The following decagon strip is the basis for our first example:
What is clever about this strip is that we have lots of freedom to modify the shapes of some of the pieces, for example:
This modification of pieces works for the decagon because the two modified pieces swap position. Whatever changes are made to one piece are also made to the other. This frequently allows one or more pieces to be combined.
Overlaying the first decagon strip with a strip of squares gives the following 8 piece dissection:
But by modifying the decagon strip we can save a piece as shown below:
As can be seen this technique can lead to some odd shape pieces.
Our next example is a dissection of the decagon and dodecagon. We show first the unaltered overlay and then the optimised overlay.
The final example is a dissection of octagon and decagon that also makes use of the variable octagon strip demonstrated here.
Note also that a T strip version of the above decagon strip is possible which can also be modified. This is used to dissect the triangle to a decagon.
Another example shows a T strip of the {8/3} octagram and a modified version:
Overlaying with the T strip for a triangle gives the following TT2 dissection:
We now modify the octagram strip to get rid of the small triangle. Here is the resulting overlay and dissection:
Similar dissections are used to dissect the pentagram, hexagram, golden rectangle and the domino to an octagram. The last two of these require a much more extreme modification. Finally, the latin cross uses a different octagram strip for modification.
This time we will look at the enneagon. It is not easy to find useful enneagon strips, but the following T strip is one I found:
But this is just one possible strip. We can vary the sizes of the two largest pieces to give a range of strips:
Look at the following overlay for a dissection of a square to an enneagon. See how the enneagon strip has been chosen to ensure that the overlay of the square strip adds the minimum number of pieces.
This trick works very well for a number of enneagon dissections. It helps that the basic strip can also be varied in a number of small ways.
For more examples of variable strips, see the hexagon strip of the dissection of square to hexagon and the heptagon strip of the dissection of square to heptagon.
It seems that there are few strips where any of the above tricks are possible, but when they are, then they can result in particularly efficient dissections.