SYMMETRY AND DESIGN

Objective:

Students will identify all lines of reflection and the fundamental region for a given design.
Students will identify the properties of shapes that tile the plane.

Mira
pattern blocks for the overhead
colored pencils or markers
protractors
Symmetry and Design Activity Sheet

• One of the underlying goals of this lesson is to help students appreciate the beauty of mathematics, in this case, the beauty of reflective symmetry.
  
• Display one of Escher's prints that contains some symmetry and talk about what makes his work so interesting to so many people.
  

DO NOT apply your marker directly to the screen. We suggest printing a hardcopy of the activity sheet directly from your browser to work on.
• Have the students use the Mira to analyze the enclosed designs and identify all lines of reflection.
  
• Of the seventeen wallpaper groups, four uses reflective symmetry exclusively. The enclosed designs show these four symmetry groups.
  

	
Fig. 1 is created by reflecting a	Fig. 2 is created by reflecting a
    60š -60š -60š triangle over its edges.	    45š -45š -90š triangle over its edges.

	
Fig. 3 is created by reflecting a	Fig. 4 is created by reflecting a
30š -60š -90š triangle over its edges.	square over each of its edges.

• A polygon will tile the entire plane if after tiling the sum of the angles at each vertex is 360š. (In Example D, regardless of the design, the trapezoid can not tile the entire plane.)
  
• A polygon with a design on it will tile the entire plane and produce a symmetrical pattern, if all the interior angles of the polygon are divisors of 180š. (In Example C, the shape of rhombus tiles the entire plane; however with the design, there are conflicts in the tiling.)
  

A B C D 

• Observe the difference in how the details move when reflected as opposed to rotated.

• Also discuss what happens if you continue reflecting (tile the plane) and whether these are special shapes that completely cover the flat surface or would any shape do the trick.

• Use Kali to create a unique design with reflective symmetry.

• Use tess (NeXT application) to create some unbelievably unique tessellation. (It was used to create the enclosed designs)

• Use Geometer's Sketchpad to make a shape, conjecture whether it will tessellate, test your conjecture using the program.

• Create a unique design by hand using dot paper.