The Release of Seven Others
In the illustrations presented thus far a common theme emerges. In each case, the first complete stable system form occurring after the original unit is comprised of that original unit plus seven others identical to itself. From that stage onward, this system continues to divide as a system, and forms the basic building units of all other systems to follow*.
* A good analogy is the development of a living cell which through repeated divisions form organized systems such as hearts, kidneys, lungs, etc. which also work together as a single creature in the end.
Thus, the original unit joins together with the seven others of its inherent potential, and together as the eight form all other “units”. Built into the very fabric of the natural division of unity process is an unmistakable and undeniable “potential”. We can even deduce some of what it looks like. It is seven others identical to the original one (whose potential they embody). They are locked together in one system. This system is modeled by the regular decahedron.
With the first division from 1 into 2, the process of actualizing this potential begins. Its culmination is complete with division into the system of eight.
This system is modeled by the regular star-tetrahedron. Thus seven (spherical) points modeled in one system represents Unity’s potential; and, eight (spherical) points in one system depict that potential actualized.
The arrangement of eight spherical points into a star-tetrahedron is seen in the illustration on the following page. These eight spheres are the same size as the seven spheres in the decahedron (depicted above). These “spheres” represent the unique domains of each individual “point” resulting from the original point dividing into 2, 4, and 8.
One of the consequences of the prohibition against “two points occupying the same location” is the establishment of the minimum line. This occurs when two points are in critical proximity to one another and can move no closer. This is the case in both the decahedron and star-tetrahedron. The distance between these points is the minimum unit of length possible and can be designated as. . . 1.0 lineal-unit.
Now if we look at these two systems as “points” and “lines” rather than spherical domains, a more distinct picture of the creatures at the heart of each system emerges. These can be seen in the illustration below.
On first inspection these appear to be two very different forms. And they are, outwardly. The star-tetrahedron is an omni-symmetrical four pointed star in appearance, and for this it was named. The asymmetrical decahedron more closely resembles a bi-valve (clam).
But the star-tetrahedron can also be viewed as the equivalent of five tetrahedronal volume domains (a central tetrahedron is encased within the four other tetrahedrons surrounding it). If these same five tetrahedronal sub-system volume domains be arranged differently, as illustrated in the drawing below, then except for the slight gap, the form of the decahedron is found to be very closely imbedded in this rearranged star-tetrahedron. Keep in mind that this rearranged star-tetrahedron is still identical to the original in both volume and surface.
It should be noted here, and it is explored in depth in later chapters, that this gap or sinus amounting to 7.3561… degrees itself embodies that minimum expression of the natural division of unity process, which as shown earlier is (1.0/0.7)2 and is equally expressed as 0.2040816… We see this in the angular description of the gap,
7.3561… / 360 = 0.0204336…
Its relation to the 360 degree whole is the same as (1.0/0.7)2 through the thousandths number place.
Up until this point, cosmologically speaking, it can be said that as a consequence of reproducing by the natural division of unity process the Origin-al unit is found to have a inherent potential comprised of seven others in the form of the decahedron. This potential is actualized with division into eight after beginning the whole process by first dividing one into two. The star-tetrahedron models this system of eight in its lowest, most quiescent state. So in a sense, the difference between “unity and its potential”, and “unity and its actualized potential” is really the difference between everything and nothing, again cosmologically speaking. So what does this difference tells us, at least from its geometry’s point of view?
A few pages back scale was assigned to these geometric forms. Each sphere’s diameter is 1.0 lineal unit. Thus each line comprising their internal system counterparts is also 1.0 lineal unit. Knowing this, it is easy to calculate their volumes. The system of seven points in the decahedron is volume 0.6030056…. The system of eight points as the star-tetrahedron has a volume equal to 0.5892556…. If these volume quantities are written another way, as they have been below, one can’t help but notice how similar they really are. We can see in these volume quantities a reflection of the similarity in their forms:
5+5√1/2/12 = decahedron volume
5×2√1/2/12 = star-tetrahedron volume
Their difference in volume is
and when packaged in the form of a regular tetrahedron provides us with direct evidence linking a system of seven becoming a system of eight to a spherical unit dividing one into two. For this tetrahedron, known in The Geometry of Form as a transitional-tetrahedron, or transit-tet, performs the same service for both transformations.
Consider the following examples. Imagine a 1.0 surface-unit in the form of a sphere. If we divide this 1.0 unit of surface 1 into 2, like cleaving an orange in half, and then reform each half-surface-unit into new spheres we will find there is an excess of volume that can’t be accommodated within the surfaces of the two new spheres. If this volume be packaged in the form of two regular tetrahedrons, then the volume of each one of them is:
Instead of dividing the surface of the 1.0 surface-unit sphere, let’s combine or fuse two of these sphere’s volumes into a single sphere. Now there is excess surface that must be discarded. If this surface be configured as one regular tetrahedron, the volume of this tetrahedron is:
And again, cleave the 1.0 surface-unit in half and place a regular tetrahedron on its cross-section such that the three tips of its base triangle share points on the sphere’s circumference. The volume of this tetrahedron, which is ideally scaled to the spherical 1.0 surface unit, is:
This ideal transit-tetrahedron with the 1.0 surface-unit sphere is depicted in the diagram below.
