A **tessellation** is the use of a geometric figure that covers a plane surface with no overlaps, or gaps between the figures.

Tessellations are also referred to as *tilings*. The example below of a brick wall and tiled floor are examples of tessellations using rectangles to tessellate a plane. Many different cultures, including ancient Sumerians and Babylonians, as well as the Roman and Byzantine Empires, have used tessellations to generate remarkable artwork in important buildings such as temples, churches, and cathedrals.

Tessellations are created using transformations of different polygons.

**Which polygons will tessellate a plane? **

Some polygons will tessellate a plane by themselves, and others will not. In this part of the lesson, you will use an interactive geometry applet from the National Council of Teachers of Mathematicsâ€™; Illuminations website to explore which polygons will tessellate and which polygons will not.

Click on the link Tessellation Creator. The top bar of this applet contains different polygons. Click on a polygon several times to get several copies of that polygon in the sketch window. Drag the polygons to see if you can arrange them so that they are non-overlapping and that their vertices line up without leaving any gaps. If you can, then the polygon tessellates. Do this for each polygon in the chart below.

Use your results to fill in a chart like the one shown below in your notes.

(Move your mouse over the table above to view answers.)

Look at the data in the chart. Which polygons tessellated? What relationship do the measures of the interior angles of the regular polygons that tessellated have to 360^{o}?

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A regular polygon will tessellate if the measure of one of their interior angles is a number that will divide evenly into 360^{o} without remainder. In other words, the measure of an interior angle of a regular polygon that tessellates is a factor of 360^{o}.

Based on what you have observed, do you think that any rectangle will tessellate a plane? Why or why not? Click to see a sample answer.

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The measure of an interior angle of any rectangle is 90^{o}. Since 90^{o} is a factor of 360^{o}, any rectangle will tessellate a plane by itself.

You may want to draw some of the tessellations of one polygon in your math journal. Click here to use the dynamic paper tool from NCTM Illuminations, Dynamic Paper, and generate a JPEG of a tessellation.