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How Common is Goldenality?


        When I first started out studying the golden ratio --- the only one that is popularly known to exist --- I started with infinite, number series: the Fibonacci series. Gradually, I worked my way up to an infinite series of sets of golden ratios. Each set composed a polynomial with integer coefficients. There was only one set of ratios as roots for each degree of polynomial. That work was difficult stuff to get the pairings correct between the different elements within each set of number series and to get the signs right so that its corresponding polynomial's coefficients would resolve to integers. Branching out into golden's existence within plane polygons wasn't easy at first either, but now in hindsight, and the way I'm attempting to sequence these "lessons", it looks a whole lot easier to digest. But it also lends the appearance of being very plain and simple --- almost too simple to care about it. That's the beauty of simplicity: it is easy to overlook.

        The following lessons will attempt to show how golden numbers can be found within isosceles triangles that are within a regular polygon. But upon looking at them a bit, it becomes apparent that star polygons within regular polygons are playing a role if all of a regular polygon's star polygons are within it, simultaneously. But then, things begin to look complicated when spokes from different stars need to be separated from their respective poly's to pair off with each other in order to form the necessary roots of that regular polygon's golden polynomial. So, using isosceles wedges is a whole lot more clean-cut in exposition, although not as elegant to look at.

        It may become apparent, shortly, that goldenality of odd-, or maybe even-, sided polygons (as well) do not restrict themselves to regular polygons. They may also be an inherent feature to irregular polygons. The trick to finding goldenality within a polygon, normally requires knowing how to match diagonals with other diagonals, or diagonals with the outer side of the polygon, in order to divide one by the other to get the right ratios to make a polynomial with integer coefficients. Then the proportions can be said to be "golden" with respect to one another, but only within the context of each set. Composites of golden polynomials can be multiplied together, just like the number of sides to numerous odd-sided polygons can be multiplied together, to get a composite polygon/polynomial. The new composite will have the root/ratios of its "factors" plus a few more unique to itself. The nonagon (9-gon, or enneagon) is a good example. It has a triangle (3-gon) within it. So it has the first golden ratio of "1" plus three others. Its polynomial is really a first degree polynomial times a third degree polynomial making a fourth degree, composite polynomial of four roots.

The Wedges of the Nonagon, Enneagon, or 9-gon

Composite 9-Gon, Nonagon, or Enneagon
Made of one triangle rotated into three positions within its circumscribed circle.
Total Angles Facing Inward Along the Circumference = 1260°
One Circumferencial Angle is 140°, or 1/9 of 1260°
Every other f in sequence is negated so that they can form a polynomial
with integer coefficients. It could just as easily be every even-numbered f,
rather than every odd. All of these proportions could also be multiplicatively
reciprocated, together at the same time. This would be correct for purposes
of proportionality and polynomial construction, but it would not be correct
for each angle's sine value.
1. Angle is 20°    its Sine is    f    0.347296355   1 ÷ 0.347296355   =    2.879385242
2. Angle is 60°    its Sine is    f  –1   1 ÷ ( –1 )   =  –1
3. Angle is 100°    its Sine is    f    1.532088886   1 ÷ 1.532088886   =    0.652703644
4. Angle is 140°    its Sine is    f  –1.879385242   1 ÷ ( –1.879385242 )   =  –0.532088886
0 = (x – 0.347296355) ´ (x – 1.532088886) ´ (x + 1.879385242) ´ (x + 1)
   = (x3 – 3x + 1) ´ (x + 1)
   = x4 + x3 – 3x2 – 2x + 1
0 = (x + 0.347296355) ´ (x + 1.532088886) ´ (x – 1.879385242) ´ (x – 1)
   = (x3 – 3x – 1) ´ (x – 1)
   = x4 – x3 – 3x2 + 2x + 1
This ®   0 = x4 ± x3 – 3x2 Minus or Plus 2x + 1   or this ®   0 = x4 Minus or Plus x3 – 3x2 ± 2x + 1  
are fairly descriptive, but this ®   0 Questionable Euivalence x4 ± x3 – 3x2 ± 2x + 1  
 
could be confusing to the uninitiated within the context of an equation,
because there is too much freedom to choose which ± coefficient ± is negative
and which is positive. Every other option must be the same choice and the
odd numbered choices must be opposite to the even numbered ones. So, the
first option at the sign of the ± x3 term has to be the opposite choice of
whatever the sign of the Minus or Plus 2x1 term is, or else the equation will not zero out.

 
 
Next Lesson: Odd-Sided Polygons
 
References:
Dividing the circumference of a circle with a star polygon.
How are polygons formed from a circle? Answer: The formation of Regular and Star Polygons use exterior angles.
A demonstration of the proof that the exterior angles of a polygon always add up to 360 degrees.
Proof for the Sum of the Exterior Angles of a Polygon

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Saturday, 13 December 2008 14:17:10 MST [an error occurred while processing this directive]