Welcome to Aquarius, Volume 10 (September 24, 2007)
The Basic Content and Meaning of “Proportion”
Brain byte: Proportion is the finite and specific set of real and natural shapes that can be formed or taken by physical matter. Proportion is, in the natural universe, the pre-cursor of number and therefore the most basic operating principle of Nature, prior to number or geometry or any form of counting or measuring or mathematical computation invented by any intelligent being. This is not “numerology” or coincidence. This is a phenomenon of Nature. Although Proportion is not the equivalent of number, it can be perceived or reflected by observation of numerical values that are found in the dimensions of geometrical shapes, especially right triangles.
Link to: (Welcome) or (Geometry Alpha Index) .
Invtrig1.cpp is a computer program that performs calculations that can be performed by construction. “Invtrig” means inverse, or reciprocal, trigonometric functions. This program shows that by following the construction procedure presented, the results include four very interesting proportional relationships. These relationships grow out of the “Alternative Pythagorean Theorem” and a specific set of conditions applied to the quadratic equation, which I present as the quadratic “construction.” The “quadratic construction” is discussed further below as reflecting the laws of Nature governing the manner in which physical matter creates the many shapes observed in the Natural World and the manner in which matter is self-organizing. This program therefore illustrates an important part of my argument that “Proportion” is the “number system” of Nature and Proportion is the sufficient cause for the self-organization of matter, which means of course that Proportion is the sufficient cause for the process whereby what we deem to be non-living matter transforms into that which we recognize as living matter or “life.” My conclusion therefore is that Proportion is “The Secret of Life in the Universe.” If my theory is demonstrated to be correct, then “squaring-the-circle,“ which is the best introduction to the meaning of “Proportion,” is monumentally important and not just the solution to an old but trivial geometric puzzle.
The “Alternative Pythagorean Theorem” is:
A = Altitude, B = Base, H = Hypotenuse. H is always greater than A, and therefore:
H = (A + N), and B is always equal to [(2*A*N) + N^2] or N*[(2*A) + N]
because H^2 = (A+N)^2 = A^2 + 2*A*N + N^2, and
(A^2 + 2*A*N + N^2) minus A^2 = 2*A*N + N^2.
AND if you want to see a very interesting reality that is part of the meaning of Proportion, take a closer look at the right triangle where A = 2 and B = 1. What is the length of N? And what is the result when you multiply H [sqrt(5)] by [(secant 18)^2 -1] ?
That is 2.236067978... times 0.105572809... . What is the value of 2*A*N + N^2 ?
What is the value of 2*A*N ? This is not just “numerology.” This is a phenomenon of Nature.
To avoid confusion: In the printed output of Invtrig1:
Positive solution is minus (b/2) plus sqrt[b^2 - (4*A*C)] smaller value in line length
Negative solution is minus (b/2) minus sqrt[b^2 - (4*A*C)] larger value in line length
The line that begins with “PS” indicates the Positive solution, the smaller value for X
that holds true for 1*X^2 + (B*X) - 1 = 0.
The line that begins with “NS” indicates the Negative solution, the larger value for X
that holds true for 1*X^2 + (B*X) - 1 = 0.
PS and NS are reciprocals of one another. Each value can be readily constructed as a line length. This is not algebra with “minus” values. Line X minus line Y is still a line, not a “minus” line.
RX[17] means the program operator selected ratio value #17 (from a list of 83 ratio values) to be one of the reciprocal pair of values.
RX[333] means the program operator selected an arbitrary value (not from the list of 83 ratio values) to be one of the reciprocal pair of values.
In any pair of reciprocal values, one is larger than the other. We get our A (Altitude) value for our “Alternative Pythagorean Theorem” right triangle, by subtracting the smaller value in the reciprocal pair from the larger value in the reciprocal pair. In this right triangle the Base B always has a value of 2. However, in the application of the standard quadratic equation, or quadratic construction, A = 1 and our B value takes the A value from the right triangle and our C value is minus 1. The results may be recognized as “Chaos Theory,” but I am arguing that it is more fundamental than a description of “chaos.” I say it is “Proportion” and that this is a description of all of Nature.
Printed output of the program Invtrig1:
Quadratic Construction solutions = -b +/- sqrt[b^2-4ac] / 2*A
Invtrig1.cpp Inverse ratios by quadratic construction (9/4/2007)
Explanatory notes from prior program InvXpN01.cpp:
InvXpN01.cpp Inverse of X = X + N (2/6/2006)
Based on X^2 + B*X - 1 = 0. Yields solution to
(1/X) = X + B, as in (0.618033988... and 0.414213562...
In InvXpN01, 02, 03, you can select an MV value.
MV=same Multiplier Value for variables A,B,C
InvXpN02 calls for selection of - C value.
InvXpN03 calls for selection of all variables, or 1.
