SOLITU Part P: A Few Branches on the Tree of Proportion
An Introduction to Evolutionary Proportion
Copyright 2015, John Manimas Medeiros
The following interesting patterns of proportional transformations are among the mathematical facts that sustained my faith in the Pythagoreans as I experimented in my search for the solution. Failure is the long path to success.
Evolutionary Proportion is a specific structure. The universe is ordered and organized. It is most likely a tree like so many things in nature are trees: river deltas, the blood vessels in the back of your hand, the veins in a leaf, the bodies of coral, the systems of human institutions, the way that people make decisions. If this is correct, then proportion is not a random process, not probability, but specificality. It is not that everything is determined but that everything is subject to boundaries. Everything that occurs is subject to the structure of evolutionary proportion. Maybe. That is the theory, and it should be tested further.
Just as light is the force, and a colorful garden is what we see, and just as vibrations are the source, and a symphony is what we hear, evolutionary proportion is the structure of shape and size of all that is, even the shapes and sizes of other forces, such as magnetism and graity and the solar wind and the waves of the oceans, and our perception of the tree of evolutionary proportion is measuring and counting. Because we want to communicate about proportion, because we must discuss the sizes and shapes of things, we name them, the sizes and shapes. We name the proportions and the names are numbers. Numbers are the names that we give to proportions. Our perception of light is form and color. Our perception of vibrations is sound and tone and danger and music. Our perception of proportion is mathematics.
As is true of our other senses, we do not all have the same sense of proportion. Some people's sense of proportion is more precise than others. Some people's ability for calculating proportions is faster than others. The sense of proportion gives us a sense of what is fair and just. This sense of proportion is what we use to create social and political artifacts, legal and social and political institutions intended to make human relations fair and just. Our social artifacts go through all of the processes of aging and fading, getting old and outdated, decaying, repair, collapse, demolition and replacement. Because people have an individual sense of proportion, like the different forms of art and activities that please them, people have a variety of viewpoints on what is fair and just. We fight because we do not all have the same sensory perceptions.
Here we go:
As I conducted my search over the years, I was constantly discouraged -- like, my internal voice asking "Why am I doing this? Am I crazy? A time waster?" But the answer was always "No, you are not crazy. You have an instinct and you are certain that it is correct. We can construct pi exactly as a straight line because proportion is everything. It is hidden, but you can find it. Don't give up. It is the key to understanding how the universe works." It was not only my internal voice, my faith in myself and in the Pythagoreans that kept me going. It was also the persistent appearances of evidence in support of Evolutionary Proportion.
The first and the last evidence:
The oldest (1990's) powerful evidence is the algorithm that I describe in The Pyramid Mark, in the Geometry link. I practiced the binary number system for performing addition and multiplication. Then I had read in Serpent in the Sky by John Anthony West that some historians believe the ancient Egyptians used the binary number system for computations like we use our decimal multiplication tables. As I experimented I came to see that if we use the square root of two as the base for our number system, we can follow a pattern of "sequential halving" and doubling as a method for constructing any fractional line length written as a decimal digit number. This method is described and shown in The Pyramid Mark.
The newest evidence is the elegant proportional wonder that became visible to my mind only when I discovered the solution: A - B = pi * 0.1118. The value 0.1118 is the square root of (1/80) or of (0.0125) times 0.99996959953790595223418246465663. Or, stated as the inverse: 0.1118 times 1.0000304013863102398967681881625 = sqrt(0.0125). All extended decimal values shown here continue to be subject to the limitation of a desktop computer -- precisely accurate only to the 9th decimal place. But look at what other relationships apply: 0.1118 times ten times 1.000030401 = the sqrt(1.25)
1.1180339887498948482045868343656, and that times 2
= sqrt(5) 2.2360679774997896964091736687313
and sqrt(1.25) + 0.5 = 1.6180339887498948482045868343656 = Phi, the Golden Section
and HG [0.1118] times 10, plus 0.5 = 1.618, and that plus 1 = 2.618
and you may recall that (20/11) * 1.44 = 2.6181818181818181818181818181818
and that times (PD/PE) = 2.6180339887498948482045868343657
1.118 * 2 = 2.236, * 2 = 4.472, * 2 = 8.944
1/8.944 = 0.11180679785330948121645796064401, divided by 0.1118
= 1.0000608036968647693779781810734, which equals
1.0000304013863102398967681881625 ^2 (This is not "numerology")
0.1118 * 12 = 1.3416, and that times 1.0000304013863102398967681881625
= sqrt(1.8) or 1.3416407864998738178455042012388, so my brain tells me there
is a pattern here. Number is a representation of Proportion, but Proportion
is not our decimal digital number system, or any cultural number system.
The number system of Nature is Evolutionary Proportion, and the binary system of halving and doubling, and the unfolding of the Fibonacci series and the function of the Golden Mean in the evolution of living things are special cases of proportional evolution, those parts of Evolutionary Proportion that work well and are repeated and become visible to us even before we see the bigger picture of Proportion is Everything.
