From the technical side it is necessary to be able to operate on Netscape, which is already a done deal. An important technical requirement is that your computer be loaded with Tessel Mania! from MECC Software and that it be configured to load some Tessel Mania!. These files must end in the suffix .tsm because the files are so designated.
For my project, to collect what I had learned this Summer at The Geometry Center, I chose Tesselations because I know a bit about paper and scissors methods and because I wanted to do some of the fancy designs without all the work.
This project is also a record of my progress through the three weeks of the Geometry Institute. Except for some of the fundamental procedures of tessellating, what is seen is what I have learned in the last three weeks
What are Tesselations? Let me show some samples, starting with very fundamental polygons and advancing to some very complicated results of "Orbifold" applications.
Examples of Tesselations
How to Tessellate
In what follows I hope to explain how each of several methods can be applied to create tesselations, starting with pencil and paper and ending with computer software.
A. FUNDAMENTAL POLYGONS
It is not too difficult to find pages with squares, equilateral triangles, or hexagons tesselating the page. One good source is Graph Paper Masters from Dale Seymour Publications . The book is a source of black-line masters and comes with permission to copy for classroom use. Students can be invited to create their own schemes for coloring the figures. Before the four-color map theorem has been learned a challenge might be made to used the fewest number of colors to color all the regions.
B. SIMPLE TRANSFORMATIONS
Now a few simple tesselation patterns will be shown that can be done with paper, pencil, and scissors.
To tesselate a field of rectangles, start with a rectangle drawn on a page. Replace one side with some simple design.
When doing this by hand, it is handy to start with a rectangle, draw your pattern down the middle, then cut slong that pattern, and join the parallel end segments.
Using the same method,
a slightly more advanced SLIDE tesselation can be made using the three pairs of opposite sides in a regular hexagon
This production took just a few minutes, once I spent an hour carefully reading the instructions and playing with the program.
ROTATION is another technique. It is shown here using an equilateral triangle.
C. WORKING WITHIN TESSELMANIA
Tessel Mania must be on your computer for this to work.
Furthermore it must be configured to be able to upload the software.
The fundamental figure I chose is one that can transform by Slide (top-to-bottom) and by Rotation about the midpoints of sides.
This is what the TM commands can do.
EDIT TILE (1) will work with or view a single tile.
TESSELATE is (2)
MAGNIFIED VIEWS is (3)
TILE MAGIC (4) shows how a tile was created
TESSELATION MAGIC (5) Shows how a tile moves to create the tessellation.
METAMORPHIC MAGIC (6) shows the entire tessellation transformed from the original tile to the final product.
When (4) or (5) or (6) are chosen a sub-menu appears which controls speed and the direction of morphing.
To begin transforming click on the thumbtack. Observe that the midpoints (of rotation) appear.
Next, move the TACK to any edge and click on it anywhere to create a point. While holding down, the TACK will transform the side as it is dragged around. Do this a few times and then get ready to Tesselate.
To TESSELATE click on COMMAND 2.
To STUDY the process of transformation click onCOMMANDS 4, 5, or 6.to STOP the action, click on the picture.
GET BACK to the single polygon by clicking on COMMAND 1.
DECORATE by clicking on the STAMP. A menu of objects will appear at the bottom. Select on by clicking, then move the arrow to the interior of the figure. The selected decoration will now be in the figure. Click if you want it to be applied there.
APPLY COLOR by clicking on the PAINT BUCKET. Then select a color by clicking . A texture, from below the colors, can also be selected. Then move to the region of the figure to be colored and click. If there is a second region it should be colored differently.
FINALLY TESSELATE. Click on COMMAND 2. The computer will select a second set of colors for decorating alternating figures.
Next are links to come Sample Tessellations from the supply within Tessel Mania. Be sure to close the screen when finished.
Web Sites, just a few, with Tesselations.
These are four web sites of hundreds that deal with TESSELATION.
Finally, when you get bored,....