Symmetry

MATH 124September 25, 2013

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

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

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

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, but are not really commonly discussed

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

Rotation symmetries in quadrilaterals

• No rotation symmetry: trapezoids, kites• 180 degree rotation symmetry: parallelograms,

rhombi, rectangles• 90 degree rotation symmetry: squares

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.