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Rotational symmetry, step by step

A shape has rotational symmetry when turning it about its centre lands it back on itself before the turn is complete.

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The question

How many times does it land on itself?

kite
lands on itselfonly at the full turn
order=1
  1. The shape is a kite. Rotational symmetry asks how it looks while you turn it about its centre — not how it folds.
  2. Its two pairs of equal sides are next to each other rather than opposite, so a turn would have to carry a long side onto a short one. Nothing short of a whole turn works. It folds in half down the middle, but folding is not turning.
  3. So only the full turn brings it back, which is what order 1 means: the shape lands on itself once in a complete turn.

How it works.

  1. 01

    The shape is a kite. Rotational symmetry asks how it looks while you turn it about its centre — not how it folds.

  2. 02

    Its two pairs of equal sides are next to each other rather than opposite, so a turn would have to carry a long side onto a short one. Nothing short of a whole turn works. It folds in half down the middle, but folding is not turning.

  3. 03

    So only the full turn brings it back, which is what order 1 means: the shape lands on itself once in a complete turn.

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