Math 445. Assignment 9

Due Wed 3/4 at the beginning of lab.

This assignment is just made up of constructions (which hopefully shed some light on the relationship between tessellations and symmetry).

There will be some study questions on tessellations as well, posted soon.

9.1, 9.2, 9.3

• (A) Construct 3 different tessellations (a) by modifying one of the standard tessellations (parallelogram or regular polygon) using the ideas in the handouts you got in lab or (b) using one of the other two methods below.
• (B) In each case, give a brief but clear explanation of what the underlying tessellation is (i.e., the regular or simple one) and what are the symmetries.
• NOTE: Animal designs are OK but NOT NECESSARY. What IS necessary is an actual tessellation by congruent "tiles". Do not just make a symmetric pattern (been there, done that last quarter); make a tessellation.

Method 1.

One important method is the method of cutting off a piece from an edge and then taping it to another edge in a systematic way. This works for "Escher" drawings.

Note: This works fine in Sketchpad, but of course the bumps have to be related by isometries so that they stay linked if one is dragged. The handout on triangles is rather clear on how to do this for one example.

Method 2.

Take a dissection of a simple shape such as a triangle and reflect this across all the sides to build up a tessellation of the plane (see the Math Day sketches).

Method 3.

Make a dynamic tessellation like the one in the handout made of two sizes of square, or the tessellation with squares and rhombi.