Slides, Flips and Turns

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K-12: Materials at high school level.


Introduction

How does one create congruent shapes? In other words what can we do to a shape so that it keeps its shape and exact size? The answer is that we can do moves called slides, flips and turns. If we want to be fancy we can even combine a slide and a flip into a slide-flip.


Slide
Flip
Turn
Slide-Flip

All of the moves we will discuss fall into one of these four categories:

  • Slide : An object is moved in one direction (the translation vector).
  • Flip : An object is reflected across a line (the reflection axis).
  • Turn : An object is rotated around a point (the center of rotation).
  • Slide-Flip : A combination of a slide and a flip, flip across a line while sliding along that same line.

These moves are sometimes called a rigid motion. They are motions that do not distort shape. Picking something up and moving it around for instance is a rigid motion, but stretching or warping it is not. Because a rigid motion does not change size or shape, it is also called an isometry, from the Greek iso (meaning equal) and metry (meaning measure or distance).

Tetris

If you have ever played Tetris, you will have used these rigid motions. Recall that the shapes used in Tetris consist of blocks made up of four squares.

Tetrominoes.svg

To do well in the game we try to line up the game pieces so that there are no gaps. We are allowed to slide the game pieces left and right, and we are allowed to flip and turn the game pieces as well. In other words we are doing geometry when playing the game.

We could play a similar game where the game pieces are all made up of 5 squares. These game pieces are called pentominoes. If we had a game like this - maybe it could be called Pentris - it would be quite a bit more difficult than Tetris. In Tetris we only have 5 different game pieces to manipulate. In Pentris we would have 12.

Pentominos.svg

You will be asked some questions about these pentominoes in the worksheets.