Lab 10

Part A. Composition and Symmetry of Wallpaper Patterns with GSP and Pasted Pix

Download these Sketchpad files:

• lab10-wallpaper.gsp
• Icons

Open the lab10 and and paste in a picture as explained.

• Make a new page in the lab10 file. Draw one point on the page and select it.
• Then go on the web and find a wallpaper pattern to download, to start with, choose one with some 90 degree rotations in it.
• Copy the picture from the web page into your sketch (with the point selected). The picture will attach to the point.
• Select the picture and get Properties>Object and turn off Arrow Selectable. This is how you get a picture into Sketchpad.
• Use the p4 tool to create a set of centers from two points A and B. Then adjust the GSP centers to see which of them fit over the picture.

Do this for some examples of p4 type symmetry and also p6 type symmetry until you feel you are seeing the relationships.

Part B. Kaleidotile

Kaleidotile is a free program originally written at the Geometry Center at the University of Minnesota. It is one of another enjoyable and educational(!) programs from Jeff Weeks at www.geometrygames.org. Many will work with kids.

Kaleidotile combines tessellations and kaleidoscopres in 3 (or more) geometries. The examples in the plane are most relevant for us now, but the software also shows wonderful animated views of truncations of polyhedra. (This can also be considered from the point of view of spherical geometry, which we will adopt in 445.) The third geometry, besides the Euclidean plane and the sphere is what is called hyperbolic plane geometry (again, this will be in 445).

A second relevant program from geometrygames.org that may be helpful for us this week is Kali, a program that will draw symmetric patterns with a "pen".

• Download the zip file containing Kaleidotile to the C:/temp directory.
• Link to Kaleidotile -- zip file for Window or sit.bin file for Macintosh
• Go into that directory and open the zip file to get the kaleidotile.exe application. Start the application.
• Start the application. Choose one of the Euclidean triangles from the Euclidean menu.
• Be sure to turn on Edges for some of the time, and try the Start button.
• Read the Classroom page for Kaleidotile (this file as an rtf file is part of the Kaleidotile package). Answer some of the questions for the Euclidean plane but do not worry yet about the other spaces that will be investigated in 445.
• Study the 3 special triangles for Euclidean space. You are welcome to explore the other features also, especially the polyhedral examples.

Part C. Napoleon's Theorem, etc.

This figure will be discussed in class later, but you can see what it is now.

Draw a general triangle ABC and construct on the sides of the triangle 3 equilateral triangles (with their centers). For example, construct a an equilateral triangle triangle FBC with center A', and also triangles on CA and AB with centers B' and C'. Suggestion: Make a tool that will construct an equilateral triangle EFG with a center P, starting from a side EF.

Construct the triangle A', B', C'. What is its shape whatever you do with ABC?

Construct the segments AA', BB', CC' and tell what you see.

You can leave the reasons as a mystery, or you can try rotating the pattern by 120 degrees around each of the centers A' and B' and see what you get. Is there a connection between this and the earlier part of the lab>

Practice Constructing

Sketchpad is a terrific tool for exploring constructions concerning isometries. For try drawing a triangle ABC (one of the special ones from Kaleidotile or any triangle) and investigating the double and triple reflections in lines of the triangle.