THE TRANSFORMATION OF ONE INTO MANY
Discovering
The Maximum Volume Potential
Of A Spherical Volume Unit
When science talks about “the big bang” they are actually talking about a singularity in the form of a solitary sphere. This sphere which represents everything that exists is divided up into a “bigillion” parts which reform into the conditions we perceive as our universe today. In a very real sense it can be said that the creation of the universe happened via the transformation of one unit into many.
This concept can be explored geometrically by imagining a single sphere repeatedly dividing its volume into a near infinite number of like sized spheres which are then closest-packed in the most economical manner. . . in a cube-octahedronal configuration. The resulting cuboctahedron will contain more volume than the original sphere since any given volume will be most economically contained in the form of a single sphere. However, as the number of spheres increases, the cuboctahedron will more and more approach a size limit. Of course, as the number of spheres forever increases the cuboctahedronal form will continue to grow in size. But this growth will progress at an ever diminishing rate, and is accounted for in the math farther and farther from the decimal point.
If this model begins with a sphere quanta-sized as representing 1.0 volume-unit (thus having a radius of 0.620350… and a 4.835975… surface-area) then the cuboctahedron which comes into being is prime-line-defined by an edge-length of 0.83056611 (which is also its radial measure)… producing a volume of 1.3504744… and a surface-area of 6.528716…
When these two forms are compared one finds that both the volume and the surface-area of the original sphere have been increased by essentially the same proportion:
S.A. Cuboctahedron / S.A. Original Sphere = 1.3500306…
Vol. Cuboctahedron / Vol. Original Sphere = 1.3504744…
The qualities of both surface and volume are equally conserved toward the end of this transformation process, revealing not only the rationing producing the “Maximum Volume Potential” of any sphere, but also the fact that it is intimately related to the “Ideal Cones of Maximum Volume”. This relationship runs deep: for example, if this cone of maximum volume form is quanta-sized by having its volume equal 1.0 unit then its base radial measure is 1.3504744…^(1/3).
If this geometric process be used as an analogy for the transformation of universe from its singularity state then we would see an original sphere dividing its volume into 2, 4, 8, etc., creating an extremely large number of tiny spheres. When the spheres have been separated from one another, scattered through space (such as the island stars or the particles comprising the stars), the particles or spheres comprising the system’s total surface area is extremely large compared to the volume of the system if remaining in its original form.
But when the spheres are contained and closest-packed rather than scattered, the volume of this system was seen to increase over the original sphere’s volume of one unit. And, its near infinite surface area when scattered becomes finite and equal to the system volume when contained in its minimum closest-packed form.
So it appears that when a sphere divides its unit volume into a near infinite number of spheres it creates an excess volume amounting to 0.3504744… This excess volume can be modeled by five decahedrons having edge-lengths of 0.4880395265… If any one of these decahedrons ejects just the right amount of volume to transform into the equivalent of five tetrahedrons arrayed around a common edge then each of these tetrahedrons has a volume of 0.13699…, or simply 0.137, identifying them as “transit-tetrahedrons”.
The amount of volume ejected in order to bring these decahedrons into the form of five tetrahedrons is 0.001598336…; and, in the form of a regular tetrahedron a volume of 0.159154… will make an edge-length of 1.3504744…^(1/3) showing this relationship to be deeply ingrained.
GEOMETRY’S
VOLUME AND SURFACE UNITS
AND THE
ELECTRON MASS-VOLUME
A sphere is the minimum, most efficiently packaged form of the 1.0 volume-unit. Its surface area (4.8359…) is so proportioned that each 100th part, if in the form of a cylinder of maximum volume’s surface, will embrace perfectly within another sphere with a volume equal to 1/1837.1173…*.
*Note: a sphere bores cylindrical paths through space-time, and a cylinder of maximum volume defines its “domain” at any specific moment in time.
In The Geometry of Form, this portion is called the “electron mass-volume” because this is almost exactly the proportioning in nature which describes the mass-ratio between an electron and average nucleon. It is also the ratio built into the very fabric of geometry whenever volume and it’s most economical packaging solutions are encountered.
