Translation, Reflection, Rotation

Date: 05/16/2001 at 10:14:02
From: MIKE
Subject: Geometry

What is the difference between slide, flip, and turn? I find it really 
hard. What are shapes - can you send me a list?

Date: 05/16/2001 at 13:01:44
From: Doctor Peterson
Subject: Re: Geometry

Hi, Mike.

Slide, flip, and turn are meant to be kid-friendly replacements for 
the technical terms translation, reflection, and rotation. You're 
supposed to be able to picture them more easily using the simpler 
words. So let's relate their math meaning to their everyday use.

If I SLIDE into home plate, I am basically moving sideways without 
turning. Here I'll slide a square from one side of the page to the 
other:

     +-----+                                           +-----+
     |     |                                           |     |
     |     |                                           |     |
     +-----+                                           +-----+

This only changes the shape's position, not its orientation (the 
direction it faces). Maybe there's a gyroscope inside the shape!

If I TURN around, I pivot so that I face a different direction. Here 
I'll turn, or rotate, a square by about 45 degrees:

                              +
     +---------+            /   \
     |         |          /       \
     |         |        +           +
     |         |          \       /
     |         |            \   /
     +---------+              +

A rotation doesn't change the shape, but it will change the position 
of parts of it, and the direction it faces. It's best to picture 
rotation as rotation in place, that is, imagine my first square as a 
piece of paper, put a pin through the middle, and rotate it round the 
pin, so the center of my second picture is actually in the same place 
as the center of the first.

If I FLIP a pancake, I'm turning it over. The same sort of thing 
happens if I look at myself in a mirror; the left and right sides 
switch places. Here I'll flip a right triangle over:

     +       :       +
     |\      :      /|
     | \     :     / |
     |  \    :    /  |
     +---+   :   +---+

Notice that I can't make this same change by sliding or turning; if 
the triangle were a separate piece of paper, I would have had to turn 
it over. If I had to keep the same side on top, I would have to 
replace it with a new copy made backward; or I could just put a mirror 
along the dotted line and only see the reflection of the original 
triangle in the new position. You can also imagine turning (flipping) 
a page in a book; the dotted line is then the middle of the book.

If you need more help, you might want to send in a problem involving 
these motions, and tell me where you have trouble with it.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/