A new 9-piece dissection of a 9-gon to a triangle!

The dissection below needs only 9 pieces to transform a regular 9-gon to a regular triangle! 3 pieces need to be turned over, but the scheme can be generalized to any {3n}-{n} dissection, needing only floor(5n/2)+2 pieces!

See below for how this dissection was found!

A general {3n}-element

By cutting out certain triangles out of a 9-gon or 12-gon (dodecagon) and reattaching them elsewhere, one gets figures that can be tesselated!
The process of building such an {3n}-element can be generalized (but only for the {18} we get such a nice tesselation as above)!

Superposing with {n}

So far I could not exploit the above tesselations, but the general {3n}-element can be superposed with {n} giving an immediate dissection (the {n} cuts the small inwards pointing sides in half):
Carefully designing a dissection based on the {3n}-element dissection superposed with the {n} gives a (3n+1)-piece dissection, like shown for n=3:
Some optimization steps give the dissection shown on top of this page! This optimization reduces the number of pieces for each neighbouring pairs of rotationally symmetric pieces by one.
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Last modified: June 1, 2001