There are 17 different symmetries
which can be used to make tessellations and wallpaper. There
are many pages with technical information about the 17
wallpaper groups, but most are difficult to understand. Here
is one that explains them in a much simpler way than most -
World of Mathematics.

I set myself the problem of finding
how many polyforms would be needed to create all of the 17
patterns, and would you believe the answer is two - a square
and a drafter (30, 60, 90 triangle). Actually, it is possible
to split each square diagonally into four tans (45, 45, 90
triangles) and reduce the vertex count of the shapes by one.