Symmetry Detective

Intro to Proofs · Axiom Academy

A square can be moved eight ways and still look untouched. Investigate those moves, and a hidden structure falls out: a group. A symmetry is a move that changes nothing Here is a square with its corners labelled 1, 2, 3, 4 . A symmetry is any rigid move — a turn or a flip — that leaves the square sitting exactly where it started, indistinguishable from before. The surprise is that there are precisely eight of them, no more, and that those eight obey laws as clean as arithmetic. Watch the square work through all eight: four turns of 90° , then four flips across the dashed mirror lines. Track the corner labels — they shuffle, but the outline never moves off itself. That moves yet stays put is what symmetry means. Eight moves, one shape. That complete list of self-maps is the whole cast of characters. Do one, then another — you always land on a third Pick a first move and a then move. The square performs them in order, and the result is announced as . No matter what you choose, the outcome is one of the same eight — never anything new. Try a turn then a flip, then swap the order: the answers differ, so order matters. This combine two, get one rule is called closure — and because order can change the answer, the structure is richer than ordinary addition. Fill the table — every cell is a symmetry

This is the written version of the interactive lesson above. See the full Intro to Proofs course.