Introduction: Golden Ratio Blocks
This is a perfect example of an easy project that goes all nerdy. Then there is a happy ending. (In a mathematical sense.)
"I'd like you to cut up this old post into blocks as bases for the centerpiece candles", my spouse says.
What could be easier?
We used an old piece of 6 by 6 wood post. (This measures 5.25 inch by 5.25 inch). You could make boxes or cut cardboard tubes to length too. Depending on the material, you will need appropriate cutting equipment. We used a band saw.
Step 1: Nerd Out
"What lengths," I ask.
Then the possibilities spin. Do I do constant steps? Binary blocks? Fibonacci series? Logarithmic? Quick google of "series". How about just constant ratios? So I made a spreadsheet to try some different combinations. The "Bar Chart" option let me visualize the "too many" choices.
The constant ratio choice alone has infinite possibilities. Arbitrarily, I chose the make the "middle one" a cube. Lets call this height B. Let the ratio from one to the next be r. Then for heights a, B, C we have:
A x r = B
B x r = C
For example if we choose r = 1.5 then we get
A x 1.5 = 5.25
solving for A we get
A = 5.25 / 1.5 = 3.5
and for B we have
B x 1.5 = 5.25 x 1.5 = 7.875
But we could choose any r.
Step 2: Add a Constraint
With too may choices, what if we required that they stack together nicely for storage. That is choose the heights so that A + B = C. In our sample case with r = 1.5 we have
3.25 + 5.25 = 8.5 which is not far from 7.875.
I used the "solver" function in my spreadsheet to iteratively find the ratio that made A + B = C. It came up with
A = 3.245
B = 5.2500
C = 8.495
And sure enough 3.245 +5.250 = 8.495
I made my measurements and cuts on the band saw, happy with my clever choice. On the way back to my desk I thought, "wait a minute. What was that ratio that the solver found?" There it was, staring me in the face
A = B / 1.6180 = A * 0.6180
C = B * 1.6180
That is 1/r = r -1
That's the very definition of the "Golden Ratio".
Step 3: More to Explore
I'm surprised that the Golden Ratio shows up in a practical problem (fitting three things into a box). You could use this fact to make three boxes that fit into one box of any size.
The more surprising thing is that this means that if you define a golden sequence G(n) defined by G(0) = 1 and
G(n+1) = tao * G(n) (where tao = 1.618033988749895...)
1.0000, 1.6180, 2.6180, 4.2361, 6.8541, 11.0902, 17.9443, 29.0344, 46.9787, ...
then G(n) = G(n-1) + G(n-2) which is the very definition of the Fibonacci series! But this looks different...hmmm
There is so much more to explore. Here is a random starting point:
Participated in the
Made with Math Contest