One cut, twelve copies
Tonight's idea
Fold a square in half, then in half again. Four parts now occupy the same quarter-square. A hole through the folded stack reaches all four parts. When you open it, each fold acts like a mirror and puts a copy on the other side.
The flips and turns that match the finished pattern are its symmetry moves. Together they form a group: combining two symmetry moves gives another one.
Away from creases, a mark through \(n\) stacked layers makes \(n\) separate copies.
A mark on a crease is a boundary case: reflected copies can meet, so there may be fewer separate holes. Copying one doodle with every move in a symmetry group and overlaying the results is a kind of averaging. Every direction is treated equally.
Watch one: four parts become one
Each fold doubles the stack: 1 part becomes 2, then 4. The opened pattern has four symmetry moves: do nothing, flip left to right, flip top to bottom, or make both flips. Both flips together make a half turn.
Your turn
- A small hole is away from every crease. How many copies appear after (a) one fold in half, (b) two folds in half, (c) those two folds and then one more diagonal fold?
- The left square has eight wedge-shaped parts. Reflect the orange dot across the lines until every wedge has one. Which mirror lines and turns match the finished pattern? Counting “do nothing” too, how many symmetry moves are there?
- In the right picture, rotate the bent arrow by a quarter turn three times and draw each copy. Do the four arrows have quarter-turn symmetry? Do they have a mirror line? Explain.
- For a circular snowflake, fold in half, fold the semicircle into three equal 60° wedges stacked together, then fold that wedge in half to make 30°. How many layers are stacked? A small triangular snip is made in the curved outside edge, away from the wedge's corners. How many separate notches appear when opened?
Stretch: can folds make the pinwheel?
Suppose a pattern is built only by reflecting one mark across the straight edges of a folded wedge until the paper is filled. Explain why the finished pattern must have mirror lines. Why can this exact method not make the turn-only pinwheel from question 3?
Answers
- (a) 2. (b) 4. (c) 8. Each extra fold puts every existing layer on another layer, so the count doubles.
- There are 8 dots, one in each wedge. Four mirror lines match the pattern: horizontal, vertical and both diagonals. Quarter, half and three-quarter turns also match it. With “do nothing”, that makes 8 symmetry moves.
- Yes, a quarter turn moves every arrow onto the next one. No mirror line works. The arrows circulate clockwise; a reflection would reverse them to anticlockwise. Copying over the four turns treats all four directions equally without adding a reflection.
- The first fold gives 2 layers. Stacking three equal wedges gives \(2\times3=6\). The last fold doubles that to 12 layers, so the snip makes 12 separate notches. The conditions matter: at a wedge corner or crease, copies could meet.
Stretch. Each wedge edge is used as a mirror: every mark on one side gets a reflected partner on the other. After unfolding, those edge lines are mirror lines of the full pattern. The pinwheel has only turns; a mirror reverses its clockwise direction. It can be made by rotating and overlaying copies, but not by this reflect-and-unfold wedge method.