Notice the ½ surface unit sphere is also depicted in the diagram. Its circumference is equal to the height of the tetrahedron and is strong confirmatory evidence indicating that this is an arrangement at the very heart of geometry’s transformational operations. Later chapters will show this to be true beyond any doubt.
The essential point to be gleaned from the above illustrations is that geometry uses the same unit and form (transit-tet) to account for the difference (in volume) between the decahedron and star-tetrahedron as it does to account for the surface and volume differences in spherical transformations rooted in unity. This shows that geometry is both aware of this relationship between the seven and eight and connects it to the transformation of the one into two.*
*In later chapters there is much more to be learned about the role of the transit-tetrahedrons in The Geometry of Form.
Studying the ratio of their volumes is another way of exploring this difference between the decahedron and star-tetrahedron, unity’s potential and potential actualized.
0.603005… / 0.589255… = 1.0233345… / 1.000000…
The equation above brings into focus another view of this difference quantity and describes it by the number sequence:
This is to say that the volume of the decahedron exceeds that of the star-tetrahedron by 0.0233345… of the star-tet’s volume. This must be considered a paramount volume characteristic describing Unity’s actualization of potential. This relationship exists eternally between these two forms. It is certainly a primal, if not the primal volume packaging characteristic of this transformational geometric system.
So it is of great interest when contemplating the structure of geometry to find that our own material universe, in terms of how “stuff” (volume/mass) is packaged by nature, likewise exhibits this very same numerical relationship. However, it is not generally recognized as 0.0233345… but in its form as a ratio between 1 and 1837, or:
and can be read as “the one dividing into two power of the relationship between Unity and 1837” . . . which ratio is of course that describing the average mass(volume) proportioning between the average nucleon and electron. This correspondence between geometry and nature is significant since these two sized packets (proton/neutron sized, and electron sized) together comprise all atoms and thus all typical matter in our material realm.
Another view of this primal relationship is threciprocal of the previous volume ratio:
0.589255… / 0.603005… = 0.977197…
This puts the focus on a portion (compared to 1.000…) measuring approximately:
In later chapters the reader will come to appreciate the depth to which this ratio is embedded throughout the early and dynamic structuring of geometry. For example, the minimum surface unit capable of geometric modeling is an equilateral triangle (since area requires at minimum no less than three points in closest proximity). If the motion of spin is imparted to geometry’s minimum surface unit, spinning the triangle into a circle of area, this motion can be quantified since it manifests as surface area. By assigning 1.000… unit to the area of the whole circle, the area of the original triangle is 0.413496… Their difference in area, 0.58650…, is the result of spinning the equilateral triangle. And the edge-length of this primitive triangle is
reflecting the same proportioning as found between the volumes of the star-tetrahedron and decahedron. Moreover, and getting back to the 1 into 2 division and its inseparable relationship to the 7 into 8 transformation, is the fact that this size triangle, if folded into the surface of a regular tetrahedron, has a volume of
and is recognized as an afore mentioned transit-tetrahedron charged with balancing the surface and volume differentials associated with (and common to) both transformations.
The 1:2 Ratio and Its Influence On
The Form of the Decahedron
Geometry shows in many ways that the form of the Decahedron is truly related to the one-into-two spherical transformation. For example, reduced to a minimum expression, the implied ideal volume of the transit-tetrahedrons (that form which balances the surface and volume accounts in the 1 into 2 spherical transformation) is 1 3 7 . If this idealized volume quantity is packaged in the form of a regular tetrahedron it will create an edge-length on that tetrahedron described by the very same number patterning as the Decahedron’s polar axis:
Edge-length of tetrahedron with volume 137 = 10.5146…
Decahedron’s polar axis = 1.05146…
It is also known that a tetrahedron with an edge-length equal to the Decahedron’s polar axis length of 1.0514622… will have a volume of 0.136998… (0.1370) which is ten times the transit-tetrahedron’s volume.
Thus in the realm of ideals, a volume of “137 units” in the form of a tetrahedron and the form of the seven pointed Decahedron share a common bond. The question then is should this be viewed as being fundamental to some system hierarchy, or merely an unrelated coincidental happenstance?
In the same sense, how should we regard the fact that the only two variations of a right triangle constructed on a 1:2 proportioning appear as the two fundamental triangles structuring the Decahedron’s form? Its surface is comprised of ten equilateral triangles each of which is really comprised of two 30-60 degree right triangles which have their hypotenuse and minor leg in a 1:2 ratio. And the angle between any two top and bottom edge-pairs is identical to the major angle in a right triangle having legs in a 1:2 ratio. Even the Decahedron’s pentagonal cross-section turns out to be a creature of this 1:2 proportion (see diagram below).
