As 2021 comes to a close, let's wrap up the last math puzzle. Recall the blog posting "Tree Puzzle Solved" from November 11th where a very special number, 2.4142135... , called the Silver Ratio, was found by embedding a small circle in four larger ones that were arranged with their centers in a square, with their edges touching, and then finding and dividing the radius of the small circle into radius of one of the large circles. This was very cool! But then it was also found by a special sequence of numbers 0, 1, 2, 5, 12, 29, 70, ....as described. This is called the Pell Sequence. Even cooler! Last we found that the Silver Ratio was found to be exactly 1 + Square Root of 2. Beyond cool! And, like the Golden Ratio, the Silver Ratio can be found in nature as well. Here's an article about the Silver Ratio and the making, of all things, taffy: http://www.smithsonianmag.com/science-nature/using-math-build-ultimate-taffy-machine-180960849/

It turns out that the Golden Ratio and Silver Ratio are only two of a series of numbers called the Metallic Ratios.

One way that these ratios can be very closely approximated is by sequences of numbers like the Fibonacci and Pell sequences already described. The next ratio in this series is the Bronze Ratio and this number can also be approximated from a sequence. Recall that the Fibonacci sequence is found by adding the previous two numbers, starting with 0 and 1. So, the next number in the sequence is 1, then 1 +1 gives the next number 2, then 1 + 2 gives 3, then 2 + 3 gives 5, then 3 + 5 gives 8, and so on for the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... . The Pell sequence is derived by starting with 0 and 1 and then multiplying the leading number by 2 and adding the previous number. So, to get the third number, multiply 1 by 2 and add 0 giving 2, the next is 2 x 2 plus 1 = 5, then next 2 x 5 plus 2 = 12, the next 2 x 12 plus 5 = 29, and so on for the sequence 0, 1, 2, 5, 12, 29, ... . Now, instead of multiplying by two, multiply by three. This would be 0, 1, 3, 10, 33, 109, ... . Now, as before, divide one number by its previous number and you get close and closer to the bronze ratio of 3.3027756... the further out into the sequence that you go. Like the golden and silver ratios, the bronze ratio is another "irrational" number because it can't be expressed as one integer divided by another and the decimal places go on forever. Turning back to the Fibonacci sequence, this is the first of the sequences and just multiplies the leading number by ONE and adds the previous number. The Pell multiplies by TWO, the bronze sequence by THREE. Get the pattern? So, we can generate a whole series of sequences by multiplying by successively higher numbers, the next by multplying by FOUR, and on and on. These series are collectively called the Metallic Sequences and give ratios that are found in various parts of the natural world.

I invite you to browse the internet and explore the world of the Metallic Ratios like the Golden, Silver, and Bronze Ratios and to think about how the first two were out there all along, the Golden being wrapped up in two sticks beneath your feet and the Silver in a group of imaginary crowns of trees up in the air! Golden and Silver Moments indeed!

We all know that there are many uses of gold and silver in our cultural history. The King James Bible mentions gold 417 times and silver 320. Both have worked their way into common vernacular- "going for gold", "worth its weight in gold", the "golden oldies", "all that glitters is not gold", and "silence is golden". Silver too- we say "silver tongue", "silver bullet", "born with a silver spoon", "silver screen", and "silver fox" with certain meanings.

And so, this "silver haired" writer will wrap up by quoting Neil Young and part of the song "Silver and Gold" http://youtu.be/cyyq5VxAXAk :

"Workin hard every day

Never notice how

the time slips away

People come, seasons go

We got something

that'll never grow old.

I don't care

if the sun don't shine

And the rain keeps pouring

down on me and mine

'Cause our kind of love

never seems to get old

It's better than silver and gold."

Thanks, Jon, for a lighthearted look at these ratios! My dad would have loved exploring them, I think.