Evidence along the way:
Several patterns discovered while generating and studying the "pi-lines" are in my memory. Let's look at the list of 26 pi-lines, keeping in mind that there are many other pi-lines that could be generated. Note that we can multiply (pi*M) * (pi*N) and get pi^2* M * N. Then, when we take the square root of that, we get pi * sqrt(M*N). So, we have a new pi-line. We could do this forever, but I did not have forever, and neither do you. But, here are a couple of patterns that I found interesting and helpful along the way:
PL = 3.125; // pi * 0.99471839432434584855552352107821
PL = 3.136773748694594219681304368847; // G[ sr(5) + 2.2] / sr(2)]
PL = 3.1399111679090046934014154186896; // Q[sr(Phi*8) * 48] / 55 ]
PL = 3.1406228624894574694952341031297; // A[2 + sqrt(5) ] - sqrt(6/5)
PL = 3.141320000468716979810539729062; // B[sqrt(37) - 1] * (1/Phi)
PL = 3.1414803893897418379555528189649; // C[sqrt(pi*B* pi *D]
PL = 3.1415964392353511387409825201656; // [pi*sr(J*G)]
PL = 3.1416407864998738178455042012388; // D[(9/5) + sqrt(9/5)] (pi*MQ)
PL = 3.1418181818181818181818181818181; // E[(1/0.55*1.728] Pi*MQ*TN
PL = 3.1426968052735445528926416093549; // V(40/9)*sr(0.5)
PL = 3.1446055110296931442782343433718; // F[sqrt(1/Phi) * 4 ] (pi*Ho)
PL = 3.1464265445104546409743605410398; // J[sr(9.9)]
PL = 3.1465153760315117097549686764014; // H[pi*G*MN*MN], (pi*H)
PL = 3.1466042100605056464977059018508; // [pi*sr(G*MX)], GX
PL = 3.14928; // pi * MT * 1.002375
PL = 3.1512640759198245358269325763168; // (pi*J) * sr(J/G)]
PL = 3.1514420146250458834693530842498; // [pi * sr(J * MX)]
PL = 3.1561090448809076493016604787293; // K[(9.9) / (pi*G) ]
PL = 3.1562872571612667838672356476287; // L[(pi*H)^2/ (pi*G)]
PL = 3.1564654795045281880219131572653; // MX[ L^2 / K]
PL = 3.2; // pi*MT*(55/54)
PL = 3.2360679774997896964091736687313; // [sr(5)+1]
PL = 3.24; // pi*MT*1.03125 or (33/32)
PL = 3.4909090909090909090909090909; // 192/55
PL = 3.6; // pi * (55/48)
PL = 3.84; // pi*MQ*SE, or pi*MQ*(11/9)*TN
[The value TN = PL / PL and = (secant 18)^4 * (9/11), or
TN = 1.0000564658184571247007181063881]
Look at PL :
PL = 3.1418181818181818181818181818181; // E[(1/0.55*1.728] Pi*MQ*TN
This (Pi*MQ*TN) means pi * D or 3.1416407864998738178455042012388
times TN or times 1.0000564658184571247007181063881
= 3.1418181818181818181818181818181 and this (pi*E) which I also called (pi * MT) has several interesting evolutions: For example: [pi*MT = pi*E]
pi * MT * (33/32) = 3.24, and (33/32) = 1.03125, and
pi * MT * (10/9) = 3.490909090909090909090909090909 = (192/55)
and for all pi-lines (pi*M), the product of (pi*M) * (10/9)
equals 3.4918853391928272809918240103943, which is (192 / 55) times
1.0002796544562786482007829196442, which = (V/MT)
Easily tested by multiplying any pi-line by (10/9), then multiply by 55 and then divide by 192 and the result will be 3.490909090 * (pi*M) / (pi*MT)
For example: (pi*V) * (10/9) = 3.4918853391928272809918240103943
which is (192 / 55) times 1.0002796544562786482007829196442, which = (V/MT)
Here are some other patterns: "Starting a table of equalities:"
pi*M * sr(cos 36) = sr(8) * . M . and pi*M * 0.9 = sr(8) * . M .
pi*M * (10/9) = 192 * M . and pi*M * (10/9) * (cos 36) = VARIES
55 * MT
A) pi*H * (10/9) * (cos 36) = sr(8)
B) pi*G * (10/9) * (cos 36) = sr(8) * . V .
C) pi*M * (10/9) * (cos 36) = sr(8) * . M .
New: pi*M * (cos 36) = . M . * sr(Phi) * 2
pi*M * sr(8) = . M . * 80 pi*M * (11/9) = 3.84 * . M .
