Student's Page: Symmetry

Symmetry is all around us. It is in the sturctures we build, the art we create, and most importantly, symmetry is abundant in nature. Let's investigate!

Symmetry has been catagorized to help us study it. As we look for patterns, we will also learn the language of symmetry. Here are a few definitions to start with. Don't worry too much about them, but they are here for reference, and will make more sense later.

A transformation is an operation which maps, or moves, a figure to a new position... We will be looking at several transformations.

An isometry is a transformation which preserves lengths. That is a figure is moved, turned, and/or filped, but it still is the same size and shape.

Reflectional Symmetry is also called line symmetry or mirror symmetry because there is a line in the figure where a mirror could be placed, and the figure would look the same.

• Think of a figure on a piece of paper, then folding the paper in two so that the two halfs match up.

  [include figure with demonstrates reflectional symmetry]

• It is possible to have more than one mirror line. [include figure with demonstrates double reflectional symmetry]

• What letters have reflectional symmetry? Does is matter where the mirror is placed? Click to Letterforms and Illusions to investigate letters further.

  See Java Demonstrations of letter reflections.

Rotational Symmetry has a center point where an object is turned about.

• Think of the center of a pizza [or orange], if it is made so that all the pieces are the same size and have the same ingredients in the same places, then the pizza could be turned and you couldn't tell the difference. This means the pizza has rotational symmetry.

• What other foods have rotational symmetry?

• It is possible for a drawing to have several centers of symmetry.

  [include figure with demonstrates multiple centers of symmetry]

  How many centers can you find?

• When will reflectional symmetry and rotational symmetry be confused? Can you find a drawing that could be either reflectional or rotational symmetry?

• Use Geometry Sketchpad to...

Translation is another type of symmetry, but it is not as apparent. Translational Symmetry results from sliding a figure a certain distance in a certain direction (vector).

• Think of using a stamp and moving across the top of a wall along the ceiling. Now stamp every 6 inches. This is translational symmetry.

• Can translational symmetry be mistaken for other symmetries? Think of transforming more than once- lie two reflections or a reflection and a rotation etc.

• Check out Tesselations and Symmetry, to see how...

Extensions:

Compositions of transformations

A Glide reflection is a composition transformation. First a figure is translated then the figure is reflected.

• Think about foot prints in the sand, and using only one foot to make the pattern.

  [include figure with demonstrates a