Fold, punch, unfold

Here is a way of making symmetric shapes: fold a sheet of paper, cut or punch it, and open it out. In the Punching Game, you fold a square sheet of paper and punch a hole through it. When you unfold it, there are two holes, one on each side of the fold, at the same distance from it. The fold is a line of symmetry of the pattern.

Why two holes? The punch goes through both layers of the folded paper at once. When you open it, each layer has its own hole, and the two holes are mirror images of each other in the fold line.

If the paper is folded twice, the punch goes through four layers. Fold a square in half vertically, then in half again horizontally, and punch once near the corner of the folded bundle. When you open it out, there are 4 holes, one near each corner of the square: every hole is reflected in both fold lines.

You can also work backwards. Two holes near the middle of the left side and the middle of the right side, at the same height, came from a vertical fold down the middle: a fold always lies exactly halfway between two mirror holes. And if you are told the line of symmetry, each missing hole is the reflection of a given hole: the same distance from the line, straight across on the other side.