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Geometry

Students define tessellations and explore which regular polygons, or combinations of regular polygons, can tessellate the plane.

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After completing this tutorial, you will be able to complete the following:

- Define a tessellation.
- Define regular and semi-regular tessellations.
- Explain which regular polygons can tessellate the plane.
- Explain which combinations of regular polygons can tessellate the plane.
- Write a tessellation code that corresponds to a given semi-regular tessellation.
- Form the regular or semi-regular tessellation given by a tessellation code.

Mathematician Robert Penrose used rhombuses to create remarkable tilings of the plane, now called Penrose tilings. Penrose tilings are a special type of tessellation, or patterns of figures that completely cover a plane without gaps or overlaps. A Penrose tiling is chiefly characterized by self-similarity and a lack of translational invariance, meaning that no two shifts of the tiling look the same and that any portion of the tiling looks similar to some larger portion. Penrose tilings have important applications to quantum physics, number theory, and geometry.

One category of tessellations is known as regular tessellations. These are constructed from one congruent regular polygon in such a way that the edges line up and the arrangement of regular polygons is identical at every vertex. This means that the interior angles of the regular polygons must divide evenly into 360°. The only regular polygons for which this is possible are triangles, quadrilaterals, and hexagons, which have interior angle measures 60°, 90°, and 120°, respectively.

Another category of tessellations are semi-regular tessellations. These use two or more regular polygons to tessellate the plane. The arrangement of these polygons is the same at every vertex. Notice that the Penrose tiling is not an example of either a regular or a semi-regular tessellation since the arrangement is not the same at every vertex. The measures of the interior angles of the regular polygons that meet at a vertex of a semi-regular tessellation must sum to 360°. See the semi-regular tessellation below.

It can be cumbersome to describe semi-regular tessellations, and drawing one each time you wish to describe one can be time-consuming. Because the arrangement around every vertex of a semi-regular tessellation is the same, we could instead simply describe the pattern of polygons in a clockwise direction. For the semi-regular tessellation in Figure 1 above, we could say that it follows the pattern: square, triangle, triangle, square, triangle. However, since the polygons are regular, we could simply write down the number of sides of the polygon: (4; 3; 3; 4; 3). This is known as the tessellation code. We can write tessellation codes for regular tessellations as well. Notice that we could have started with a different polygon in Figure 1, such as the triangle at the top left of the green vertex. In this case, the code would be (3; 4; 3; 4; 3). Both of these codes describe the same tessellation. This means that tessellation codes are not unique.

There are only eight semi-regular tessellations:

Approximate Time | 30 Minutes |

Pre-requisite Concepts | Students should be able to define a plane figure and a regular polygon; recognize regular polygons; explain translation, rotation, and reflection in the plane; apply transformations of regular polygons in the plane; and know the sum of interior of polygons. |

Course | Geometry |

Type of Tutorial | Concept Development |

Key Vocabulary | polygons, regular tessellation, semi-regular tessellation |