The Reflecting
Light Model

for The Extended
Euclidean Algorithm

expanded from its source in
The Divine Proportion
          The Fibonacci rabbit breeding model is limited to producing only the quadratic golden beautiful polynomial of roots:

x₁ = 1.618... and x₂ = 0.618...
Also known collectively as: phi, tau, and the golden: section, ratio, or mean.
          The male honeybee genealogy model is mathematically more generic by allowing expansion into higher degree golden beautiful polynomials if we don't mind entertaining the idea of multiple genders. For example: a three gendered biology would require three different genders of bees to sexually reproduce themselves. They would not be two opposites and a hermaphrodite (that would be a binary situation), but a triad: no bee would have a totally opposite partner and every bee would be partially (one-third) different and similar to each of its other two partners. This is what you would call a real group marrage. A minor point: it also requires working your way backwards within an individual male's family tree.
          The reflecting light model is simpler in that it doesn't require mentally modeling anything extra-ordinary. It simply looks at the situation of light pasing through panes of sandwiched glass and tracking the different alternate pathways the lightbeam could take based on: the number of panes of glass and the number of consistent reflections through out its journey. The lightbeam is only allowed to reflect from interior surfaces. Requiring a fixed number of reflections through out is a little bit extra-ordinary, but not as difficult to imagine as alternate breeding situations. All of this will lead up to a formula for iterating number series and what to do with them. This will be divided into five sections----five polynomials of degrees one through five.
          It is necessary to track each surface the lightbeam bounces off of by labeling all surfaces (with a letter name). Although the two outside surfaces on either side of the sandwich have their own letter name, surfaces on the inside of the sandwich must share letter names as if they were one surface instead of two. Then a chart is drawn up grouping the light's pathways into seperate groups of paths based on which surface the light first struck (two panes of glass would result in two groupings; it is a one -to- one coorespondance). By counting the total number of pathways possible within each group at each reflection increment, and adding up their running totals, a pattern imerges that can be formulized for every single number of glass-panes and number of reflections up to infinity as a limit.

First Degree: x 1

No Reflections >>>>>>>>>>>>

One Reflection >>>>>>>>>>>>

Two Reflections >>>>>>>>>>>

Three Reflections >>>>>>>>>>

Ad Infinitum.............

          Now a chart can be made up showing the trend and a formula that duplicates most of the action (skipping the first case of no reflections):

(R)eflections
R: 0, 1, 2, 3, 4, 5, 6, 7, ...,
b: 0, 1, 1, 1, 1, 1, 1, 1, ...,

Initial Value for:
b₁ = 1

Self-Looping Formula:
b_(R+1) = b_R

So at first,
b₁ = 1

Then,
b₂ = 1 = b₁ = 1

Then,
b₃ = 1 = b₂ = 1

Then,
b₄ = 1 = b₃ = 1

Ad Infinitum.............

Further, a polynomial can be built using the repetitive results of this series:

x = b = 1

0 = x 1

Second Degree: x² x 1

No Reflections >>>>>>>>>>>>

One Reflection >>>>>>>>>>>>

Two Reflections >>>>>>>>>>>

Three Reflections >>>>>>>>>>

Ad Infinitum.............

          Now a chart can be made up showing the trend and a formula that duplicates most of the action (skipping the first case of no reflections):

(R)eflections
R: 0, 1, 2, 3, 4, 5, 6, 7, ...,
b: 0, 1, 1, 2, 3, 5, 8, 13, ...,
c: 0, 1, 2, 3, 5, 8, 13, 21, ...,

Initial Values for:
b₁ = 1 & c₁ = 1

Self-Looping Formula:
b_(R+1) = c_R
c_(R+1) = c_R + b_R

So at first,
b₁ = 1
c₁ = 1

Then,
b₂ = 1 = c₁        = 1
c₂ = 2 = c₁ + b₁ = 1 + 1

Then,
b₃ = 2 = c₂        = 2
c₃ = 3 = c₂ + b₂ = 2 + 1

Then,
b₄ = 3 = c₃        = 3
c₄ = 5 = c₃ + b₃ = 3 + 2

Then,
b₅ = 5 = c₄        = 5
c₅ = 8 = c₄ + b₄ = 5 + 3

Then,
b₆ = 8  = c₅         = 8
c₆ = 13 = c₅ + b₅ = 8 + 5

Ad Infinitum.............

Further, a polynomial can be approximated using the progressive results of this series:

x₁ = (b   c) = 1.6180339....
x₂ = (c   b) = 0.6180339....

0 = x² x 1

3^rd Degree: x³ 2x² x 1