https://en.wikipedia.org/wiki/Talk:Silver_ratio

Expanding the definition of the Silver ratio to include varieties of the Sacred Cut.

[[File:Sacred cuts v2b - command prompt output.svg|thumb|Sacred cuts v2b - command prompt output]]
Since ...
[[File:Silver ratio octagon.svg|thumb|left|120px|Silver ratio within the octagon]]
[[File:Search-for-silver-ratios-v2b PERL-code.pdf|thumb|PERL code]]
* The silver ratio of 2.414..., and its negated reciprocal of -0.414..., can form two solutions for a polynomial <math>(x^2 - 2x - 1)</math> in one unknown similar to this property of the Golden ratio <math>(x^2 - x - 1)</math>.
* The silver ratio can be found as a relational proportionality among the lengths of certain diagonals versus the length of one side of an equilateral polygon of an even-number of sides as is already included within this article and displayed on the left.
* Thus, a few more varieties of silver ratios appear via a computer search for quadratic polynomials in one unknown of integer coefficients whose solutions are the proportions among the diagonals and side of an equilateral polygon of 4x__? number of sides in which the variable of __? is a prime number greater than unity.
* We've already been given the first and simplest of this series: the octagon of 4x2 sides. Yet, there exists at least another unique set of solutions for another prime: the number 3.
* Three solutions for a 4x3=dodecagon ...
* <math>1 + \sqrt{3}</math> is one of two solutions for <math>x^2 - 2x - 2</math>
* <math>2 + \sqrt{3}</math> is one of two solutions for <math>x^2 - 4x + 1</math>
* <math>\frac{(2 + \sqrt{3})}{2}</math> is one of two solutions for <math>2x^2 + 2x - 1</math>.
* The third possibility of a 4x5=icosagon possesses no unique set of solutions. They are all repetitious variations of the Golden Ratio.
* But, these three sets of solutions are the only three my feeble computer could discover before I gave up waiting for anything else to appear.
* So, the revised definition of a Silver ratio could be that it is formed from the solutions to a quadratic polynomial in one unknown whose solutions are ratios of the internal and external diagonals of an equilateral polygon of 4x__? number of sides in which the variable of __? is a prime number and greater than unity.
* All of this is displayed within the screenshot of the DOS command prompt outputs in the upper right.
* The software code for discovering these two additional varieties of the silver ratio were executed in PERL (displayed in the lower right). -- ~~~~