V 9 MT
pi*M * sr(2) = . M . * 40 .
add to this list: (pi*M) * sr(sec 36) = 192 * M
55 * Q
Also: (pi*D) * (sec 18)^4 = 3.84 one can explore 3.84 relation to other pi-lines
(sec 18)^4 = (11/9) * 1.0000564658184571247007181063881
(pi*H)^2 = 1.0000564658184571247007181063881 * 9.9
(pi*E) / (pi*D) = 1.0000564658184571247007181063881
and: (sec 18)^2 * 2.5 = 3.0557280900008412143633053250749
= (pi*F)^2 * 0.30901699437494742410229341718282 [sinc 18]
and: see (pi*V), 10 / (9*V) = pi * sqrt(0.125), and 9/40 = 0.225
and: (40/9) = 4.4444444444444444444444444444444
that times (D/F)^2 = 4.4360679774997896964091736687313 = [ sqrt(5) + 2.2]
and: look at (pi*F) = 3.1446055110296931442782343433718
while 55/90 = 0.61111111111111111111111111111111
and (55/90000) + 1 = 1.0006111111111111111111111111111, + 2 + (72/500)
I have hundreds of research documents, each about 30 pages long, each with explorations of proportions of pi-lines and other values of interest.
such as: (pi*MT) * 0.3375 = 1.0603636363636363636363636363636
inverse 0.94307270233196159122085048010974, * (pi*MT)
= 2.962962962962962962962962962963 (inverse of 0.3375)
Have you noticed that 3.1418181818181818181818181818182 = (864 / 275)?
and: (55/48) = 1.1458333333333333333333333333333
times 3.1418181818181818181818181818182 = 3.6
(80/75) = 1.0666666666666666666666666666667
inverse = 0.9375, times 3.84 = 3.6
It looks like "numerology" but it is evolutionary proportion. The proposition of evolutionary proportion is that the proportions we observe ARE NOT simply the outcome of a set of natural forces but ARE IN FACT the single most fundamental force of Nature. In other words, the shapes and sizes of all things is caused by Evolutionary Proportion. AND, mathematics is our perception of evolutionary proportion. That is what mathematics IS, a perception and a personal skill. Mathematics is personal, like an emotion or personality trait, not universal.
This is why some people say "I'm not good at math." They say that because it is an accurate element of self-awareness. They know that they do not possess the math skills that others do. And, this is similar to any statement about personal skills, such as "I cannot sing" or "I am color blind" or "I can knit faster than most people" or "I love alpine skiing" and so forth.
and: (pi*G) = [sqrt(5) + 2.2] * sqrt(0.5) is the same as sqrt(2.5) + sqrt(2.42)
and: sqrt(sec 36) = (10/9) * (MT/Q), the same as:
1.111111111 * 1.000607346 and (MT/Q) = (F/V)
note (MT/Q) is (E/Q), in the pi-line list (pi*MT) = (pi*E)
and: (pi*M) * 0.9 = (M/V) * sr(8)
(pi*M) * (pi*MT) = sr(8) * (192/55) * . M . AND
(pi*M) * (pi*V) = sr(8) * (192/55) * . M .
complexities of equalities:
(pi*H) * (10/9) * (cos 36) = sr(8) and therefore: (pi*M) * 0.9 =
sr(8) * (M/V) = (pi*H) * (10/9) * (cos 36) * (M/V)
(pi * MT) * (10/9) = (192/55) and sr(8) * (10/9) = (pi*V)
and (pi*MT) * (pi*V) = sr(8) * (192/55)
sr(8) denominator series:
sr(8)/V = 0.9 * pi
sr(8)/F = sr(cos 36) * pi
sr(8)/H = (cos 36) * 10/9 * pi
sr(8)/D = 0.9 * (V/D) * pi
= 0.90030252245908817867181052349698 * pi
= 0.9 * 1.00033613606565353185756724833 = 0.9 * (V/D)
sr(8)/MT = 0.9 * (V/MT) * pi
= 0.90025168901065078338070462767978 * pi
= 0.9 * 1.0002796544562786482007829196442 = 0.9 * (V/MT)
sr(8)/G = 0.9 * (V/G) * pi = 2.8327723285795317134639799997497
(V/G) = 1.0018882638830471224770463536466
* 0.9 = 0.90169943749474241022934171828192
Also: (pi*F)^2 = 9.8885438199983175712733893498502
+ 8 = 17.88854381999831757127338934985
= 16 * 1.1180339887498948482045868343656
or = 2 * 8.9442719099991587856366946749251 [ = sqrt(0.8) * 10]
M * 10 = pi * sr(0.125) * M
Therefore: pi * M * 10 = pi^2 * sr(0.125) * M
Let M = D, then M * 10 = 1.1107377520931496717605939575593
This material is very long and time consuming to re-visit to extract the most interesting examples of proportional equalities. Some time I may assemble the many sets of notes and copy them to a CD data disc and make that available for a reasonable price.
Link back to: (Journey List) or (Welcome) page links or (Mindstream) of J. Manimas or (JM Magazine 2015) or back to (SOLITU Contents).