Symmetry: slides, turns and flips

前置知识

1 项必需的前置知识（一直追溯到底）

你将学会

• Identify lines of symmetry and the order of rotational symmetry of shapes.
• Perform slides, turns and flips on a grid and describe them precisely.
• Find a move that undoes a given move, and explore what happens when two flips are done one after the other.

定义

• A line of symmetry divides a shape into two mirror-image halves.
• A shape has rotational symmetry of order n if exactly n turns within one full revolution (including the full turn) leave it looking the same.
• A slide moves every point the same distance in the same direction; a turn rotates about a fixed point; a flip reflects in a line.

定理

• A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.
• Slides, turns and flips do not change lengths or angles.
• Flipping twice in the same line returns every point to where it started.
• Two flips in parallel lines give a slide, and two flips in crossing lines give a turn about the crossing point.

常见误解

• Thinking the diagonals of every rectangle are lines of symmetry; this is true only for squares.
• Believing a flip and a half turn always have the same effect.

通向

可选发现

• Groups The symmetries of a shape can be combined and undone; together they form a group, such as the 8-element dihedral group of a square.
• One thing after another Doing one move after another, and undoing a move, are early examples of composing and reversing actions.