Department of Mathematics, Statistics
and Computer Science
Wim Ruitenburg's Fall 2017 MATH 1300-101
Last updated: November 2017
Comments and suggestions: Email wim.ruitenburg@marquette.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 class we combined rigid motions of the square.
We describe rigid motions by how they move the corners (labeled A, B, C, D
counterclockwise).
- In the book on pages 334 through 339 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 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.
You may use search engines to find their names, and more.
Example Problems
- Recommended problems from Chapter 11 of the book:
2, 6, 17
- Describe the symmetries of the square (label its vertices as A, B, C, D
counterclockwise similar to the equilateral triangle in class).
- We label the corners of an equilateral triangle counterclockwise by A,
B, and C.
Let m stand for the reflection which moves ABC to BAC.
Let r stand for the rotation which moves ABC to BCA.
Write the reflection which moves ABC to ACB as a product of m's and r's
(remember that you perform such products from right to left).
- Recommended problems from Chapter 11 of the book:
47
- 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.
- How many symmetries does a tetrahedron have?
How many of these are proper, and how many are improper?
- When we look at ourselves in the mirror, left and right are always
reversed, but top and bottom are not.
Why?