Math Monday: Tubular Trefoils

by Glen Whitney

At Bridges 2013, the MoMath team learned from Tom and Koos Verhoeff how to make this lovely mathematical structure:
Here’s how to do it from a single sheet of 11″ x 17″ cardstock. Begin by cutting six 17″ long by 1.5″ wide strips.


Then make a mark every 1 19/32″ along one long side starting from the corner; on the parallel side of the same strip, start 17/32″ over and mark off every 1 19/32″. Draw lines connecting each mark to the nearest mark on the parallel side:


Mark all six the same way, making sure to slant in the same direction on all of them, and cut off the excess bits on either end to produce six strips of ten rhombuses each:


Now cut three of the strips in half, and cut the other three into a piece with just two rhombuses and a piece with eight, and arrange all of the resulting strips in order 5, 8, 5, 2, 5, 8, 5 2, 5, 8, 5, 2:


Now fold every line drawn on the 5-long strips as a valley fold, and every line on the other strips as a mountain fold:


Tape each rhombus of the 5-long and 8-long strips to the one three later in the strip to produce triangular tubes:


Finally, adjoin the tubes in order so there are no gaps between successive rhombuses — each piece has a sort of “jaw” formed by the end rhombuses which exactly mates with the end of the next tube. The two-long pieces also tuck into place at the end of the previous tube. Here’s what it looks like with four pieces used:


And seven pieces:


And the whole thing:


For the mathematical background of this tubular trefoil construction (for example, these rhombuses have the ratio of the long diagonal to short diagonal equal to the square root of two), and a whole bunch more “recipes” for building cool things out of triangular tubes, see this page that Tom Verhoeff put up. If you’re inspired to create something with this simple but versatile method, send pics to

This article first appeared on Make: Online, September 16, 2013.

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