For example, exactly 125 of these 1/1837.1173… volume-units, together in a single form as a cube, produce the 1.0 surface-unit as the cube’s surface. The edge-length of this cube is
(The cube root of 1.8371173…) / 3
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MODELING “DIFFERENCES”
WHEN SPINNING THE
TRANSIT-TETRAHEDRONS
When the 1.0 surface unit in the form of a sphere divides its surface into two new spheres casting out excess volume in the form of two transit-tetrahedrons in the process, the volume of the system stays the same as the originating unit-surface-sphere’s volume, i.e. 0.0940… even though surface area is added in the amount of the two transit-tet surfaces: 2(0.413…).
This is a comparatively static view of the transformation. For if “spin” is added to the now four geometric forms (two 1/2-surface-unit spheres and two volume accounting t-tets) not only does the systems surface increase even more, but its volume increases as well, since the tetrahedrons spin into cones.
The static view begins with 1.0 surface unit in the form of a sphere having a volume of 0.0940… and ending with two half-surface-unit spheres (0.5) and two tetrahedrons having 0.413… surfaces. Surface area of the system increased from 1.0 areal unit to 1.827…
Introducing spin to the geometric forms doesn’t change the sphere’s surface or volume. But both qualities change when the tetrahedrons are sent spinning into cones. Their static volumes of 0.013770… each become 0.0333… And their surface measure increases from 0.413… to 0.683… making the system’s totals quite different from the static view. The original 0.0940 volume increases to 0.1330960… and the original 1.0 surface unit has the two cone surfaces in addition (which includes their circular bases) making the surface component 2.36760…
It should be noted here, that if the “ideal” T-tet with volume of 0.0137467… be substituted for the “actual” T-tet above, then the system volume increases to 0.132980… and the system surface component becomes 2.3660254…
Now both of these quantities of surface and volume arising from the dynamic transformation of the 1.0 surface-unit sphere, are also found to be the necessary differences in surface and volume which arise when the 1.0 volume-unit sphere is transformed.
For example, this ideal “spin”-resultant system volume of .0132980… is exquisitely scaled to .0132514… which is the volume of a sphere having the exact surface area necessary for a sphere equal to 1.0 volume– unit to be able to divide its volume into two new spheres. Moreover, the surface area of these four spin-generated volumes if modeled as the volume of a single sphere is 1.259921… which itself is the number that precisely describes the rate of surface growth for any spherical unit continually dividing its volume into two new spherical units.
The sketch above illustrates that when dividing the spherical surface unit, whether its volume or its surface area, the resulting system-surface areas are the same (1.259921…) when “spin” considerations are applied and the “ideal” form of the Transitional-Tetrahedron is used.
Another example begins with the sphere equal to 1.0 volume-unit. Remember, a sphere represents geometry’s form which encompasses the most volume possible using the least surface area. It is the geometric form which most economically packages volume. Its exact opposite form is the tetrahedron having the most surface with the least volume. These two forms anchor the range of omni-symmetrical distortions of a volume unit. And 1.0 volume-unit in the form of a sphere, by distorting into a tetrahedron, increases its surface area by 2.3696… surface-units; and as was shown above, 2.3676… surface-units resulted from our dividing the surface of the sphere equal to 1.0 surface-unit into two new spheres (packaging its excess volume into two spinning transit-tetrahedronal cones).
The ideal pattern to which all of the transit-tetrahedrons are modeled is “The Ideal Transit-tetrahedron”. This is that tetrahedron which perfectly fits within or upon the circular cross-section of the 1.0 surface-unit sphere. The height of this “ideal” tetrahedron is exactly the same length as the diameter of the ½ surface-unit sphere. The diagram below illustrates these relationships.
The volume of this “ideal” transit-tet when spun into a cone becomes 0.03324519 which is exactly the volume of the ½ surface-unit sphere. The four forms (two ½ surface unit spheres and two spinning “ideal” T-tets) now have a total combined system-volume of 0.132980… And, in the form of a “hemisphere” creates a surface area equal to 1.0 surface-unit.
On the other hand, if this 0.132980… volume be divided into two, and each portion packaged in the form of the “Ideal Cone of Maximum Volume” (where its circular base is an inclusive surface component) then each cone’s surface will measure 1.0 surface-unit. Its circular base is equal to the circular cross-section of the 1.0 surface-unit sphere. This is cone number 3 in the drawing to the left. What’s more, the height of this cone is equal to the height of two of the ½ surface-unit spheres stacked atop one another, as the drawing depicts. And of course these two sphere’s volumes equal the volume of this “ideal” cone. Note that the drawing also shows the two other “Ideal Cones of Maximum Volume” which clearly shows the deep relationship these three cones have to the geometry of one sphere dividing its surface area into two new spheres.
