Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

**Nehushtan****Member**- Registered: 2013-03-09
- Posts: 957

So what possible transformations are isometries of the plane? You might think that there were a whole bunch of them: rotations, reflections, and translations. (A reflection followed by a translation is sometimes called a glide reflection.) In fact the picture is simpler than that: it turns out that rotations and translations can be built up from reflections alone! A translation in a certain direction is simply a reflection in two axes perpendicular that direction, while a rotation about a point O is a reflection in two axes through O.

Hence any translation is a reflection in two axes perpendicular to the direction of translation whose distance apart is half the distance to be translated.[list=*]

[*]

[/list]

Hence any rotation is a reflection in two axes through the centre of rotation whose angular separation is half the angle to be rotated through.

*Last edited by Nehushtan (2017-11-20 05:47:30)*

**240** books currently added on Goodreads

Offline

Pages: **1**