TESSELLATING ON A PLANE AND SPHERE

 

Performance Task Description
Your class has been asked to teach a small group of elementary school students about the Moorish Culture. Your job is to teach the students about the art found in their temples. Moors are Islamic: The Islamic religion does not allow pictures (idols) in their temples. Instead, the temple walls and floors are covered with beautiful, intricate mosaics or tessellations. The word tessellate comes from a Latin word which means small stone.

To prepare for this lesson, you will gather and make demonstration materials that show examples of these mosaics. Look in books and/or on the internet to find examples and then make two tessellations of your own: one of the plane (Euclidean Geometry) and on on a sphere (non-Euclidean Geometry). Your demonstration materials will include written explanations of the steps you followed to make the tessellations and a comparison chart of the two types. Your teacher will give you more detailed instructions to help you understand how to tessellate on a plane and on a sphere.

Remember the utimate goal is to gather and make materials that you will use to teach others how to imitate the Moorish tessellations on a plane and sphere. You also wish to engender and appreciation for their beauty.
Tessellating On A Plane
The first tessellation you will make will be on a flat plane. It should completely cover a sheet of paper. The shape with which you will tile can be any of the three shapes that could be used to make a pure tessellation, but it must be modified using at least one of the transformations we learned about in class. You may use traditional instruments like a compass and a straight edge, or you may use Geometer's Sketchpad to create the tile and subsequent tessellation. The tessellation must be colored so as to make it esthetically pleasing.

After you have completed your flat tessellation, you must include written step-by-step instructions about how you developed your tessellation. Pictorial examples or diagrams of each step would be most helpful.
Tessellating On A Sphere
After you have completed tessellating on a flat surface, consider the following:

  1. What are the three types of regular polygons that can be used to create a pure tessellation on a plane? Do you think these will work on a sphere? How many sides are needed to form a shape on a sphere?
  2. What kind of transformations can be used to modify these tiles, so that they can still cover the fundamental region without overlapping or leaving any gaps?
  3. Does the size of the tile matter on a flat plane? (Remember a sheet of paper is not the entire plane). Will the size matter on a sphere?
After you obtain the sphere you wish to tessellate, you may want to experiment with rubber bands to see what kind of shapes can be tiled. Next you will need to devise a spherical ruler and protractor for it. Once you have decided upon a shape, modify it using one of the transformations we learned about in class. Then tessellate your sphere. Color your spherical tessellation.

After you have completed your spherical tessellation, you must include written step-by-step instructions explaining how you developed your tessellation.

Finally, you will make a comparison chart, comparing the construction and characteristics of flat and spherical tessellations.