L2 · Rotation Symmetry and Order of Symmetry
“I can test a shape for rotation symmetry by turning it around its centre point, I can state its order of symmetry (how many times it overlaps itself in one full turn), and I know central symmetry means it overlaps after a half turn (180°).”
Push a pin through the centre of a square and turn it slowly through one full turn, 360°. Step 1: After a quarter turn (90°) the square overlaps itself exactly - same outline, corners on corners. Step 2: It overlaps again at 180°, again at 270°, and again at 360° back at the start. Step 3: Count the overlaps within the full turn: 4. The order of symmetry of a square is 4. Step 4: Now a rectangle (longer than wide). At 90° the long side would stand upright - no overlap. At 180° it overlaps. At 270° no; at 360° yes. Count: 2 - order 2. The smallest working turn is 360° / order: for the square 360 / 4 = 90°; for the rectangle 360 / 2 = 180°. Say it plainly: the turn test is the fold test's partner - fold for mirrors, turn for spins. And if the ONLY overlap is the full 360° turn itself, the shape has no rotation symmetry, because every shape matches itself after a complete turn.
The order of symmetry counts…
An equilateral triangle turns about its centre. At which turns does it overlap itself, and what is its order? -> It overlaps at 120°, 240° and 360° - order 3. (Smallest turn 360 / 3 = 120°.)
"A shape overlaps itself at 72°, 144°, 216°, 288° and 360°. Its order of symmetry is ___" -> 5
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.