### HSED422/MSED456: What is a Tessellation?

According to the Wikipedia, a tessellation is a collection of plane figures that fills the plane with no gaps or overlaps. The definition of tessellation at the Math Forum web site requires all the plane figures to have the same shape. Older definitions stipulate that the shapes all be square, or made out of clay. What definition of tessellation do we want to use in this class, and what properties of tessellations are we interested in?

A nice collection of tessellation-related definitions can be found at http://www.spsu.edu/math/tile/defs/definitions.htm.

Properties of Tessellations

Symmetric/Periodic: the tessellation is self-symmetric; it's made up of a repeating motif or fundamental region.

Monohedral/Isohedral: All tiles are the same shape.

Regular: The tessellation is periodic and tiles are congruent regular polygons.

Semiregular: The tessellation is periodic and all tiles are regular polygons.

Asymmetric/Aperiodic: The tessellation does not repeat itself. Some mathematicians have discovered tiles from which it is impossible to construct a symmetric tiling!

Reptile: All tiles are the same and each tile can be decomposed into a number of smaller copies of itself.

Other: The images below show other collections of plane figures that cover the plane. Do they deserve the title "tessellation"?