Part B. An example of wallpaper groups and symmetric patterns over the whole plane.

The first part of this exercise will be to create some examples of repeating symmetric patterns from an original square "tile". The second part will be an analysis of the symmetries of each pattern.

Constructing the Tile

In a new sketch, construct this pattern by constructing a square and shading in the triangle with vertices at C, D and the midpoint of BC. Save this sketch. You will use it as the basis of several experiments.

Pattern 0. Translations

In a new sketch, copy the figure from above and paste into the new sketch. Now mark segment AB as vector and translate the whole figure. Select the two shaded interiors (no points or segments!) and Define Transform with name T_AB.

Now mark segment BC as vector and translate point A to get a new A'. Select A and A' and Define Transform as T_BC. The two custom transforms are in the Transform menu and can be selected using Control-1 and Control-2.

• Translate the whole figure by BC.
• Then translate the whole figure by AB.
• Continue until the window is fairly full and a pattern emerges. You may want to resize ABCD.

Analysis of Pattern 0

You can do this on the screen or by printing it out, but BE SURE TO CHOOSE "Scale to Fit Page" if you do this. Assume the pattern is continued "to infinity".

• What are the centers of rotational or point symmetry?
• What are the vectors that define translations that are symmetries of this pattern?
• What are the lines of mirror symmetry or invariant lines of glide symmetry?

Pattern 1. All sides as mirrors

In a new sketch, copy the figure from above and paste into the new sketch. Now mark segment AB as mirror and reflect the whole figure. Select the two shaded interiors (no points or segments!) and Define Transform with name R_AB.

Now mark segment BC as mirror and reflect point A to get a new A'. Select A and A' and Define Transform as R_BC. Continue in the same way naming reflection across side CD and side DA so that the four custom transforms are in the Transform menu and can be selected using Control-1 through Control-4.

• Reflect the whole figure across BC.
• Then reflect the whole figure across CD. Then reflect across DA.
• Continue until the window is fairly full and a pattern emerges. You may want to resize ABCD.

Analysis of Pattern 1

You can do this on the screen or by printing it out, but BE SURE TO CHOOSE "Scale to Fit Page" if you do this. Assume the pattern is continued "to infinity".

• What are the centers of point symmetry?
• What are the vectors that define translations that are symmetries of this pattern?
• What are the lines of mirror symmetry or invariant lines of glide symmetry?

Pattern 2. Two corners at centers of rotation.

In a new sketch, copy the figure from above and paste into the new sketch. Now mark A as center and rotate the whole figure by 90 degrees. Select the two shaded interiors (no points or segments!) and Define Transform with name A90.

Now mark B as center and rotate point A by 90 degrees to get a new A'. Select A and A' and Define Transform as B90.

Now you can transform using A90 and B90 using Control-1 and Control 2.

• First, rotate the original shape by A90 until nothing new appears.
• Then rotate the whole resulting shape by B90 until nothing new appears.
• Then rotate the whole resulting shape by A90 until nothing new appears.
• Continue until the window is fairly full and a pattern emerges. You may want to resize ABCD.

Analysis of Pattern 2

You can do this on the screen or by printing it out, but BE SURE TO CHOOSE "Scale to Fit Page" if you do this. Assume the pattern is continued "to infinity".

• What are the centers of 90-degree rotational symmetry?
• What are the centers of point symmetry that are not also centers of 90-degree rotational symmetry?
• What are the vectors that define translations that are symmetries of this pattern?
• What are the lines of mirror symmetry or invariant lines of glide symmetry?

Pattern 3. A corner rotation and a mirror

In a new sketch, copy the figure from above and paste into the new sketch. Now mark A as center and rotate the whole figure by 90 degrees. Select the two shaded interiors (no points or segments!) and Define Transform with name A90.

Now mark segment BC as mirror and reflect point A to get a new A'. Select A and A' and Define Transform as R_BC.

Now you can transform using A90 and R_BC using Control-1 and Control 2.

• First, rotate the original shape by A90 until nothing new appears.
• Then reflect the whole resulting shape by R_BC until nothing new appears.
• Then rotate the whole resulting shape by A90 until nothing new appears.
• Continue until the window is fairly full and a pattern emerges. You may want to resize ABCD.

Analysis of Pattern 3

You can do this on the screen or by printing it out, but BE SURE TO CHOOSE "Scale to Fit Page" if you do this. Assume the pattern is continued "to infinity".

• What are the centers of 90-degree rotational symmetry?
• What are the centers of point symmetry that are not also centers of 90-degree rotational symmetry?
• What are the vectors that define translations that are symmetries of this pattern?
• What are the lines of mirror symmetry or invariant lines of glide symmetry?

More Patterns

There are more ways to fill the plane with this square pattern, though some may turn out to be the same old ones in disguise. Here are some suggestions.

• Generate the pattern with rotation by point symmetry, placing centers at each of the midpoints of the sides of ABCD.
• Make one transformation rotation by 90 degrees with center A and the second rotation by 180 degrees with center B.
• In the horizontal direction, define a glide reflection with vector AB and mirror line = perpendicular bisector of AD. Add to this translation by AD or else a glide reflection in the vertical direction.