1. Symmetry
MATH 124

2. Reflection symmetry Also called line symmetry
Appears in early elementary school
The reflection line, or line of symmetry, divides the object in two congruent parts. If the object were folded over the reflection line, the two congruent parts would fit each other exactly
A shape can have multiple lines of symmetry

3. Rotation symmetry
A shape has rotation symmetry if it can be rotated (turned) around a fixed point until it fits exactly on itself (or the space it previously occupied). The fixed point is called the center of symmetry
Every object has 360 degree rotation symmetry, so we don’t consider that one

4. Translation symmetry
A shape has translation symmetry if it can be translated (i.e. slid or shifted) to land on itself (or on the space it previously occupied)
Equivalently, an image has translation symmetry if it can be divided by lines into a sequence of identical figures
Technically, only infinite objects can have translation symmetry

5. Symmetry
A symmetry of a shape is any movement that fits the shape onto the same set of points it started with
There are other types of symmetry – we will discuss one more next week

6. Reflection symmetries in quadrilaterals
Zero lines of symmetry:
parallelogram, general trapezoid
One line of symmetry:
kite (diagonal), isosceles trapezoid (line connecting midpoints of bases)
Two lines of symmetry:
rhombus, rectangle
Four lines of symmetry:
square

7. Rotation symmetries in quadrilaterals
No rotation symmetry:
trapezoids, kites
180 degree rotation symmetry:
parallelograms, rhombi, rectangles
90 degree rotation symmetry:
squares

8. Symmetries of regular polygons
A regular n-gon has n lines of symmetry. If n is even, then the lines of symmetry connect opposite sides and opposite vertices. If n is odd, then the lines of symmetry connect vertices with opposite sides.
A regular n-gon has rotation symmetry of the order n. The smallest angle of rotation possible is 360/n, and any multiple of that angle will also work. For example, a 10-gon can be rotated 36, 72, 108, 144, 180, 216, 252, 288, 324, 360 degrees.