Golden Ratio Blocks

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Introduction: Golden Ratio Blocks

About: I make my living inventing stuff for various companies, and developing my own inventions. But my true love is inspiring kids to build cool stuff. I do this by mentoring High School FIRST robotics teams, an...

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?

Supplies:

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.

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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:

https://www.mathsisfun.com/numbers/nature-golden-r...

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    7 Discussions

    0
    jeanniel1
    jeanniel1

    5 weeks ago

    This is PERFECT - and not because it's a Golden Ratio! Ha ha! I love the way you colored the blocks, and the thinking behind the size. Bravo! BTW - the colors remind me of the outcome my team used but our's were food grade dye and alcohol as the carrier. Gave a nice watercolor-y touch.

    0
    secretshedfreak
    secretshedfreak

    8 weeks ago on Step 3

    Struggling to type this comment after you melted my brain, how did you colour the wood with the rainbow effect?

    0
    CreativeTinker
    CreativeTinker

    Reply 8 weeks ago

    Painting them was a different matter. My spouse and the twins are very crafty when it comes to painting and decorating. Me not so much. I'll ask.

    0
    CreativeTinker
    CreativeTinker

    Reply 7 weeks ago

    They used tubes of acrylic paints. They continuously mixed new colors as the advanced around the blocks. Most of the mixing occurred right on the blocks instead of a painters pallet.

    0
    secretshedfreak
    secretshedfreak

    Reply 7 weeks ago

    Ahhh, great finish! Thanks for getting back to me.

    0
    BIGDaddy504
    BIGDaddy504

    7 weeks ago

    Is this surface only? Colors look great though

    0
    KristiE7
    KristiE7

    8 weeks ago

    I love this!!!!