AM=multiplier value for variable A only
BM=multiplier value for variable B only
CM=multiplier value for variable C only
C= -1 times CM times MV, MVs can all be 1
BS= B*B, FAC= 4*A*C, DE= 2*A
TR= determinant (BS-FAC) which equals BS+FAC
when C = -1. TM= square root of determinant TR
NS= negative solution, absolute value of (-B-TM)/DE
= (B+TM)/DE. PS= pos solution, abs val (-B+TM)/DE
NQ=NS*NS, and with A=1, NQ= B*NS + (abs val C)
In InvXpN01 and 02, A=1 and NQ= B*NS + (abs val C)
In InvXpN03, A varies, and BD=B/DE, TD=TM/DE and
BD + TD = NS, and there is an effect of the A variable.
With Invtrig1.cpp, the algorithm shows how we can construct
trigonometric ratios or any reciprocal pair: sine-cosecant
or cosine-secant or tangent-cotangent and square roots of these.
Key procedure that Altitude A in a constructed right triangle
is the positive difference between the values of the reciprocal pair
and the Base B = 2 and C = (-1)
Key procedure is to make DE (2*A) always DE = 2 and the Altitude
A of the right triangle is the B value in the quadratic construction,
while the B Base of the right triangle is always equal to 2.
Therefore the solutions are (H+A)/2 and (H-A)/2 and B^2 is always
equal to the Pythagorean Theorem difference of (2*A*N + N^2)
where H-A = N, N being the value added to A to get H.
In this program this N added value is labeled as NA.
There are four interesting results that follow:
1) (H+A)/2 and (H-A)/2 are simple line-length constructions
from the right triangle that equal the two solutions
from the quadratic construction with B=reciprocal difference.
2) 1/X = (X+B) for smaller Pos solution, which is the
B value in the quadratic, A in the right triangle.
3) NS^2, larger Neg solution squared = (B*NS)+1.
4) In the triangle, H-B, the result of laying down the hypotenuse
is the A value squared for the algorithm to yield the square
roots of the prior solutions, and the same four results.
RX[17] = TRG = 1.111785940503, GRT = 0.899453719974
A=TRG-GRT = 0.212332220529, B=2.000000, H= 2.011239660477
MV=1.000000, B=2.000000, NA=1.798907, NXR=0.011240
Check rt tri: A= 0.212332220529, H= 2.011239660477
Comparison: (H+A)/2 = 1.111785940503, (H-A)/2 = 0.899453719974
Main quadratic construction:
PS = X^2 + 0.212332220529*X -1 = 0.899453719974
TR=4.045084971875 TD=1.005619830238 -BD=0.106166110264
NS = X^2 - 0.212332220529*X -1 = 1.111785940503
TR=4.045084971875 TD=1.005619830238 +BD=0.106166110264
NS^2= (0.212332220529 * 1.111785940503) +1 = 1.236067977500
RX[43] = TRG = 2.894427191000, GRT = 0.345491502813
A=TRG-GRT = 2.548935688187, B=2.000000, H= 3.239918693812
MV=1.000000, B=2.000000, NA=0.690983, NXR=1.239919
Check rt tri: A= 2.548935688187, H= 3.239918693812
Comparison: (H+A)/2 = 2.894427191000, (H-A)/2 = 0.345491502813
Main quadratic construction:
PS = X^2 + 2.548935688187*X -1 = 0.345491502813
TR=10.497073142515 TD=1.619959346906 -BD=1.274467844094
NS = X^2 - 2.548935688187*X -1 = 2.894427191000
TR=10.497073142515 TD=1.619959346906 +BD=1.274467844094
NS^2= (2.548935688187 * 2.894427191000) +1 = 8.377708764000
RX[333] = TRG = 1.500000000000, GRT = 0.666666666667
A=TRG-GRT = 0.833333333333, B=2.000000, H= 2.166666666667
MV=1.000000, B=2.000000, NA=1.333333, NXR=0.166667
Check rt tri: A= 0.833333333333, H= 2.166666666667
Comparison: (H+A)/2 = 1.500000000000, (H-A)/2 = 0.666666666667
Main quadratic construction:
PS = X^2 + 0.833333333333*X -1 = 0.666666666667
TR=4.694444444444 TD=1.083333333333 -BD=0.416666666667
NS = X^2 - 0.833333333333*X -1 = 1.500000000000
TR=4.694444444444 TD=1.083333333333 +BD=0.416666666667
NS^2= (0.833333333333 * 1.500000000000) +1 = 2.250000000000
RX[333] = TRG = 2.500000000000, GRT = 0.400000000000
A=TRG-GRT = 2.100000000000, B=2.000000, H= 2.900000000000
MV=1.000000, B=2.000000, NA=0.800000, NXR=0.900000
Check rt tri: A= 2.100000000000, H= 2.900000000000
Comparison: (H+A)/2 = 2.500000000000, (H-A)/2 = 0.400000000000
Main quadratic construction:
PS = X^2 + 2.100000000000*X -1 = 0.400000000000
TR=8.410000000000 TD=1.450000000000 -BD=1.050000000000
NS = X^2 - 2.100000000000*X -1 = 2.500000000000
TR=8.410000000000 TD=1.450000000000 +BD=1.050000000000
NS^2= (2.100000000000 * 2.500000000000) +1 = 6.250000000000
End of Invtrig1.cpp Inverse ratios by quadratic construction (9/4/2007)
A careful reading of the opening text describes the previous programs labeled as InvXpN01, InvXpN02, and InvXpN03. The most important fact about this explanatory text, in case it is confusing, is that all of these programs and the Invtrig1 program apply the quadratic equation or quadratic formula as what I call “the quadratic construction.” Performing the quadratic construction means using the compass and straightedge to construct line lengths that are numerically equal to the two solutions (larger = Negative solution) (smaller = Positive solution) of the quadratic equation. Review:
