Golden Ratios of Fermat-Primed Polygons

  • TotAng is short for the Total of All Internal Angles Along each Polygon's Circumference
  • 3-Gon is an Equilateral Triangle, 5-Gon is a Regular Pentagon, etc.....
  • Each entry of this table is based on one of the five Fermat Primes: 3, 5, 17, 257, and 65,537, which are the first five terms of the Fermat series, derived by the formula:
    22k + 1 = 3, 5, 17, 257, 65'536, 4'294'967'297, etc....
  • k is the simple integer progression, of: 0, 1, 2, 3, 4, 5, 6, 7, etc....
  • For example, the middle 2 is first raised to the upper k-th power, then the bottom 2 is raised to that result before adding 1:
    20 = 1, 21 = 2, 2 + 1 = 3 (the first fermat prime).
  • f is the Greek letter "Phi", often used to designate "golden" ratios. Each of the roots listed below are f proportions within their "polynomial set".

              From the sixth term (where k = 6), and upwards to the infinith term, all successive Fermat numbers may be composite. Polygons composed of composite Fermat Numbers are unconstructible using only straight-edge and compass. Consequently, any polynomials which are generated from the golden roots of unconstructible polygons will be unsolvable. Polynomials that are solvable in one unknown yield solutions for x of the form: x = some formula of numbers, such as the quadratic roots of the 5-Gon: x = (-1 ± Ö5) / 2. We could still generate an approximation of roots from the infinite class of unconstructible polygons of odd number of sides using trigonometry similar to what is being done below, but I'll skip over them here.

              The Phi-roots that are shown below can be approximated using many different methods. Two methods are possible using the information gleaned from the data below. One way is by using triangles (what I sometimes like to refer to as: "wedges") within regular polygons of odd numbers of sides. Each polygon is oriented with one of its sides along our visual-bottom putting one of its angles at its top. All of the triangles would then share their apex angle (the angle at their top) at the top of their surrounding polygon. Their bases would be shared along the polygon's base and along diagonals that horizontally cut across their associated polygon. The length of the base of each triangle is divided by the length of its slope. This makes each triangle's base equivalent to the f proportional value for that triangle while its slope is equal to the value of 1. Another way is to use trigonometry's sine function on one-half of each triangle's apex angle. From a few premises, is derived the formula:

    Sine of a Half Wedge Opposite ÷ Hypoteneuse = sine (An_Angle_in_Radians)
    Hypoteneuse = 1 = Referencing Unit of Measurement
    Opposite = sine (An_Angle_in_Radians)
    Opposite = f ÷ 2
    f ÷ 2 = sine (An_Angle_in_Radians)
    An_Angle_in_Radians = 2p ´ (An_Angle_in_Degrees° ÷ 360°)
    f ÷ 2 = sine (2p ´ (An_Angle_in_Degrees° ÷ 360°))
    An_Angle_in_Degrees = Apex_Angle° ÷ 2
    f ÷ 2 = sine (2p ´ Apex_Angle° ÷ 2 ÷ 360°)
    f ÷ 2 = sine (p ´ Apex_Angle° ÷ 360°)
    f = 2 ´ sine (p ´ Apex_Angle° ÷ 360)
     
     
     
    Additionally, every other f must be signed negative or else its resulting polynomial won't have integer coefficients.

    The following output is from a Perl program (61 Kb) that I updated on Thursday, 10 October 2002.
     

     
    Triangle

    3-Gon, TotAng = 180°   The circumferencial angle is 60°
    1. Angle is  60°      its Sine is   f  =  ± 1       its Reciprocal is   1 ÷ f  =  ± 1

    Its Polynomial of All f Roots is

    x  Minus or Plus  1 = 0

    Its Reciprocal Polynomial of All Reciprocal f Roots is

    x  Minus or Plus  1 = 0

    Test result for this polynomial ® If  0  loosely equals zero, then this test proves positive.
     

     
    Pentagon

    5-Gon, TotAng = 540°   The circumferencial angle is 108°
    1. Angle is  36°      its Sine is   f  =  ± 0.618033989       its Reciprocal is   1 ÷ f  =  ± 1.618033989
    2. Angle is  108°      its Sine is   f  =  Minus or Plus 1.618033989       its Reciprocal is   1 ÷ f  =  Minus or Plus 0.618033989

    Its Polynomial of All f Roots is

    x2  ±  x  –  1 = 0

    Its Reciprocal Polynomial of All Reciprocal f Roots is

    x2  ±  x  –  1 = 0

    Test result for this polynomial ® If  1.11022302462516e-016  loosely equals zero, then this test proves positive.
     

     
    Heptadecagon
     

    17-Gon, TotAng = 2700°   The circumferencial angle is 158.8235294117°
    1. Angle is  10 10/17°      its Sine is   f  =  ± 0.184536719       its Reciprocal is   1 ÷ f  =  ± 5.418975724
    2. Angle is  31 13/17°      its Sine is   f  =  Minus or Plus 0.54732598       its Reciprocal is   1 ÷ f  =  Minus or Plus 1.827064741
    3. Angle is  52 16/17°      its Sine is   f  =  ± 0.891476712       its Reciprocal is   1 ÷ f  =  ± 1.121734294
    4. Angle is  74 2/17°      its Sine is   f  =  Minus or Plus 1.205269273       its Reciprocal is   1 ÷ f  =  Minus or Plus 0.829690114
    5. Angle is  95 5/17°      its Sine is   f  =  ± 1.478017834       its Reciprocal is   1 ÷ f  =  ± 0.676581822
    6. Angle is  116 8/17°      its Sine is   f  =  Minus or Plus 1.700434271       its Reciprocal is   1 ÷ f  =  Minus or Plus 0.588085066
    7. Angle is  137 11/17°      its Sine is   f  =  ± 1.864944459       its Reciprocal is   1 ÷ f  =  ± 0.536208998
    8. Angle is  158 14/17°      its Sine is   f  =  Minus or Plus 1.965946199       its Reciprocal is   1 ÷ f  =  Minus or Plus 0.508660919

    Its Polynomial of All f Roots is

    x8  Minus or Plus  x7  –  7x6  ±  6x5  +  15x4  Minus or Plus  10x3  –  10x2  ±  4x  +  1 = 0

    Its Reciprocal Polynomial of All Reciprocal f Roots is

    x8  ±  4x7  –  10x6  Minus or Plus  10x5  +  15x4  ±  6x3  –  7x2  Minus or Plus  x  +  1 = 0

    Test result for this polynomial ® If  2.1538326677728e-014  loosely equals zero, then this test proves positive.

    If the test result is a number in scientific notation showing some absolute value less than 10 decimal digits, then this isn't too bad. In fact, it's pretty good since this program rounds to 10 digits.
     
     
    Next Lesson: Salt is a physical precursor to planetary archetypes.

     

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    Thursday, 10 October 2002 10:04:36 Pacific Daylight Time