So far we have looked at dissections from strips and tessellations as two distinct methods, but in fact this is not the case. The following is a typical strip dissection:
But the overlay diagram is just a simplification of the following:
The strip dissection is in fact a simplification of overlaid tessellations. The advantage of understanding this is that we can often now modify the tessellation. So in this example we can merge the two hexagon pieces to make a larger piece:
The new hexagon tessellation requires the small piece to be turned over, and this is a common feature of these tessellations. The new dissection also requires nine pieces so, so far, no advantage has been gained. But instead of this octagram strip we can now try a different one:
This strip is not usually very useful as it is rather cramped for space, but we will now try combining it with our new hexgon overlay:
Because we enlarged the largest piece of the hexagon tessellation we have managed to fit all the small pieces of the octagram inside, although only just. The resulting dissection has now saved a piece.
Finally we show the new dissection. Because we didn't actually need to modify the octagram tessellation, just the strip we will just show a simplified version of the overlay diagram.
Next we look at a PP2 dissection of a hexagon and octagon using the following two strips.
Here is the overlay.
Looking at the highlighted areas we can see that the dissection has 10 pieces, which is too many. Next we produce the full tessellation from the strips.
Now we isolate each tessellation and try to improve it.
Each modification now requires a piece to be turned over. Now we overlay the two new tessellations in the best way possible. Here is the final overlay and dissection.
Our final dissection has only 8 pieces, so these modifications have been very worthwhile.
Our next example is based around a dissection of a triangle to an octagon based on the following two strips:
We have not used this octagon strip before as it usually generates dissections with too many pieces.
Here is the overlay resulting from overlaying these strips:
This eight piece dissection was discovered by Harry Lindgren (1961). Now we produce the full tessellation from the strips.
By recombining some triangles and by turning over a piece we get the following tessellation. We highlight the same area to make it clear how it has changed.
We have successfully reduced the number of pieces to seven. Here is the final dissection:
Note that there are better seven piece solutions that do not require turning over a piece here.
There is a dissection of an octagon to an enneagon that is based on the same octagon strip as we use here, but this time it is not necessary to turn over a piece.
There are many other ways that we can save pieces when we look beyond simple strip dissections and instead look at and modify the tessellations that they represent.