This 0.132980…volume, as a single spherical volume was seen to create a 1.2599210… surface area, which is the system surface area after the 1.0 surface-unit sphere divides its volume into two new spheres. This additional 0.259921… surface increment is modeled by a tetrahedron equal to ½ the volume of the volume 0.01370… actual surface-accounting transit-tetrahedron. The height of this ½ transit-tet, which accounts for surface when dividing the volume of the 1.0 surface-unit sphere, is 0.31629… and 0.31626… is the radius of the sphere accounting for surface when dividing the volume of the 1.0 volume-unit sphere.
This sphere’s 0.132512… volume seems to be obviously modeled on, and is commensurate to the 0.132980… system volume resulting from the division of the spherical 1.0 surface-unit and resultant spun-into-cone transit-tetrahedrons.
SPINNING TRANSIT-TETRAHEDRONS AND THE
HYPOTHETICAL “ONE UNIT” SPHERE
Built into geometry is an intimate relationship between the transit-tetrahedrons and what I have come to call “The Hypothetical One Unit Sphere”. Simply put, that unique sized sphere which has its qualities of surface and volume sum total “One Unit” when added together. Keep in mind that this is somehow a very different expression of the “Unit” in that it is not dimensional such as the familiar lineal, areal, and volumetric expressions of what is essentially the same “fundamental unit”. None the less, geometry recognizes and embraces this special spherical unit placing it prominently in its hierarchy of geometric characters.
For example, if the “ideal” Transit-tet is modeled by four closest packed spheres, which center-points define the tetrahedron’s four vertices, then each of these spheres surfaces measure ¾ of 1.0 surface-unit. If one of these spheres be pulverized by repeatedly dividing its volume into more and more, and smaller and smaller spherical packages eventually a limit will be reached where the volume of these spheres closest-packed together into one form (i.e. a cuboctahedron, the form springing from the closest packing of uni-radius spheres) no longer registers any more meaningful growth regardless of how many more repeated divisions. For any sphere, this is its “Maximum Volume Potential”, which is 0.082480… for this “ideal” T-tet sphere. Geometry’s “actual” volume accounting Transit-tetrahedron’s “Maximum Volume Potential” is 0.082623… and, 0.082622… is the volume of the sphere equal to “One Unit”.
An even simpler example: the volume of this “actual” Transit-tetrahedron is 0.013770… If six of these volumes are combined into one form its volume is 0.082623… This is also the volume of the afore mentioned hypothetical “One Unit” sphere (to better than 0.99998… fine).
So when the 1.0 surface-unit sphere divides its surface area into two new spheres, casting out two spinning tetrahedrons of excess volume, the two resulting cones have surfaces which together total 1.36760… surface-units. Transform this surface area into a regular tetrahedron’s surface and its volume is 0.08268… Again, this is the volume of that special “One Unit” sphere (to better than 0.999… fine).
If this same 1.36760… surface-unit be transformed into a star-tetrahedron’s surface its volume is o.o7956… And,0.07957… is the volume contained by the 1.0 surface-unit in the form of a square rolled into a cylinder (another fundamental model of “unity”).
USING TRANSIT-TETRAHEDRONS
TO MODEL THE PROTON
AND NEUTRON MASSES
Two similar yet different transit-tetrahedrons account for transformational differences when involving the “one-surface-unit sphere”. If two of these spheres fuse volumes into one larger sphere, excess surface area is ejected in the amount and form of one Transit-tet with volume = 0.0137020… Or, if instead, one of these spheres divides its surface into two new spheres then excess volume is ejected in the amount and form of two transit-tets with volume = 0.0137706…
Midway between the two transit-tets above are two other transit-tetrahedron volumes: the “ideal” T-tet with volume = 0.0137467…; and, the transit-tet which arises in the transformation of a 7 point system into an 8 point system (the actualization of Unity’s potential) with a 0.01375000… volume.
It should probably be noted that the proportionate volume relationships among these very special geometric characters is very nearly the same as that arising between the masses of the proton and neutron.
Proton_ = 1836.152 = 0.9986….
