Dissection models are normally made up of loose pieces. When knocked, the pieces become disarrayed. One solution to this problem is to have frames for the pieces to fit into. An alternative solution is for the pieces to be interlocked as for a jigsaw puzzle. This is the purpose of locked dissections.
No simple method seems to give best results for these dissections. Twist hinge dissections can be a good starting point, but can frequently be improved upon. This makes these dissections an interesting challenge to find.
Here is the ordinary dissection of a triangle to square:
The pieces of the above dissection can be joined by hinges. Now each hinge can be replaced by an extra piece that can be turned over to produce a locked dissection. There are four hinges but it is only necessary to add three pieces. Here is the result:
The disadvantage of conversion of hinged dissections is that each hinge adds an extra piece to the dissection.
Certain dissections derived from overlaid tessellations can give very efficient locked dissections. Here is an example:
Note that a piece does not have to lock onto others if it is held in place by the surrounding pieces. This is the case for the central square of the following dissection:
With normal dissections I can often be confident that no further improvements can be found, but that is not the case here. So there are lots of opportunities to find improvements or to find new dissections. Generally I am looking for dissections of the polygons and polygrams that require twelve pieces or less.
Note that for locked dissections I do not follow my normal convention of showing when pieces are turned over as these pieces are very common in these dissections and can be rather small.