Good dissections can be made when two shapes can both be easily dissected to form tiles that repeat to form tessellations where both tessellations repeat with the same regularity and with the same symmetry.
The first tessellations that we will look at all repeat with a square tiling and have the same symmetry and so can be overlaid.
Here is the square itself:
This is a tessellation of the dodecagon:
This is a tessellation of the octagram. This can be varied in a number of ways.
The tessellation of the dodecagram can also be varied in a number of ways. If pieces are turned over, it is possible to form a tessellation that saves a piece.
The tessellation of the domino is obvious enough and be varied greatly, but here we tile it to ensure that it has the same symmetry as the above tessellations.
The tessellation of the greeek cross is very simple:
Here are the results of overlaying the square, dodecagon and greek cross tessellations.
The octagon does not naturally form a tessellation, but dissecting it to do so allows it to easily overlay the above tessellations.
Here is an example.
The following tessellation of the dodecagram has a different symmetry pattern. Only the square and greek cross tessellations have the same symmetry, although there are different tessellations for the dodecagon and octagram that can be used.
In the previous overlays the dots of the two tessellation must have the same orientation but need not be aligned. (Normally they are as it results in more symmetric dissections). But now the dots must be aligned. Overlaying with squares gives the following:
Here is a different tessellation of the dodecagram:
And here is a tessellation of the domino with the same square repetition:
Here is the resulting dissection:
So far all our tessellations have had square based tiling. The following two have hexagonal based tiling.
Here is the hexagon tessellation:
And here are two tessellations of the {9/3} enneagram. The second one turns over two pieces in order to save a piece.
Overlaying the hexagonal tessellation on these gives the following dissections:
The following is another dissection with a hexagon based tiling. Here is a tessellation of the {12/2} dodecagram.
Overlaying with the hexagon tessellation gives the following dissection:
We could also overlay this dissection on top of a tessellation of the {9/3} enneagram above, but the resulting dissection has too many pieces for this to be wothwhile.