Thursday, February 25, 2010

Zomadic Alchemy: The Triangle Question

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The Question
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Given a length of square stock and a standard compound miter chop-saw how would you make an equilateral triangle using only 45 degree cuts?
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 This is a woodworking math puzzle I invented. For three years now no one I've asked
has been able to answer the question.
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The Hints
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You need only set up the saw once
You may fashion a simple length stop
You will make exactly four cuts

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The Answer
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Set the saw to a 45 degree x 45 degree compound cut as shown.
Cut off the tail of the stock.
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Rotate the stock and off cut 180 degrees - this is critical.
partial rotation shown

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Rotate stock 180 degrees
completed rotation shown

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Use off cut as a length stop.
Slide stock to the stop.

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Make the second cut.
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remove first part
rotate stock 180 degrees and slide to stop

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With exactly four cuts you can generate three parts.
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Place the three parts flat on a surface

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Rotate the two side parts by 90 degrees
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Rotate one part around the fold axis.

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Fold the second part over by 90 degrees.
Here is an equilateral triangle made from only 45 degree cuts.
That's It!


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There is More!
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It's interesting that the two angled sides
of the triangle are not the same.
Depending on the respective orientation from these triangles 
one can make edge truncated octahedrons and tetrahedrons.
Consequently it follows that one can generate an octet-truss.

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Octahedron and Tetrahedron

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An octet-truss generated from a single part.



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Photos
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These alder frames are glued and nailed.
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Three triangles preparing to merge.
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The start of a tetrahedron.
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The start of an octahedron.
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Truss components
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1 comment:

Andre Gosselin said...

Thanks a lot for this puzzle. I was playing to create tetrahedron lamp fixture and was going exactly the way you described in this puzzle.