Adam Ennes

The Poligonal Sequence of Common Ratios

By Adam N. Ennes

Euclid defined geometry as we teach it to this day.  He did a fine job indeed, for we hold much knowledge due to his discoveries.  Without his intuitive nature, I would not be typing this today.  So before I continue, I would like to thank Euclid for his contributions to geometry.

            There is one part of his work that I felt differently about.  When I was a sophomore at Oak Park & River Forest High School, my current teacher was explaining the relationship of internal to external angles of a shape.  That is more specifically, how to determine the external angle of a shape (polygon), given the internal angle.  This is Euclidean geometry, as below;

            The internal angle of this triangle is 180 degrees, and according to Euclid, it has an external angle of 360 degrees.

Euclid’s formula for determining the external angle is as follows;

180-60=120

120*3=360

             This particular method of determining the external angle of a Triangle can also be viewed like this.

Since we are measuring with 180 degrees, in theory we are

measuring with a triangle, when we

use Euclid’s method.  If we measured as such with

shapes 3-10, our I/E angle ratio sequence would look like this.

(I=internal angle/E=external angle)

.5, 1, 1.5, 2, 2.5, 3, 3.5, 4

              In all relevancies, the progression of .5 in this sequence is akin to the progression of 180 degrees per shape in the sequence of internal angles.  But why allow the external angle to remain at 360 degrees?  Shouldn’t the external angles of polygons grow in relation to the growth the internal angles?  I believe that they do.  If we measure with a circle (360 degrees), instead of a triangle (180 degrees), we find that the external angle (E) progresses with the internal angle (I) to form a sequence that continues for infinity without passing 1 as a result.  Here is how the triangle will look if we use a circle to measure the external angles.

I=180  E=900

360-60=300

300*3=900

180/900=.2=1/5=I/E

              Here is one formula we can use in order to prove the above angles;

180+900=1080 (this is also the E for a square(4))

1080/360=3

*The following sequence is just a different way of measuring the external angle of a polygon, viewed in two fashions.  It is not to say Euclid was wrong, but rather, I like to say he was thinking too linear………For, one could measure ‘E’ with any shape, but you will only get this sequence by measuring with 360 degrees.*

*Note that in the 4 Primary shape sequences, the triangle sequence is the pentagon sequence’s I/E ratio progression and vise versa.*

            So what does it all mean?  To me, it clearly takes infinity, and places it between zero and one.  A radical notion indeed, never the less it forms a sequence which is filled with ratios that are common, others are as well, but are not recognized as being so.

Pick any whole number.

            24?

            Ok.

           A 24 sided polygon (which may look circular from afar) has in internal angle of 3960 degrees.

I=3960

Which in the form of Euclidean geometry, would yield an E=360, & an I/E Ratio of 11(=3960/360) or .090909090909(=360/3960).  This is congruent to 11/1 or 1/11.  Interestingly enough 1/11 is measured by nines (.090909) and 1/9 is measured by elevens (.111111).  Anyway, Euclid’s sequence deserves more study, but I find this following method to be more interesting, and not as sporadic, though I will search for similar patterns in sequence of polygons I/E and E/I ratios.

            Here is how the polygon 24 would look as it does in the PSCR.

(24-2)*180=3960=I (same as Euclid)

(24+2)*180=4680=E

Note that 3960/360=11, which is the simplified I of Polygon 24, and that 4680/360=13, which is the simplified E of Polygon 24.

            (I/E)=(.846153846538461538)=(11/13)=(Polygon 24).

              This is described by the polygonal sequence of common ratios as being in the square sequence.  Which brings us to the point of determining the I/E ratio of any whole numbered Polygon, using the circle (360) as our unit of measurement as opposed to the triangle (180).  The following 3 formulas are for finding the I/E based on whether it came from the triangle, square, pentagon, or hexagon.  Which from now on I will refer to as the 4 primary shapes.

All odd polygons (Triangle and Pentagon Rooted)

(N-2)/(N+2)=I/E

Half of all even polygons (Square Rooted)

((N/2)-1)/((N/2)+1)=I/E

Half of all even polygons (Hexagon Rooted)

((N/4)-.5)/((N/4)+.5)=I/E

            The Polygonal Sequence of Common Ratios is just one way of grasping the infinite.  This is an exploration into the unknown.  This knowledge is meant to be known by all.  There is a relevant simplicity that is most like remembering one’s own name.  Until I taught myself this, I had no comprehension of the relationships between shapes, fractions, & decimals.  I could not discern between 1/3 and 3/7.  Now I know; and many will as well, soon enough.  I believe that is sequence can be taught to all children from ages 5 up, with a definite emphasis on shapes in general in the earlier years, especially the 4 primary shapes.  Within and without these 4 primary shapes lays infinity.  For every shape comes from and is striving to become a circle. Our perception proves this.

