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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 dissection is based around the first one here. Compare to see how this one was found.
Locked dissections would be fairly boring if they could be found simply by converting twist hinged dissections into locked dissections. This dissection is an example of one that was not discovered in this way. The best known twist hinged version of this dissection requires 11 pieces so this is a four piece improvement. It shows the possibility that many locked dissections can be improved.
Greg Frederickson’s third book mentions a twist hinge dissection of a pentagon to a hexagon in 13 pieces that can be directly converted to a 13 piece locked dissection. But using a totally different method I arrive at the above 10 piece solution.
Greg Frederickson’s third book shows a twist hinge dissection of a pentagon to a decagon. A simple conversion gives this locked dissection. Unfortunately half the lugs are rather too small.
The same method will work for dissecting any {n} to a {2n} in 2n+1 pieces.
This version requires the same number of pieces but removes the small lugs. Unlike the previous version, this method does not generalise.
Greg Frederickson’s third book shows a twist hinge dissection of a pentagon to a pentagram. A simple conversion gives this locked dissection.
The same method will work for dissecting a {p} to a {p/q} in 2p+1 pieces, but only for lower values of q.
This derives from the normal dissection of a pentagon to decagram.