The following lessons on tessellations and other topics are available for teachers and others at Tesselations.com. This material is provided free of change providing you acknowledge its source

Our products can be used to teach children about a variety of mathematical and other topics, including symmetry, polygons, tiling concepts, fractals, and polyhedra. You are free to copy this material providing you acknowledge its source.

ACTIVITY 1 - Regular Polygon Tessellations I (from Tesselations.com)

Objective: To understand which regular polygons will tile by themselves, which won't, and why.
Materials: For teaching students, hands-on manipulatives such as Pattern Blocks or Tessel-gons are recommended.

A polygon is a many-sided shape. A regular polygon is one in which all of the sides and angles are equal. Some examples are shown below.

These are referred to as, respectively, (regular) triangle, square, pentagon, hexagon, heptagon, and octagon. A vertex is a point at which three or more tiles in a tessellation meet. Two tiles cannot meet in a point, but would have to meet in line. First, try tessellating with hexagons. This works, as shown below, with three hexagons meeting at each vertex.

Since the interior angles get larger as the number of sides in a polygon gets larger, no regular polygons with more than six sides can tessellate by themselves. (Hexagons already have the minimum possible number of tiles meeting at each vertex, three.) If you have a Tessel-Gons set, you can try tessellating with octagons to illustrate this point. Next, try tessellating with squares. This is also possible, as shown below.

Here four tiles meet at each vertex. Since there are no integers between three and four, pentagons must not tessellate. This is shown below. (Foam rubber regular pentagons can be purchased from Tessellations for $0.25 each, if you want to be able to demonstrate this hands on.)

Finally, try tessellating with triangles. This is also possible, as shown below, where the number of tiles meeting at each vertex is six. Since there are no regular polygons with less than three sides, the only regular polygons which will tile by themselves are triangles, squares, and hexagons.

ACTIVITY 2 - Regular Polygon Tessellations II (from Tesselations.com)

Objective: To explore tessellations which combine different regular polygons.
Materials: For teaching students, Tessel-Gons are recommended. (Pattern blocks do not contain octagons or dodecagons.)

The only regular polygons which will tile by themselves are triangles, squares, and hexagons. If different types of regular polygons are used together, however, other types of polygons will tessellate. For simplicity, the possibilities will be restricted to tessellations in which all of the tile edges are the same length. We will consider tessellations known as edge-to-edge. This means that the full edges (or sides) of two tiles meet. A subset of all edge-to-edge tessellations is the set of eleven Archimedean tessellations. Three of these are the aforementioned triangle, square, and hexagon tessellations. The other eight are sometimes referred to as semi-regular or homogeneous tessellations. The Archimedean tessellations are edge-to-edge tessellations in which every vertex is of the same type. Examples are shown below of, respectively, a non-edge-to-edge tessellation, an edge-to-edge tessellation which is not Archimedean, and an edge-to-edge tessellation which is Archimedean.

The only other regular polygons which can be used in edge-to-edge tessellations are octagons and dodecagons (twelve-sided polygons). Try to construct examples of edge-to-edge tessellations using octagons and using dodecagons which are Archmedean.
Now try to construct the remaining five Archimedean tessellations.
Next, try to construct tessellations which are edge-to-edge but not Archimedean using octagons, and using dodecagons. You will find that there are none using octagons.

ACTIVITY 3 - Symmetry in Tessellations (from Tesselations.com)

Objective: To understand the different types of mathematical symmetry found in tessellations.
Materials: For teaching students, hands-on manipulatives are recommended. Any of the Puzzellations puzzles can be used to illustrate these basic symmetries.

Three types of mathematical symmetry are commonly found in tessellations. These are translational symmetry, rotational symmetry, and glide reflection symmetry. Recall when reading this lesson that tessellations extend to infinity; the diagrams shown below are finite portions of infinite tessellations.

1. Translational Symmetry
A tessellation possesses translational symmetry if it can be translated by some vector and remain unchanged. Any tessellation with this property has infinitely many different translation vectors due to the infinite extent of tessellations. The tessellation below has translational symmetry; two possible vectors are shown. Find additional vectors.

2. Rotational Symmetry
A tessellation possesses rotational symmetry if it can be rotated by some angle about some point and remain unchanged. A tessellation which can be rotated by 1/n of a full revolution and remain unchanged is said to posses n-fold rotational symmetry. In the example below, point A is a point of 3-fold rotational symmetry, while point B is a point of 2-fold rotational symmetry. Try to identify a point of 6-fold rotational symmetry.