That milk bottle cap, your steering wheel.  All things we consider to be a circle is still a polygon.  A bottle cap is an obvious one.  That ceramic bowl though sure looks round……like a circle, but on a molecular level is still an unknown polygon.  Now it is known, its’ existence in-between the shapes attempt to be circular.  If we can teach this to the world, human kind will realize that science is proving what spirituality has revealed to human kind throughout the ages past.  That is not to say that we have nothing else to learn.  With this theorem this sequence that organizes chaos, one can do many things.

               I will start with an interesting pattern derived from finding each Fibonacci sequence number’s shape root (that is, which primary shape it derived from) and listing it in order.  Like so;

0,1,1,2,3,5,8,13,21,34,55,89,144…………

For the sake of pattern recognition, we’ll use letters to represent the shape root, T=Triangle, S=Square, P=Pentagon, and H=Hexagon

S(0),P(1),P(1),H(2),T(3),P(5),S(8),P(13),P(21),H(34),T(55),P(89),S(144).............

This pattern repeats itself infinitely throughout the continuation of the Fibonacci sequence.

Another use for this sequence is to apply a letter to each shape, thereby creating a codex.  ‘A’ for the Triangle, ‘B’ for the Square, all the way to ‘Z’.  This creates a codex that only those who know the sequence could use.

A Re-listing of Relevant Formulas, and Rules.

E=exterior angle of shape in ratio form unless otherwise stated

I=interior angle of shape in ratio form unless otherwise stated

       N=polygon in question

1.  All odd polygons (Triangle and Pentagon Rooted)

(N-2) / (N+2)=I/E

2.  Half of all even polygons (Square Rooted)

((N/2)-1) / ((N/2)+1)=I/E

3.  Half of all even polygons (Hexagon Rooted)

((N/4)-.5) / ((N/4)+.5)=I/E

4.  N= ((I+E)) / 360 (In whole angle form)

5.  E=I+720 (In whole angle form)

6.  Every polygon is the result of the shape 4 sides less than itself, and the precursor to the shape with 4 more sides than itself.  From 3 - ∞.

7.  Every simplified I/E Ratio is in relation to the shape(s) before and after itself.  By means of the # of sides and the I/E Ratio, of the shape(s) before and after itself.  For the Triangle and Pentagon sequences, both the polygonal and I/E Ratio progress by +4.  The Square and Hexagon sequences progress by +4 in terms of sides, but the Square’s I/E progression is maintained by +2, and the Hexagon’s by +1.

            Rule 6 is reinforced by rule 7 , and now we can simply know what each shape’s I’E is by adding 4, 2, and 1.

Pseudo Conclusion.

            Also as a result, in my mind at least (please let me know your opinion), this theorem will go on to prove that every thing without, is within. As well as how everything that is within is without.  Even though our universe is gargantuan beyond proportion, it is within you, and all as well.  With this theorem, one can forget about numbers entirely and view the simplified form of infinity.  This infinity, which in its existence is striving to become and came from a circle, exists in between 0 and 1. 

            A reference to the cosmos perhaps?  Or?  A new way to teach decimals, ratios, addition, subtraction, multiplying, dividing, pattern recognition, probability, computer programming, communications……  The possibilities may be as endless as the Polygonal Sequence of Common Ratios itself.  It is my aspiration to teach this to people of all ages.

Imaginary Shapes

            An Imaginary Shape is any number with a decimal represented as if it is a polygon with an Internal and External angle (I/E). 

            If Pi was a shape it would have to be an Imaginary Shape, for we can not in literal terms, draw a shape with 3.14159265 sides.  But we can imagine it like so.

N=Pi

((N-2)*180))/ ((N+2)*180)=(I/E)

I=205.4866776

E=925.4866776

I/E=.2220309407

E/I=4.503876789

E*I=1

            Thus we have found Pi’s I/E and E/I. Which in terms of the main sequence fits in right between the Triangle (3) and the Square (4).  Pi is 3.14159265 which is just .14159625 more than 3 (Triangle), but not more than 4 (Square). 

*The bigger the number of any shape (real or Imaginary) the bigger the decimal equal to I/E, vice versa. Any number, decimal or not, will fit between any 2 polygons, unless it is a number less than 3, then it fits between 0 and 3 (Circle and Triangle).*