing within. Or secondly, it could be that “actual” surface was “created” as the sphere distorted into the tetrahedron. In this case the observer would see no visual change to the appearance of the volume unit’s surface even though it has increased by nearly fifty percent.