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 symmetry.
The first tessellations that we will look at all repeat with square 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. The second of those below turns over two pieces in order to save a piece.
The tessellation of the domino is perhaps not so obvious:
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, dodecagram and domino 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:
So far all our tessellations have had square symmetry. The following two have hexagonal symmetry.
Here is the hexagon tessellation:
And here are two tessellations of the enneagram {9/3}. The second one turns over two pieces in order to save a piece.
Overlaying these gives the following dissections: