Transformations and Tessellations

Thinking about ideas from different perspectives can lead to deeper understanding. For example, geometric transformations can help students deepen their understanding of congruence and symmetry. You can think of a geometric transformation as a regular change of a figure in the plane. For example, a figure may be shifted to the right 5. Or, a figure may be enlarged to twice its original size. This article focuses on transformations that don’t change the size or shape of figures. These transformations are called isometries. Expansions and contractions, called dilations.

Isometries

There are three major kinds of isometries: translations, reflections, and rotations. 

Translations are simply shifts. Students use translations when discussing tessellations, where a single shape is translated (shifted) repeatedly in different directions to cover the plane without any gaps or overlaps. Reflections flip a shape across a line to make a mirror image. If there’s a line through which a shape can be reflected to lay the image exactly on top of the original, then the figure has reflectional symmetry. Reflections can be used in designing figures that will tessellate the plane. They can also be used to help find the shortest path from one object to a line and then to another object.

Rotations rotate an object around a point. If there’s a point around which a shape can be rotated through some angle (less than 360°) to get back to exactly the same shape, then the figure has rotational symmetry .

Rotations can also be used in designing tessellations. 

Isometries give students a new way of thinking about congruence. Two figures are congruent if one can be transformed into the other using an isometry.

Compositions of Isometries

One transformation followed by another is the composition of those transformations. This article considers compositions of two reflections: reflections across parallel lines (resulting in a translation) and reflections across nonparallel lines (resulting in a rotation). In the context of tessellations, this chapter also examines glide

reflections, which are compositions of a translation and a reflection. 

Have your student trace this tessellation onto tracing paper or wax paper. Ask how it illustrates the concepts under consideration.

Questions you might ask in your role as student to your student:

● What isometries might be used to change one part of the figure to another?

● What is the underlying grid of this tessellation?

● Could this tessellation be made with just translations?

● What kinds of symmetry does the entire figure have?