Basic Four Constants (B4C) Circle Expansion/Contraction Ratio Coordination

Again and again in nature one finds that things live and ‘work’ because of the fudge factor, the ‘lawful inexactitudes’, the colel of numerology. The inexact recurrence cycles of the planets, not even explained by nested levels as Steiner pointed out in the 1920’s, and re-affirmed to exist by chaos theory investigations in the late 20th century are good examples. Human art from traditional indigenous sources uses the imperfect pattern device commonly, as for example in a Navajo rug pattern that has closure but for a loose thread, to allow consciousness focused by the design an out to find a higher level or octave. These examples are all evidences of the dynamic presence of shocks in the octaval processes of existence.

The inexact but close coordination of the circle expansion /contraction ratios for the basic four (real domain) math constants that form a group from several different perspectives illustrate an example of a Gurdjieffian TI-DO shock or bridging energy in an octave process. The TI-DO shock fills the ascending TI interval to the next octaves’ fundamental
DO to exactitude, thus completing it. These constants were coordinated by ancient builders when they erected structures based on 2 orthogonal axes using the math relations of phi, the golden section, and two for the north-south axis math relations and e, base of the natural logarithms, and pi for the east-west axis. The PI Great Pyramid at Giza, Egypt and the Parthenon in Athens, Greece, are two examples. I have also found that the B4C are coordinated by a rule of exponents such that a new constant is created which is the least-mean-square error optimized value of the number which is simultaneously a root of each of the 4 constants where the index of the root is very close to an integer of 3 digits or less. This optimized value I designate HC (Heleus’ constant) equal to 1.0060427, which is simultaneously about the 80th root of phi, the 115th root of 2, the 166th root of e, and the 190th root of pi. HC approximations and its integral and simple rational powers abound ubiquitously in tables of math constants ( such as Steve Finch’s). The basic 4 constants also coordinate by summing to just over a straight angle when represented as circle expansion/contraction ratios created by stacking tangent circles of that common ratio inside a characteristic angle and bringing all four angles to a common origin. When this is done, the sum exceeds a straight angle by about 3.57 degrees, which as shown here, characterizes a TI-DO shock.

In G. Spencer Brown's epochal math book, The Laws of Form (and considering the author’s Buddhist leaning, the title may as well have been The Loss of Form), makes a case that the basic object and experience in math is distinction, to make one, or have one.
Modeling this as a straight line orthogonal bisected yields two right angles above and two below the original horizontal line relating two points of view. The right angle is thus the first and foremost object created as the distinction. It also as angle represents the fullness of a range of choices included by the distinction.

Here the constants have been grouped so that pairs come near as possible to sum to a right angle. The angles for phi and pi sum to about 89.5 degrees and the angles for 2 and e sum to almost 94 degrees, so adding to a straight angle plus about 3.57 degrees. I have been able to demonstrate systems thinker and Bell Helicopter inventor Arthur Young’s conjecture (The Reflexive Universe) that e and pi are ‘somehow the same number’. Indeed, a fairly simple demonstration shows how they are orthogonal phase-conjugate partners, or the same entity seen from one side or the other of a right angle. In the illustration, you see they form nearly a right angle as measured between their bisectors. Likewise, phi and 2, but so far I have found no similar demonstration for their orthogonality.

In an octaval process, the fundamental note is doubled in frequency ascending to the high DO, thus high to low DO frequencies have a 2:1 ratio. According to Gurdjieff, there are two places in the process of eight distinct stops where normal progression from step to step breaks down, needing external energy or a shock to proceed, eliminating relative hazard or chaos. These occur between the MI and FA and TI and DO notes which in a Western scale are half-steps instead of whole steps. In a natural or just intonation scale, the ratio of the TI-note to low DO is 15/8 or 1.875:1. Since phi, e, and pi are respectively an irrational number and two transcendentals, of the four basic constants only 2 is an integer, and so admits of more possibility of adjustment as a simple entity since the others are defined by rigid operations. When the adjustment is made shrinking (narrowing) the 2-angle so the four sum to exactly a straight angle, the ratio of circle radii in adjacent circles in the 2-angle drops to 1.87260738197 which is barely under the 1.875 just intonation natural ratio by one part in 782.66, covering the TI interval very well, and thus reducing the octave 2:1 ratio to effectively TI at 1.875. This means the original excess of the sum of the basic four constants’ angles of about 3.57 degrees beyond the straight angle represents the TI-DO shock in process completing it and coordinating the constants about a common center.


Michael Heleus ----- Oct. 3, 2003

B4C Main Basic Real Constants Part II


It often happens in searching for hidden significant
connections between basic math entities like the basic
four real constants phi, 2, e, and pi that
unexpectedly a connection will appear that in
retrospect seems a natural development expectable in
due course. An example connecting these four constants
by exponents in a different way than in part I
follows:

Solve
(Phi^x + 2^x + e^x + pi^x - RTANG = 0, x) | x =
.445704478798

where RTANG (=3 + 2A(3/2)) is the ratio of radially
adjacent tangent circles as in Part I included in a
right angle. When RTANG is turned inside out by taking
its inverse RTANG^-1, and substituted into the above
equation, x becomes -4.5412907205. Approximations to
the absolute value of that are important in balancing
radial motion with rotary motion. Logarithmically,
i.e., musically, comparing the two x's absolute values
with the larger as numerator yields 1.8725564444. This
value is to 1 part in 36,761.7937266  the radially
adjacent circle ratio in the
adjusted-to-make-a-straight-angle angle including the
formerly 2-ratio radially adjacent and tangent circles
of part I . Hardly a coincidence since the same four
constants in the context of right and straight angles
are involved.

  Since '+' is one-dimensional grouping and 'x' is two
dimensional grouping., we now push the envelope a bit
to illustrate a near-invariance under orthorotative
transformation, i.e.., dimension shift, by
substituting 'x' for '+' in

Phi^x + 2^x + e^x + pi^x - RTANG = 0

 Then, if RTANG^-1 is substituted for RTANG in a
second x -equation, the x values resulting are + and -
.531093692179, respectively. The former we'll call A.
A = 17/32 to a part in in 3397.74227114. The
reciprocal of A =A^-1 is 1.88290694227 B. We just saw
the value 1.8725564444 derived from the x values of
the 1-D '+' or summed variety of the B4C exponential
equation, which is to 1 part in 180.914625312 equal to
B from the multiplied variety of the equation, so
demonstrating the near-invariance under orthorotation
of the '+' and 'x' varieties of the
B4C-equivalent-to-a-right-angle equal exponential
equation.


Michael Heleus   Nov. 25, 2003

MICHAEL HELEUS UPDATE....MARCH 16, 2008

The constant linking B4C, that is phi, 2, e, and pi by being nearly

the same root from a two or three digit integer inverse power of the

constants such that when the constant, HC, is raised to each of the

integer powers one of the B4C constants results to better than 1 part

in about 1000. When least mean squares error optimized, the number is

1.00604272345--compare that to the current most elegant,simplest and

most accurate approximation just found (3/2008):

31^(570^-1)=1.00604272299

which is good to a part in about 2.2 billion. 31 closely approximates

phi cubed and 31 also is the number of great circles relating to an

icosahedral system.