Neutron 1838.683
0.013750 = 0.9985…
0.013770
0.013746 = 0.9982…
0.013770
MODELING THE ALPHA PARTICLE
In The Geometry of Form
The Alpha particle is the nucleus of the element helium, as well as a form of radiation. It is comprised of two protons and two neutrons and is a very stable particle. So too are other atomic nuclei which can be considered agglomerations of multiple Alpha particles.
It is possible this stability is related in part to the fact that four spherical volumes are most efficiently packaged in a tetrahedronal array. This is significant since a tetrahedron is not only a complete structural form, but also because it is geometry’s minimum requirement for a three dimensional form. And most importantly, it is the most stable of all geometrical arrangements.
As atomic nuclei, protons and neutrons collectively are called “nucleons”. Modeling their masses as spherical volumes produces a corresponding average mass-volume for a nucleon of 1837.418162. . . with respect to the electron’s mass-volume of 1.0 unit*. The radius of a sphere so scaled is 7.5981255. . . This creates an edge-length on the Alpha particle’s internal structural tetrahedron measuring 15.196251. . . Close scrutiny shows that this tetrahedron’s proportions are patterned on fundamental geometric ideals based in unity and rooted in the very heart of geometry.
*1997 data for proton’s mass: 1836.152668…; neutron’s mass: 1838.683655…
The Alpha particle’s internal structural tetrahedron is depicted in the previous diagram. It is, to a 0.9998. . . degree of congruence, patterned on the natural distribution of 1.0 surface-unit. This tetrahedron is in essence the manifestation of that “ideal” 1.0 surface-unit form magnified in length by the power of 10, making its’ surface 100 times, and its’ volume 1000 times its’ patterned “ideal”.
Deriving The Ideal Patterns
Geometry’s minimum expression of a surface-unit is the form of an equilateral triangle. This is because three points is the minimum requirement for enclosing an area. The triangle is equilateral because the points are most economically closest packed. By spinning this triangle around the center of its area a circle of greater area is formed in the process. The now two areas can be compared and units defining each can be assigned. If the circle of area be considered 1.0 surface-unit, then the area of the equilateral triangle is:
0.413. . .
Likewise, if the three-dimensional tetrahedron (geometry’s minimum volumetric form defined by four points) is spun into the form of a cone, and this cone be quantified as 1.0 volumetric-unit, then the tetrahedron’s volume, like the spun triangle, is:
0.413. . .
This ratio is built into the geometry of these forms and holds true regardless of their size. Moreover, these proportions as surface and volume are special quantities favored by The Geometry of Form.
For example, as the equilateral triangular area, it is the amount of surface difference that geometry must account for when two 1.0 surface-unit spheres fuse volumes into a single sphere. And if this triangular area be folded into the surface of a tetrahedron, then two of these tetrahedronal volumes account for the excess-volume if one of the 1.0 surface-unit spheres divides its surface into two new equal sized spheres. Evidence that geometry especially likes accounting with this packaging scheme is the fact that this tetrahedron with a surface equal to
0.413. . .
has the three vertices of its base triangle commensurate to the 1.0 surface-unit sphere’s circumference, while its height measure is the diameter of each of the
two ½ surface unit spheres (making for the arrangement at left).
In the case of the tetrahedron volume 0.413. . . , each triangular face equals 1.0 surface-unit. And when this tetrahedron spins into a cone, its volume becomes 1.0 unit. And the height of this cone (and tetrahedron) is also the diameter of a sphere having a 1.0 unit-volume. Now, when this tetra-volume is divided into eight equal portions in the form of regular tetrahedrons each of their surfaces will measure 1.0 surface-unit. Arrange the eight tetrahedrons so that they share a common vertex forming a vector equilibrium (the formal structure defining the closest packing of spheres*) and the largest sphere that can be held within its confines has a 1.0 unit-volume. The smallest sphere that can embrace this vector equilibrium has a volume of 1.837. . . and is seen to be the same numbers patterning the average nucleon.
* Equal-radius-spheres closest packed together form an internal matrix consisting of tetrahedrons and octahedrons. Each sphere has an omni-surround
of twelve other spheres. However, the maximum possible density occurs only in the tetrahedronal components of the structure.
So, in the geometry of four spheres, each with a volume of 1837. . . , in their most efficient packaging form we find a tetrahedronal structure having a volume of
413. . .
with triangular faces of 100 areal units, and a 1000 unit “spun into cone” volume.
