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The small figures in the table above indicate the overhead compared to an ordinary dissection. They give an indication of where improvement may be possible. Is it possible to reduce this overhead to zero?
This is based on the usual four piece dissection of a triangle to a square. In Greg Frederickson’s second book he describes how to convert hinged dissections to twist hinge dissections. By replacing each twist hinge by a lug and a lock (also called tab and blank), twist hinge dissections can be converted to locked dissections. This is an example.
Unfortunately, the conversion of a hinge to a twist hinge adds a piece, and since we have replaced three hinges this dissection requires three extra pieces compared to the unlocked version.
This dissection is based around the first one here. Compare to see how this one was found.
Greg Frederickson’s second book shows a twist hinge dissection of a triangle to a hexagon. A simple conversion gives this locked dissection. Replacing twist hinges by lugs and locks does not add extra pieces so this is an efficient dissection.
The same method will work for dissecting any {n} to a {2n} in 2n+1 pieces.
This dissection is based around the first one here.
This dissection is derived from the dissection here.
This dissection is from the normal 7 piece dissection.
I am rather pleased with this dissection: it is simple and elegant. Also, it was an unexpected find. It is an unusual locked dissection as it does not require any pieces to be turned over.
This dissection is based around this dissection. Note that there is a very short edge on the left edge of the triangle just above the spike.
This dissection is from the normal six piece dissection. The large overhead of five pieces strongly suggests that this dissection can be improved.