DISCOVERING SYMMETRY WITH KALEIDOTILE

SYMMETRIES IN THE PLANE

ACTIVITY SHEET

The exercises contained in this activity require the KaleidoTile program developed by Jeff Weeks. If you do not yet have a copy of this software, you may wish to download KaleidoTile.

For more information regarding this activity and its objectives, see the accompanying Teacher's Guide.

Start up the KaleidoTile program on your computers and stop the polyhedron in the Tiling window from spinning. In the Symmetry Group window, choose the button labeled
(2, 3, 6). The resulting pattern tiles the plane. In the Basepoint window, move the base point to the center dot.

Find as many lines of reflection as you can in the pattern in the Tiling window. If you would like a hard copy of the pattern on which to sketch, choose Print in the File menu when you are in KaleidoTile.
When you have finished, choose dual edges from the What to Draw window. Dual edges appear in the pattern.
What is another name for these lines?

Without changing KaleidoTile's settings, complete the following exercises.

Definition: In KaleidoTile, the base triangle is the triangle with the darkest outline.

FIG.1
The triangles adjacent to the base triangle are the darkest triangles in this figure.

How are the triangles adjacent to the base triangle like the base triangle, and how are they different?
KaleidoTile starts with the base triangle and uses it to tile the plane. Explain why this base triangle has the same name as the "base triangle" we used when experimenting with the mirrors in Lesson 13.
Check your answer: Click on Demo under the Help menu at the top of the screen. Were you right?

Notice that each vertex of the base triangle touches a number of similar triangles.

Example

For each of the three vertices, count how many triangles meet at that vertex. The current pattern has 12 triangles at the left-most vertex, 4 at the middle vertex, and 6 at the left-most vertex. Start a table recording this information.
Create four different patterns by moving the base point, but without changing the symmetry group. For each pattern, count the number of triangles meeting at each vertex. Add this information to the table.
Examine your table. What is the pattern? What relation do these numbers have to the triple (2, 3, 6)?

In the More Symmetry Groups window, choose the settings ¼ / 2, ¼ / 4, ¼ / 4 and hit Set. Make sure the dual edges remain on.

Repeat what you did above, but with this new symmetry group. Specifically,

Look at four different patterns by moving the base point, but without changing the symmetry group.
For each pattern, count the number of triangles meeting at each vertex of the base triangle. Create another table recording your information. Here is a pattern to get you started; 8 triangles meet at each of the right and left vertices of the base triangle, and 4 meet at the middle vertex.

How do these numbers relate to the triple (2, 4, 4)? Compare the two tables. What is the pattern?

Challenge topic

Measure the angles of the base triangle.

How do these numbers relate to the triple (2, 4, 4)?
How do they relate to the number of triangles meeting at each vertex?
What is the sum of these angles?
What happens to these angles if you move the base point around, but do not change symmetry groups?

Return to the Symmetry Group (2, 3, 4), and measure the angles of the base triangle.

How do these numbers relate to the triple (2, 3, 4)?
For any particular pattern, what is the relationship between the angles of the base triangle and the symmetry group name?
What is the sum of the angles for any base triangle in the (2, 3, 4) symmetry group?

Predict which symmetry groups will result in a planar pattern.Check your answer by selecting these symmetry groups on KaleidoTile.