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Problem of the Week
Problem B
Tessellation Consternation

A tessellation is a tiling pattern in which two-dimensional shapes fit together with no overlaps or gaps. A regular tessellation uses congruent shapes.

Below are three examples of tessellations, using right triangles, parallelograms, and two slightly different birds, respectively. The first two are regular tessellations, the third is not.

A tessellation of congruent right-angled isosceles triangles in two different shades.A tessellation of congruent parallelograms in three different shades.A tessellation of two slightly different bird shapes in two different shades.

A circle is an example of a shape that does not tessellate, since circles cannot be tiled without creating overlap or gaps:

Five circles are placed so that 3 are side by side in a row and the 2 others are side by side in another row touching the first row. There are small gaps between the circles.

  1. For each shape below, name the shape and then decide whether or not it can be tessellated. Note that shapes may be rotated, reflected, and/or translated in order to make them tessellate.

    Hint: For no gaps, what must the total angle be where vertices meet?

    1. A triangle with three sides that are equal in length and one angle labelled as measuring 60 degrees.

    2. A triangle with two sides that are equal in length. The third side is shorter than the two equal sides.

    3. A four-sided polygon with all sides equal in length and one angle labelled as measuring 90 degrees.

    4. A five-sided polygon with all sides equal in length and one angle labelled as measuring 108 degrees.

    5. A six-sided polygon with all sides equal in length and one angle labelled as measuring 120 degrees.

    6. An eight-sided polygon with all sides equal in length and one angle labelled as measuring 135 degrees.

  2. On a piece of cardboard or sturdy construction paper, draw an irregular quadrilateral (that is, a quadrilateral with all four sides of different lengths) and cut it out. Trace it on a piece of paper, then continue to move and retrace it to see whether it will tessellate or not.

    Suggestion: If desired, use a digital tool instead of tracing by hand.

Theme: Geometry & Measurement