Clearly the Alpha particle’s structural tetrahedron is patterned on these formative fundamental geometric relationships which themselves are ultimately based on second and third “powerings” of the same One Unit. But even the first power expression of The Unit , a line one unit in length, shows geometry’s built in bias for mass-volume relationships on the order of the electron and nucleon. For a single line one unit in length, in the form of the edges of a regular tetrahedron, creates a volume of:
1/1833
There is proof that geometry likes this fundamental unit of length manifesting as a tetrahedron. If three of these tetrahedrons are stacked atop one another they will exactly equal the height or edge-length of 1.0 surface-unit in the form of a cube. Furthermore, this cube’s volume can be sub-divided into 125 smaller cubes each of which have a volume of:
1/1837
Again, the same rationing as that between the electron’s mass and average nucleon’s mass.
It can be said unequivocally that built into geometry’s inherent structuring is a fundamental mass-volume unit the size of nature’s electron. We saw that the surface unit, configured as a cube, contains one hundred-twenty-five 1/1837 volume units. Moreover, 27 of these 1.0 surface-unit cubes, together in one cube, produces a volume of:
1.837
which is the volume of that very special sphere in geometry embracing the vector-equilibrium containing the 1.0 volume-unit sphere. It is also the model upon which the Alpha particle’s size and structuring is patterned.
φ
MODELING THE DIFFERENCE IN MASS
BETWEEN THE PROTON AND NEUTRON
If a proton’s mass is indeed 1836.152701 times an electron’s mass; and likewise, a neutron’s mass is 1838.683662 times the size of an electron, then using simple subtraction it can be determined that their difference is 2.530961 “electron masses”.
Both are collectively known as “nucleons” and thus a nucleon is, on the average, about 1837.418182 times the electron’s mass. An electron could be considered as being about 1/1837.418182 the size of the average nucleon. This can also be expressed as 0.000544242. Therefore, 2.530961 of these quantities is
0.001377455
and is seen to be directly related to The Geometry of Form’s transit-tetrahedron 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.
Modeling this difference in mass between the two nucleons occurs in the following manner. First, divide the transit-tetrahedron’s volume into two separate packages, into ½ volume “t-tets” (which are themselves models of surface differential when a spherical surface of one unit divides its volume into two new spheres). Each of these 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 a “star-tetrahedron”) having each of its five sub-tetrahedronal chambers model of this mass difference between the two different sized nucleons.
*
THREE VIEWS OF THE STAR-TETRAHEDRON
Another way of modeling this mass difference is to assign the value 1.000 to the proton’s mass. The electrons mass comparatively is .000544617… making the neutron’s mass 1.001378404… showing again the excess mass (.00137…) being analogous to a power of the transit-tetrahedron. And if the neutron be considered the 1.000 mass unit, the proton becomes 0.998623493… and the electron 0.000543867… In this case 2.530961 “electron masses” equals 0.001376507…, or one-tenth the volume of the 0.013770607 transit-tetrahedron.
Its probably also worth noting here that the mass ratio between the electron and alpha particle, 1/7294.299…, when expressed as 0.0001370, again shows the influence of the (surface accounting) transit-tetrahedron since its volume is 0.013702….
APPENDIX
Listed Below Are Some Of The Other Topics From Some Thoughts On Universe:
Infinity As A Single Point (Walking Into The Sphere)
What About Thought
Thought Is The Observer
Modeling The Cosmic Egg
The View From A Far vs. The View Close At Hand
The Role Of Number Place
Quanta-sizing The View
The Kinship Between The One And The Seven
Unity and Its Lowest Common Denominator
The Relationship Between The Sphere And The Decahedron
Edge-Spinning The Minimum Surface Unit
The Forms Of The Transit-Tetrahedron
The Transit-Tetrahedron As Fundamental
The Transit-Tetrahedron Viewed As A Four Sphere System
The Transit-Tetrahedron And The Divine Proportion
Modeling A Dilemma (The Dual Nature Of Light)
Deca/Star-tet Proportions Same As Focal Point Of Geometry
Number Patterns As Conceptual Expressions
Surface And Volume As Co-Equal Qualities
Modeling The Original Sphere
The Number Pattern 1837 And The Regular Tetrahedron
The Cuboctahedron (modeling The Transit-Tetrahedron)
The Relationship Between The Two And The Five
The Thousandth Part
The Evaluation Of The Cosmical Number
From Many Into One (The Certainty Of Global Domination)