In the context of the “one-into-two” influence on the structure of the Decahedron is the blatant correlation between the number seven itself (the number of points or spheres defining the Decahedron’s structure) and Unity. Numerically “1 / 7” depicts this relationship and is equally expressed as 0.142857142857142857… But neither shows the formative influence of the 1 into 2 natural division of unity process that is revealed in the view expressed below:
0.02 X 7 = 0.14
0.0004 X 7 = 0.0028
0.000008 X 7 = 0.000056
0.00000016 X 7 = 0.00000112
0.0000000032 X 7 = 0.0000000224
0.000000000064 X 7 = 0.000000000448
0.00000000000128 X 7 = 0.00000000000896
= 0.14285714… = 1 / 7
The chart above reads: “the sum of 2 hundredths times seven, plus 4 ten-thousandths times seven, plus 8 millionths times seven, plus . . . etc” an so on, showing the unmistakable imprint of the natural division of unity process on the relationship between unity (or one) and seven.*
* In another section, the number seven is shown to be co-equal to the number one when the base-ten/decimal number system is looked at with respect to it organizational structuring.
The Sinus Tetrahedron
Imagine what might be termed the negative image of the form of the five tetrahedrons arranged around a common edge. That image would be defined by the “gap” which is left open, and which separates this form from that of the regular decahedron. When the edges of the tetrahedrons measure 1.0 unit in length, the distance across the gap between the two vertices of opposing tetrahedrons is 1/9 unit, or in decimal form, 0.111111… The volume of this irregular tetrahedron is then 0.016004…
It is essential to understand that this specific volume is born out of the relationship between the seven and eight (1.0 unit diameter) sphere systems, and because of this, it should harbor characteristics within its own composition reflecting this history of its origin. This is of course a foundational notion underlying the use of models throughout this text.
This special volume quantity from the irregular sinus tetrahedron can be transformed into the shape of a regular tetrahedron having an edge-length of 0.514006…
When this tetrahedron is compared to a decahedron equal in polar axis length to this tetrahedron’s edge-length (the two forms are scaled to the same unit and thus are congruent) then this particular decahedron will have its edges measure 0.488848… (see diagram below).
Now picture this decahedron spitting out just the right amount of volume to become the equivalent of five tetrahedrons arrayed around a common edge, just as we began with in the original decahedron’s transformation. Its polar axis of 0.514… becomes 0.488848…as a common edge of the now five regular tetrahedrons. The volume of each of these tetrahedrons is 0.013767667… and as 0.013770… is identified as a Transit-tetrahedron. . . i.e., that form balancing surface/volume differences in the one-into-two spherical, and seven into eight system transformations.
The Sinus tetrahedron has a further link to the characters in the transformational scenario from which it was born. Again, its volume in regular tetrahedronal form provides this link through its edge-length of 0.514006… This is because a cube so scaled (edge = 0.514360…) has a volume equal to 0.1360827… which if evenly divided into two new cubes creates 1.000… surface-unit on each of these two cubes.
The preceding evidence certainly must be indicative of some system at work regarding these forms and their relationship to one dividing into two, and a seven point system becoming eight.
THE NUMBER 137
“ There will be a physics in the future that works when hc/e^2 has the value 137 and that will not work when it has any other value . . . Now there is no known reason why it should have this value rather than some other number. Various people have put forward ideas about it, but there is no accepted theory. Still, one can be fairly sure that someday physicist will solve the problem and explain why the number has this value.” *
* Quoted from an 1963 article in Scientific American by the great physicist Paul Dirac.
“One cannot escape the feeling that these
mathematical formulae have an independent
existence and an intelligence of their own,
that they are wiser than we are,
wiser even than their discoverers. . .”
My quest for the missing images at the start of the “Big Bang” scenario led me to ask how geometry handles the division of one thing into two . . . an action seemingly more reasonable for the primal singularity’s initial transformation, rather than going from one thing to bigillions with no intervening transformations.
When I modeled the singularity as a “unit of surface” in the form of a sphere, and divided that surface into two new spheres, I realized there was excess volume. The unit of surface as two spheres can’t hold as much volume as that same surface unit in the form of one sphere. At that time, this was a new revelation for me and I wondered if in some way geometry itself accounts for this excess volume.
Mitosis” was the answer I was seeking, and I had found it only after a long journey into the geometric structure of form. The bronze sculpture depicted below embodies the principal components of this one-into-two transformation. The tetrahedron is one of two equal packets which together represent that quantity of “excess volume”. The larger circle is the cross-section of the initial sphere, and the smaller circle is the cross-section of either of the two new spheres.
All of these same components are also depicted in the print image below.
The dynamic of geometry dividing a spherical surface into two new spheres is exposed in MITOSIS, the graphic image illustrated above.
Later I would discover the beautiful dances amongst and between the primal forms of geometry. It was Mitosis’s proportioning set to a particular scale that provided the key opening the doorway into what I have since come to call “The Geometry of Form”.