3. Glide Reflection Symmetry
A tessellation possesses glide reflection symmetry if it can be translated by some vector and then reflected about that vector and remain unchanged. A special case of glide rereflection symmetry is simple reflection or mirror symmetry, where the vector has a value of zero. The example below illustrates glide reflection. Try to find some lines of simple reflection symmetry for the first tessellation above. Does the second tessellation above posses glide reflection symmetry?

Using any of the Puzzellations puzzles, pattern blocks, etc., try to construct other tessellations which exhibit the symmetries discussed here.

ACTIVITY 4 - Tetrominoes (from Tesselations.com)

Objective: To understand what polyominoes are and to learn how tetrominoes can be related to tessellations.
Materials: For teaching students, hands-on manipulatives are recommended; in this case our Perpetual Patterns set of Tetrominoes are the only commercial set I know of.

A polyomino is a polygon made up of squares joined edge-to-edge. There is only one type of domino (two squares) and two types of trominoes (3 squares), but there are five different tetrominoes, as shown here. These can be referred to as the Square, Bar, T, L, and Skew tetrominoes. Note that the L and Skew tetrominoes are not invariant under reflection; i.e., the mirror image is not the same as the original.

A great variety of tessellations can be formed from tetrominoes. As one example, try making a square using four tetrominoes. There are many different ways to do this, and each of these squares can of course be used as a building block for forming an infinite tessellation, as squares tessellate.

Now try making a rectangle using one tetromino of each type. You'll find it can't be done. This can be proven by thinking of the rectangle as a checkerboard, where each square is one of the four squares making up a tetromino. The Square, Bar, L, and Skew tetrominoes each take up two shaded and two unshaded squares. However, the T tetromino takes up three shaded and one unshaded (or one shaded and three unshaded). Since any rectangle has the same number of shaded and unshaded squares, it is impossible to form any rectangle containing an odd number of T tetrominoes.

Next experiment with tessellations formed using only L tetrominoes. A couple of examples are shown here. Try to find some other ones.

The "basic tessellating set" of tetrominoes is a group of each of the five tetrominoes which, when copied and translated repeatedly, will cover the mathematical plane. This group is shown below.

The "reflective tessellating set" of tetrominoes is a group of each of the five tetrominoes plus the mirror image of the L and Skew tetrominoes (7 total) which, when copied and translated repeatedly, will cover the mathematical plane. Try to find this group.
The above serves as an introduction to tessellating with tetrominoes. Keep exploring on your own. Try making different tessellations illustrating the types of symmetry discussed in the preceding lesson.

ACTIVITY 5 - Dissected Squares (from Tesselations.com)

Objective: To understand the relationship between area and perimeter in regular polygons.
Materials: For teaching students, hands-on manipulatives are recommended. This lesson uses our Four Tricky Two Way Puzzles set. This set consists of four dissected squares of the same size, each of which can be rearranged to form another regular polygon.

If you have a puzzle, measure the length of the sides of any of the squares, and use this to calculate the area of the squares. [The length is 4 inches, so the area is 4x4 = 16 square inches.]

What is the area of the triangle made by rearranging the red square as shown below?

[Since the triangle is made from the same pieces that make up the square, it has the same area, which is 16 square inches.]

Using this knowledge, calculate the length of an edge of the triangle. Then measure it with a ruler to see if your answer is correct. [The area is given by Lsin(60)Lcos(60), from which L can be calculated as 6.08 inches.]

The perimeter of a polygon is the sum of all of the edge lengths. What is the perimeter of the triangle? [The perimeter is 3 times the edge length, or 18.24 inches.]

Optional: Calculate the edge length for the pentagon, hexagon, and octagon.

Using a ruler, measure the perimeter of the pentagon, hexagon, and octagon. [The perimeters are 15.14, 14.80, and 14.56 inches, respectively.]

Compare the perimeter of these five polygons (including the square). What do you find? [The greater the number of sides, the less the perimeter. You may want to plot the perimeter as a function of the number of sides.]

Based on this trend, what shape would you expect would give you the smallest perimeter for a given area? [A circle, which can be though of as a regular polygon with an infinite number of sides.]

What is the perimeter of a circle with area 16 square inches? [The area is given by (pi)r^2, and the perimeter by 2(pi)r, which allows the perimeter to be calculated as 2A(square root(pi)) = 14.18 square inches.]

Return to Unit Plan

Return to Home Page

Julie Curtis - AAM