Golden Ratios of Odd-Sided 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.....
  • 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".

              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 result must be signed negative or else its resulting polynomial won't have integer coefficients.

    Additionally, every other result 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

    Prime 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

    Prime 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  0  loosely equals zero, then this test proves positive.
     

     
    Heptagon

    Prime 7-Gon, TotAng = 900°   The circumferencial angle is 128.5714285714°
    1. Angle is  25 5/7°      its Sine is   f  =  ± 0.445041868       its Reciprocal is   1 ÷ f  =  ± 2.246979604
    2. Angle is  77 1/7°      its Sine is   f  =  Minus or Plus 1.246979604       its Reciprocal is   1 ÷ f  =  Minus or Plus 0.801937736
    3. Angle is  128 4/7°      its Sine is   f  =  ± 1.801937736       its Reciprocal is   1 ÷ f  =  ± 0.554958132

    Its Polynomial of All f Roots is

    x3  Minus or Plus  x2  –  2x  ±  1 = 0

    Its Reciprocal Polynomial of All Reciprocal f Roots is

    x3  Minus or Plus  2x2  –  x  ±  1 = 0

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

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

    Composite 9-Gon, TotAng = 1260°   The circumferencial angle is 140°
    The 9-Gon has its own unique roots, plus those of the: 3-Gon.
    1. Angle is  20°      its Sine is   f  =  ± 0.347296355       its Reciprocal is   1 ÷ f  =  ± 2.879385242
    2. Angle is  60°      its Sine is   f  =  Minus or Plus 1       its Reciprocal is   1 ÷ f  =  Minus or Plus 1
    3. Angle is  100°      its Sine is   f  =  ± 1.532088886       its Reciprocal is   1 ÷ f  =  ± 0.652703645
    4. Angle is  140°      its Sine is   f  =  Minus or Plus 1.879385242       its Reciprocal is   1 ÷ f  =  Minus or Plus 0.532088886

    Its Polynomial of All f Roots is

    x4  ±  x3  –  3x2  Minus or Plus  2x  +  1 = 0

    Its Reciprocal Polynomial of All Reciprocal f Roots is

    x4  ±  2x3  –  3x2  Minus or Plus  x  +  1 = 0

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

     
    Undecagon

    Prime 11-Gon, TotAng = 1620°   The circumferencial angle is 147.2727272727°
    1. Angle is  16 4/11°      its Sine is   f  =  ± 0.284629677       its Reciprocal is   1 ÷ f  =  ± 3.513337092
    2. Angle is  49 1/11°      its Sine is   f  =  Minus or Plus 0.830830026       its Reciprocal is   1 ÷ f  =  Minus or Plus 1.203615624
    3. Angle is  81 9/11°      its Sine is   f  =  ± 1.309721468       its Reciprocal is   1 ÷ f  =  ± 0.763521118
    4. Angle is  114 6/11°      its Sine is   f  =  Minus or Plus 1.682507066       its Reciprocal is   1 ÷ f  =  Minus or Plus 0.594351144
    5. Angle is  147 3/11°      its Sine is   f  =  ± 1.918985947       its Reciprocal is   1 ÷ f  =  ± 0.521108558

    Its Polynomial of All f Roots is

    x5  Minus or Plus  x4  –  4x3  ±  3x2  +  3x  Minus or Plus  1 = 0

    Its Reciprocal Polynomial of All Reciprocal f Roots is

    x5  Minus or Plus  3x4  –  3x3  ±  4x2  +  x  Minus or Plus  1 = 0

    Test result for this polynomial ® If  -4.9960036108132e-015  loosely equals zero, then this test proves positive.

     
    Prime 13-Gon, TotAng = 1980°   The circumferencial angle is 152.3076923076°
    1. Angle is  13 11/13°      its Sine is   f  =  ± 0.241073361       its Reciprocal is   1 ÷ f  =  ± 4.148114905
    2. Angle is  41 7/13°      its Sine is   f  =  Minus or Plus 0.709209774       its Reciprocal is   1 ÷ f  =  Minus or Plus 1.410020048
    3. Angle is  69 3/13°      its Sine is   f  =  ± 1.136129493       its Reciprocal is   1 ÷ f  =  ± 0.880181358
    4. Angle is  96 12/13°      its Sine is   f  =  Minus or Plus 1.497021496       its Reciprocal is   1 ÷ f  =  Minus or Plus 0.66799308
    5. Angle is  124 8/13°      its Sine is   f  =  ± 1.770912051       its Reciprocal is   1 ÷ f  =  ± 0.564680781
    6. Angle is  152 4/13°      its Sine is   f  =  Minus or Plus 1.941883635       its Reciprocal is   1 ÷ f  =  Minus or Plus 0.514963915

    Its Polynomial of All f Roots is

    x6  ±  x5  –  5x4  Minus or Plus  4x3  +  6x2  ±  3x  –  1 = 0

    Its Reciprocal Polynomial of All Reciprocal f Roots is

    x6  ±  3x5  –  6x4  Minus or Plus  4x3  +  5x2  ±  x  –  1 = 0

    Test result for this polynomial ® If  -5.55111512312578e-015  loosely equals zero, then this test proves positive.

     
    Composite 15-Gon, TotAng = 2340°   The circumferencial angle is 156°
    The 15-Gon has its own unique roots, plus those of the: 3-Gon, and 5-Gon.
    1. Angle is  12°      its Sine is   f  =  ± 0.209056927       its Reciprocal is   1 ÷ f  =  ± 4.783386117
    2. Angle is  36°      its Sine is   f  =  Minus or Plus 0.618033989       its Reciprocal is   1 ÷ f  =  Minus or Plus 1.618033989
    3. Angle is  60°      its Sine is   f  =  ± 1       its Reciprocal is   1 ÷ f  =  ± 1
    4. Angle is  84°      its Sine is   f  =  Minus or Plus 1.338261213       its Reciprocal is   1 ÷ f  =  Minus or Plus 0.747238275
    5. Angle is  108°      its Sine is   f  =  ± 1.618033989       its Reciprocal is   1 ÷ f  =  ± 0.618033989
    6. Angle is  132°      its Sine is   f  =  Minus or Plus 1.827090915       its Reciprocal is   1 ÷ f  =  Minus or Plus 0.547318139
    7. Angle is  156°      its Sine is   f  =  ± 1.956295201       its Reciprocal is   1 ÷ f  =  ± 0.511170297

    Its Polynomial of All f Roots is

    x7  Minus or Plus  x6  –  6x5  ±  5x4  +  10x3  Minus or Plus  6x2  –  4x  ±  1 = 0

    Its Reciprocal Polynomial of All Reciprocal f Roots is

    x7  Minus or Plus  4x6  –  6x5  ±  10x4  +  5x3  Minus or Plus  6x2  –  x  ±  1 = 0

    Test result for this polynomial ® If  -2.63122856836162e-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.
     

              There is a short-cut for determining the coefficients. They form a convenient pyramid. Some coefficients, represented by the blue lines, are carried down from above. The others, represented by the red lines, are summed up:

    Pyramid of Coefficients for Odd-Sided Polygons up to the 15-gon


     
     
    Next Lesson: Fermat-Primed Polygons

     

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