quadratic formula: -b (+/-) sqrt[b^2 - (4*A*C)] / (2*A), and the two solutions are:
1) Negative solution = -b, - sqrt[b^2 - (4*A*C)] / (2*A), this solution is TD + BD
2) Positive solution = -b, + sqrt[b^2 - (4*A*C)] / (2*A), this solution is +TD - BD, where...
TD = sqrt[b^2 - (4*A*C)] / (2*A), and BD = b / (2*A). All lines, no minus lines.
This is explained in greater detail in What is Proportion, really? Volume 8.
The next paragraph of explanatory text explains that we are constructing a right triangle with a value A, but when we apply the “quadratic construction” we make B equal to that A in the quadratic formula and A=1, and C = -1. The very interesting four results are then listed:
1) (H+A)/2 and (H-A)/2 are simple line-length constructions
from the right triangle that equal the two solutions
from the quadratic construction with B=reciprocal difference.
2) 1/X = (X+B) for smaller Pos solution, which is the
B value in the quadratic, A in the right triangle.
3) NS^2, larger Neg solution squared = (B*NS)+1.
4) In the triangle, H-B, the result of laying down the hypotenuse
is the A value squared for the algorithm to yield the square
roots of the prior solutions, and the same four results.
The crucial concept here that requires repeated articulation, is that this relatively simple mathematical program illustrates the principles that Nature applies to create shapes, to create molecules, to create cells, to create living tissues, to create living things.
Sufficiency of Proportion Theory is based on the postulate that geometry is not mathematics. This will sound strange and illogical to many people at first, because we are accustomed to thinking of geometry as a branch of mathematics. However, this custom is logically invalid. Mathematics is a language. Mathematics is comprised of written linguistic characters that are symbols of something other than themselves. The equal sign, for example (=) is a symbol that means “is equal to” just as the word “bigger” means “greater than” some other designated object, and in Mathematics, the symbol “>“ as in “A>B” means that item A is “greater than” item B.
Geometry is not a language. Although we use mathematical language to talk about the realities of geometry, geometry itself is not a symbolic language. A triangle does not represent anything other than itself. The same is true for any square, circle, or any other plane or solid shape. Geometry is in fact a description of physical reality based entirely on the shapes of objects. That is what geometry is, and that is not mathematics.
Geometry is physical reality. Mathematics is a language invented by humans. Like any language, mathematical language can be amended or expanded. Geometry, and Proportion, can only be discovered but can never be invented or altered in any way by human thought or action.
What Proportion is, really, is what Nature does with its precise tools that is the same as what humans can do with a compass and straightedge. With a compass and straightedge, we can
1) Compare the sizes of two or more line lengths or angles without measuring either against any established or absolute standard of size. Nature has no “meter” or “yard” or “degree” in storage somewhere, no standards of weight, volume or unit of time.
2) Halve or double the size of a line or angle, and perform sequential halving.
3) Add or subtract one line length from another, or one angle from another, and in a series of steps apply addition and subtraction to accomplish sequential shaping.
Proportion is the finite and specific set of real and natural shapes that can be formed or taken by physical matter. Proportion is, in the natural universe, the pre-cursor of number and therefore the most basic operating principle of Nature, prior to number or geometry or any form of counting or measuring or mathematical computation invented by any intelligent being. Nature’s intelligence and all that Nature is and does is accomplished by virtue or Proportion. Nothing more is needed.
However, since number is a natural subsequent product of Proportion, any intelligent being can perceive Proportion through the mechanism of a number system. That is what I am doing here on my web pages, presenting an introduction to proportion, presenting the re-discovery of Proportion -- because I believe it was once understood by humans on Earth in the past.
Link to: (Welcome) or (Geometry Alpha Index) .