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Solution to the Gamma Riddle
This diagram represents the numbers by which one must multiply the fractional factorials, one at a time, to progress in unit steps from left to right along the X axis. It can be seen that there is a singularity at zero, where the multipliers stop being negative and start to be positive. So below zero, we have alternately a positive and then a negative result from the multiplication - a negation function. Above zero, the results are all positive. The negation function suggests a sinusoid at frequency πx, with a quadrature component in the imaginary axis. The reciprocal of the fractional factorials clearly shows the oscillation of the real component. The riddle is to find the unreal one. | ||
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The trouble begins when we examine the negation function. It is a real number to the power of a real number. We have not specified an imaginary axis in the expression, and expect NATURE to deliver the correct hand of the spiral in the imaginary axis. We expect NATURE to IMAGINE. It is quite different with the Euler spiral. Here the expression is already complex: Nature replies “If you INSIST on imagining these things, then by YOUR standards it is a left-hand spiral”
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There is INVERSE SCALING SYMMETRY in the Gamma function about the point x=0.5 (x=-0.5 for fractorials). Thus, if there is a spiral for negative values, it will be modulated in amplitude by the reciprocal of the Gamma function symmetrically about 0.5. This leads to the following figure: |
However, this is a diagram of a figure that - as we will find out -
DOES NOT EXIST, so the problem of whether
it actually goes to amplitude zero at x=0 need not be solved.
The singularity at zero is shown by a red dot. This is a right-handed
spiral, and as we pass through zero we find ourselves copying a
HALF-SPIRAL and the singularity, by
means of a positive multiplier.
That half-spiral and singularity is then copied again and again.
The following shows two such half-spirals:
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If the spiral had been of the opposite hand (left-handed), we would still be copying half-spirals into the positive X domain: |
So the Gamma function has been WRECKED in the positive domain, and no longer bears any resemblance to what we started with: |
Nature has shown that you cannot CHOOSE
between Sin(πx) and
-Sin(πx)
modulated by the envelope, but must have BOTH.
Thus the left-hand twist of the spiral is cancelled by the right-hand
twist, which is equal and opposite - and the spiral
DOES NOT spiral.
If we had the number sequence 1, 4, 16, 64
we would say it is exponential in 4. A half-step of the exponentiation
process would involve the SQUARE ROOT OF 4,
just as a third-step would involve the cube root &c.
So a half-step would deliver either 1, 2, 4, 8, 16,
32, 64 for a square-root that is 2, or 1, -2, 4,
-8, 16, -32, 64 for a square-root that is -2.
The negation function is exponential in -1. Thus, the half-step is multiplication
by i1 or by -i1. But
nature refuses to allow this choice.
A set-theoretical explanation is that we started with the set of
REAL and never introduced anything
unreal. The set of real does not contain any element of the unreal,
so nature will not put it there.
Another way of seeing it is that the Reals are what the author calls
MONAL, requiring a single-number
(UNARY) specification. The Complex are
BINAL, requiring a two-number
(BINARY) specification, due to an
additional freedom of movement in complex space (in the i axis).
There is nothing in the original specification of the FACTORIALS,
1, 1, 2, 6 &c. to define the vector in the imaginary axis.
In passing, a study was made of the problem of multiple solutions,
and the difficulty was defined as a UNARY
result being delivered when the solution is still
BINAL. One of the parameters has been
left UNDEFINED, thereby misleading the
mathematician into believing that the one solution is the only one.
The conclusion for the problem of multiple solutions, therefore,
is that all DEGREES OF FREEDOM must
be closed off before one can trust an answer in the complex mathematics.
The conclusion for the Gamma riddle is that
EITHER the Gamma function OR the
complex mathematics or BOTH are artefacts
of Man.
The Gamma function is incompatible with a complex negation
function because the factorial rule will put half-turns of spiral
and singularities into places where they do not belong.
The sinusoid of frequency πX that is seen in the reciprocal of the
Gamma function needs an explanation other than a negation spiral in
complex space.
If it can be shown for other functions unrelated
to the Gamma function, and having a negation function, that Nature rejects
the imaginary axis in this way, then it can be said that Nature rejects
complex mathematics as a human product.
The result of this research is that it has been shown that complex
mathematics is a human artefact in the context
of the Gamma function.
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