Getting started: Go to BoxerMath's Tessellation Tool, which will allow you to build tessellations and other designs by attaching the corners of various shapes.

Note: BoxerMath's Tessellation Tool will open in a second window. Take a moment to adjust the two windows so that you can work with the applet and also view the text of the activity.

1. What polygons do you see? Can you find more polygons? (Hint: click on the brown arrow just above the Reset button.)
2. What does the Reset button do?
3. What happens when you click on the Help button?
4. Can you move a polygon to the main area? How?
5. Change a color and describe how you did it.
6. BoxerMath's Tessellation Tool lets you choose between translate and rotate. What is the difference between a translation and a rotation?

Challenge: Can you use triangle(s) and the translate feature and/or rotate feature to cover the plane with no gaps or overlaps?

1. How does your tessellation of an equilateral triangle compare to the other students in your class? How does your tessellation compare to the two samples shown below?

        

2. How did you make your tessellation? Did you use one triangle and both the translate and the rotate feature? Did you use two triangles and the translate feature?
   
   
3. If you are limited to only one of the triangles, can you make a tessellation? Can you make a tessellation of triangles no matter which of the four triangles is available to you?
   
   
4. Not counting the colors, what is similar about these two tessellations? What is different? [Hint: Try to use the words rotation and/or translation in your explanation.]

        

5. Four triangles can be chosen from BoxerMath's Tessellation Tool palette of polygons. Each of them can be rotated to "match" the others. Can you explain the connections between the pairs of triangles shown below?

   

Manipulatives: Use the triangles in a set of activity pattern blocks to make a tessellation. (Paper activity pattern blocks are available on the Web; see Hand Made Manipulative Instructions by Margo Mankus.)

Revisiting the activity: Now that you have tessellated with BoxerMath's Tessellation Tool and with activity pattern blocks, return to the Tessellation Tool to think about using only one equilateral triangle.

If we know that there are 360 degrees in a circle, and if we think of using just one triangle (and the rotation feature) to make our tessellation, how many equilateral triangles do we need to fit together? How many degrees are in the interior measure of the angles that meet?

 Formalizing the mathematics:

 Focus on either of the two units:

   

1. What is the interior measure of the angles of a triangle?
2. Look at the vertex of one triangle within a tessellation of triangles. How many triangles, in all, are touching at that vertex?
3. What is the sum of the angles that share that vertex?

 Before returning to BoxerMath's Tessellation Tool, predict whether a square will tessellate by answering these questions:

1. What is the interior measure of the angles of a square?
2. Think of the vertex of one square within a tessellation of squares. How many squares, in all, are touching at that vertex?
3. What is the sum of the angles that share that vertex?

    Test your prediction. How did you do?

 Repeat the process with a hexagon. Again make a prediction by asking yourself:

1. What is the interior measure of the angles of a hexagon?
2. Think of the vertex of one hexagon within a tessellation of hexagons. How many hexagons, in all, are touching at that vertex?
3. What is the sum of the angles that share that vertex?

    Test your prediction. How did you do?

 Now that you have considered the cases of the equilateral triangle, the square, and the regular hexagon, can you complete the following chart?

Column 1	Column 2	Column 3
name of polygon	degrees of the interior measure of each angle	360 degrees divided by # in Column 2
equilateral triangle	60	360 / 60 = 6
square	90
regular pentagon
regular hexagon
regular heptagon
regular octagon

 Can you fill in the blanks in the following statement?

In a tessellation the polygons used will fit together with their angles arranged around a point with no gaps or overlaps. When using just one polygon (for example, only equilateral triangles), the interior measure of each angle will need to be a factor of _____ degrees (meaning that ____ degrees can be divided evenly by that angle measure). The only regular polygons that qualify are the __________________, ___________________, and ___________________.

Assessment:

Choose any polygon other than a triangle, square, or hexagon. Illustrate and explain why it will not tessellate regularly in the Euclidean plane.

Math Forum Resources

[Information and Lessons]  [Problems of the Week]  [Dr. Math] 

What Is a Tessellation?

Definition and illustrated explanation of tessellations.

The Four Types of Symmetry in the Plane

Rotation, translation, reflection, and glide reflection, with illustrations and problems for consideration.

Repeated Reflections of an "R"

Lesson to draw a design with reflectional symmetry and rotational symmetry.

Tessellation Links

Links to other sites on the Web.

Tessellation Tutorials

A series of tutorials that teach students how to tessellate.

Selections from Math Forum Problems of the Week:

MidPoW: Tiling Triangles - posted March 11, 2002

Given the dimensions of a large triangle, find the dimensions of the 25 small triangles that tile it.

MidPoW: A Surfer's Tessellation - posted April 30, 2001 [Pow-teach discussion]

Shane is re-tiling his surf shack. How many of each kind of tile will he need, and what does a big tile cost?

GeoPoW: Transforming a Triangle - posted November 13, 2000

Given two triangles on a coordinate plane, find a pair of transformations that will map the first triangle onto the second triangle.

GeoPoW: Rotating a Triangle - posted April 30, 2001

Given a triangle on a coordinate grid, find the new coordinates of the vertices if you rotate the triangle 90 degrees about the origin.

Selections from Ask Dr. Math Archives:

Symmetry

What is symmetry?

Tessellation

Are there any non-regular convex polygons with more than four sides that can tessellate?

Tessellation Proof

I am looking for the proof of the statement that only three regular polygons tessellate in the Euclidean plane.

Tessellations and Symmetries

How do you make a tessellation with a rotation, reflection, and translation all in one shape?

Types of Tessellations

I am doing a project on tessellations. Can you explain some of the mathematics behind them?

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