Some polygons do not tessellate easily. An example is the octagon:
Combining this with a tessellation of squares gives eight pieces. But instead the octagon tessellates very easily if we combine it with a small square:
We can form another tessellation by combining with the same small square with a square:
Since these two tessellations repeat in the same way we can overlay them to form the following dissection:
What we have done is to create a dissection of a square with a small square to an octagon with the same small square. But we can throw away the small squares giving a simple dissection of a square to an octagon.
This requires only five pieces, which is a good improvement.
This method is not often used, but when it is, it gives very nice dissections. Some examples are the dissections of hexagon to dodecagon, octadecagon or hexagram.