Transformations and Symmetry

Transformations

• Reflection• Translation• Glide Reflection• Rotation• Scaling (not topological)

A transformation turns a geometric figure into another by one of the actions above.

Reflection

The blue triangle is reflected across line m (called the line of reflection).

Translation

Sliding a figure. A translation could be accomplished by several reflections.

Glide Reflection

The figure reflects across the line, and slides forward.

Rotation

The blue image is rotated about the point p.

Scaling

The blue star has been scaled upward by a multiplying factor. Scaling (changing size) is

not a topological transformation.

Symmetry

•Our basic idea of symmetry is 2-fold symmetry

•A figure whose halves are mirror images of each other over a fold line

•We could think of this as being able to transform half of the image into the other half by reflection

Which are the lines of symmetry?

Only m

Which are the lines of symmetry?

L, M, N, O

Which are the lines of symmetry?

M, O

Which are the lines of symmetry?

T,R

Symmetry (II)

• More complicated types of symmetry arise if we consider objects obtained from reflection, PLUS other transformations such as rotation

• Various patterns have been characterized as symmetry groups

Conway Notation

• Is one way mathematicians use to describe various symmetry groups

• Can get very complicated!

Kali• We’re going to look at some

symmetry groups using the program Kali

• Keep an eye out for two basic notations:

– An integer (1,2,3) denotes a ROTATIONAL symmetry

– An * denotes a reflection as well

KALI

The title picture …

…was created in kali, then colored in with paint

Symmetry in Nature

Chinese Rose

Symmetry in Nature

Crab

Symmetry in Nature

Starfish

Symmetry in Art

Quilt

Symmetry in Art

Pennsylvania Dutch Hex Sign

Symmetry in Art

Persian Carpet

Symmetry in Art

Ukranian Painted Easter Eggs

Symmetry in Art

Mosaic Tile (Iran, 14th C.)

Symmetry in Art

A Kaliedotile