Math 487 Lab 8, 11/20/2002

Experiment: This is a link to some GSP tools for drawing symbols for rotation centers. It may download correctly! Please try. It seems better to save it to disk and then open in Sketchpad.

Part A of the lab consists of generating a series of patterns with squares with a triangle inside. You can save these, or print them, or you can copy them to graph paper by hand. You should make careful note of the relationship between the isometries that generate the symmetric figure and the pattern of the rotation centers and the mirrors, etc.

Part B of the lab asks you to generate a freer symmetric pattern by starting with two or 3 isometries (such as two rotations) and then generating a pattern with a blob.

Part A. 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. Make more pages with this figure in your document from File > Document Options > Add Page > Duplicate Page 1. Save this sketch. You will use it as the basis of several experiments.

Pattern 0. Translations

Start with a fresh page with the square figure above. Select A and B (not the segment) and Mark AB as vector. Then translate the whole figure by AB (you may want to leave out the segment AD and points A and D from this to avoid duplication).

• Then translate the new figure again and again until you have a row of 8 squares.
• Now mark segment BC as vector and translate the row of 8 squares by BC to get a new row.
• Translate the new row by BC also and continue until you have 8x8 squares with the triangular pattern inside each. You may want to resize ABCD by dragging A or B if the pattern is too large for the screen.

Analysis of Pattern 0

Assume the pattern is continued "to infinity".

• What are the centers of rotational or point symmetry?
• What are the vectors that define translation symmetries of this pattern?
• What are the mirror lines of line symmetry?
• (Optional for this lab -- answer later.) What are the invariant lines of glide reflection symmetry?

Pattern 1. All sides as mirrors

Start with a new page with the same single square with the triangle inside.

• Now mark segment AB as mirror and reflect the whole figure.
• Now mark segment BC as mirror and reflect the whole figure in BC.
• Then reflect the whole figure across CD. Then reflect the whole figure across DA.
• Continue reflecting until you have 8x8 squares. You may want to resize ABCD.

Analysis of Pattern 1

Assume the pattern is continued "to infinity".

• What are the centers of rotational or point symmetry?
• What are the vectors that define translation symmetries of this pattern?
• What are the mirror lines of line symmetry?
• (Optional for this lab -- answer later.) What are the invariant lines of glide reflection symmetry?

Pattern 2. Two corners at centers of rotation.

Start with a new page with the same single square with the triangle inside.

• Now mark A as center and rotate the whole figure by 90 degrees.
• Rotate the original shape by A90 two more times. (Why would more times add nothing to the pattern?)
• Now mark B as center and rotate the whole figure by 90 degrees. (Note: if you look carefully, it is often the case that you can get the same resulting figure by selecting and rotating only a piece of the whole.)
• Then rotate everything by B90 two more times.
• Then rotate everything by A90 3 times.
• Continue until the figure contains an 8x8 squre of square patterns and nothing new emerges in the 8x8 pattern
• You may want to resize ABCD.

Analysis of Pattern 2

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 translation symmetries of this pattern?
• What are the mirror lines of line symmetry?
• (Optional for this lab -- answer later.) What are the invariant lines of glide reflection symmetry?

Pattern 3. A corner rotation and a mirror

Start with a new page with the same single square with the triangle inside.

• Now mark A as center and rotate the whole figure by 90 degrees 3 times.
• Now mark segment BC as mirror and reflect the whole figure in BC.
• Rotate the while figure by A90 again 3 times.
• Then reflect the whole resulting figure in BC.
• Continue rotating and reflecting until the figure contains an 8x8 squre of square patterns and nothing new emerges in the 8x8 pattern by continued rotation or reflection.
• You may want to resize ABCD.

Analysis of Pattern 3

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 translation symmetries of this pattern?
• What are the mirror lines of line symmetry?
• (Optional for this lab -- answer later.) What are the invariant lines of glide reflection symmetry?

More Patterns from the Square (NOT TO BE DONE IN LAB 8. Go on to Part B.)

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 (point reflection), placing centers at each of the midpoints of the sides of ABCD.
• Make one transformation be rotation by 90 degrees with center A and the second be rotation by 180 degrees with center B.
• Define a glide reflection with vector AB and mirror (invariant) line = perpendicular bisector of AD. As a second isometry, add translation by AD or else a glide reflection in the vertical direction.

Part B. Patterns from an irregular Motif

Example 1 - 60 degrees

Draw an irregular polygon (a "blob"). Also draw two points A and B. Then generate a pattern by repeatedly rotating everything by 60 degrees with center A or with center B until a good-sized pattern of blobs appears with nothing new appearing in the region of interest as you continue rotating with center A and B.

Note: You should experiment with various relative sizes and positions by dragging A, B and the vertices of the blob. See whether you can work with the shape or centers so there is little or no overlap of blobs.

Find and CONSTRUCT examples of all the kinds of centers of rotation of this pattern. Find enough so that all others can be constructed by translation. Find the translation vectors also.

Example 2 - 72 degrees

Draw an irregular polygon (a "blob"). Also draw two points A and B just as before. Repeat the same experiment but this time use rotations of 72 degrees.

Note: You should experiment with various relative sizes and positions by dragging A, B and the vertices of the blob. See whether you can work with the shape or centers so there is little or no overlap of blobs.

What happens that is different from Example 1? What kind of pattern emerges? What can you say about where the centers of rotation are located?