Department of Mathematics, Statistics
and Computer Science
Wim Ruitenburg's Fall 2012 MATH 1300-101
Last updated: November 2012
Comments and suggestions: Email wimr@mscs.mu.edu
Book, chapter 11 on symmetry
This book chapter mostly considers rigid motions of the plane.
Examples are
- Reflections (improper).
- Rotations (proper).
- Translations (proper).
- Glide reflections (improper).
Usually we only need a little information about points and lines to uniquely
determine the whole rigid motion.
Symmetries of an object are those rigid motions that move the object back onto
itself.
- When we combine or reverse rigid motions, we again get rigid motions.
- In the book on pages 410 through 415 we see finite objects whose groups
of rigid motions are called Z_1 or Z_2 or Z_3 or Z_4 and so on, or are called
D_1 or D_2 or D_3 or D_4 and so on.
The book calls them symmetry types.
Here Z_n has n elements, for example Z_4 has 4 elements.
Here D_n has 2n elements, for example D_4 has 8 elements.
- All rigid motions on finite objects in the plane are reflections or
rotations.
Rotations can be built from two reflections one after the other.
- Translations may be considered as rotations about a center lying in a
direction at an infinite distance.
- We consider the collection of symmetries of objects in space.
For example, the cube has 48 symmetries, of which 24 are proper and 24 are
improper.
There are 5 famous very symmetric objects like the cube, collectively known as
the platonic solids.
Example Problem(s)
- Recommended problems from Chapter 11 of the book:
2, 6, 15
- How many symmetries does a tetrahedron have?
How many of these are proper, and how many are improper?
- Recommended problems from Chapter 11 of the book:
44
- Consider the square.
We label the corners counterclockwise by A, B, C, and D.
When we rotate the square half a circle, A moves to C, B moves to D, C moves to
A, and D moves to B.
Describe a first and a second reflection which, when we perform the first and
then the second, result in this same rotation.
- When we look at ourselves in the mirror, left and right are always
reversed, but top and bottom are not.
Why?