Golden, Silver, Bronze, and Aluminum Ratios.

We've all heard of the golden ratio during our days of playing with math.  We've learned of its applications, its origin, everything, and we teach it to students.  Basically it is the ratio of the shorter side of a rectangle of length one to the longer side whose length is (1 + sqrt5)/2 or 1.618

We've seen it in ancient architecture such as the Parthenon, or the Pyramids of Giza. It is said that Leonardo Di Vinci's "Vitruvian Man" illustrates the golden ratio.  In addition, it is said he applied the golden ratio to his "Last Supper" Painting.  Even the Fibonacci sequence and pentagons are tied to the golden ratio.

If you read yesterday's column, I mentioned something about the silver ratio and has nothing to do with money.  It turns out there are at least three more ratios out there that are not seen as frequently.

The first is the silver ratio which is when you cut off two squares and still have a rectangle left with the same ratio as the first.  Or you get one 1 x 1 square, a 1 x sqrt 2 square and a rectangle of 1/ (1 + sqrt2).  because the whole rectangle is 1 x 1+sqrt2.  In other words, its 1:1.4.

In addition, the silver ratio is revered in Japan as the most beautiful ratio through history. Their name for it is "Yamato-hi", meaning Japanese ratio. They've used it in their architecture, statues of Buddha, and in the art of  flower arranging.

Another one, I'd not heard of is the Bronze ratio or mean.  It is the ratio of a side of 1 to the other side of (3 + sqrt13)/2 from a rectangle.  The idea of this one is to cut the rectangle into three parts instead of two.  This would be the third in the Metallic Mean family.

The Metallic Means which include the golden and silver ratios then bronze, copper, aluminum, etc, all based on the quadratic formula of x^2-px-q = 0 so that you get a solution of p + sqrt(p^2 + 4q)all divided by 2.  If p = 1 and q = 1 you get the golden ratio or if you use p =2 q = 1 you get the silver ratio.

So by changing the values of p and q, you get a different ratio.  So far though, only the silver and golden ratios are found in real life.  The other ratios seem to be theoretical only.  One interesting fact.  someone completed a survey of historical taffy pulling machines provided an interesting ratio based on the lengths the machines could pull the taffy.  The data showed some of the machines pull length are based on the silver or gold ratio.

Let me know what you think.  I'd love to hear.  Have a great day.