“We adore chaos because we love to produce order. My work is a game, a very
serious game.”
-M.C. Escher, mathematician and highly-acclaimed artist, known for tessellations
Discovering
Tessellations
PTMT:Chapter 6, Section 3
I.
Introduction
Picture in your mind a typical kitchen floor. If you imagined one composed of ceramic tile, you were most likely envisioning a floor akin to a grid of squares. Regardless of the shape, however, most of these floors create tessellations. According to Preparing to Teach Mathematics with Technology: An Integrated
Approach to Geometry (PTMT), “a tessellation is a set of polygonal regions that cover the plane without
gaps or overlaps.” Thinking of a tile floor is an excellent example to introduce one to tessellations, because it is easy to see that the floor would not be very well made if the individual tiles overlapped or did not fit together. The previous page contains art done by the mathematician M.C. Escher. Mr. Escher was known for his work with tessellations, most commonly using the subject of birds. In this exploration, we will be using more simple polygons, but it is very interesting to see how such a simple concept can make such a complex piece of art.
Although each polygon in a tessellation does not necessarily have to be the same size, it is more common to see congruent polygons used in a tessellation. PTMT defines a regular tessellation as “a tessellation that is created using congruent copies of a single regular polygon.” But, do all regular polygons create tessellations? Using the dynamic geometry environment Geometer’s Sketchpad, we will explore this problem and find how to create our own tessellation as well. Note that this work can also be completed in other DGEs such as Geogebra and NCTM’s Core Math Tools, but we will specifically be focusing on GSP.
II. Exploring Regular Tessellations
First, we need to explore regular polygons and whether or not they can all tessellate. What is a regular polygon? A regular polygon is one in which all sides are congruent and all angles are congruent. Although there are numerous regular polygons we could explore, we will be focusing on a triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon, all of which hold regular polygonal properties. To begin, let’s consider the interior angles of each of the previously listed regular polygons. Do you suppose there is a relationship between the interior angle measure and whether or not the shape will tessellate? If so, why?
Given the sketch below, we have constructions of each of these regular polygons and their interior angles. Which do you think will tessellate? Why?
mB''''AB = 144.00° mA''''AB = 140.00° mBAB' = 135.00° mOBA = 90.00° mBAB' = 128.60° mNBA = 120.00° mBAK = 108.00° mA'AB = 60.00° A' B'''' A'''' B' B' N K L M O B A B A A B A B A B A B A B A B
Now we will try to tessellate each regular polygon to test our theories. Using the techniques we have already learned (translations, reflections, rotations, dilations), we will transform the polygons in an attempt to cover the plane without any corners or sides overlapping.
As shown here, the equilateral triangle, square, and regular hexagon each tessellated. Their angle measures, 60°, 90°, and 120°, respectively, each share something in common. What is it? First, let’s consider a vertex of one of the equilateral triangles in its tessellation. Notice that this point can be looked at as a center of rotation for all of the other triangles. As a result, the interior angle measures of each of the triangles must add up to 360°. We know that 60 is a divisor of 360, so the equilateral triangle will tessellate. The same holds for the square and regular hexagon.
mB''''AB = 144.00° mA''''AB = 140.00° mBAB' = 135.00° mOBA = 90.00° mBAB' = 128.60° mNBA = 120.00° mBAK = 108.00° mA'AB = 60.00° A' B'''' A'''' B' B' N K L M O B A B A A B A B A B A B A B A B
So what conclusions can we draw about tessellations thus far? After completing the previous activity, we found that not all regular polygons will form regular tessellations. We also learned that interior angle measures play a key role when one wishes to create a regular tessellation, in that one must find a relationship between it and 360°.
III. Creating Tessellations
While regular polygons seem to form the nicest tessellations because they are very uniform in their construction and transformations, tessellations can also be created using irregular polygons. Here, we will explore how to create one starting with a simple parallelogram using GSP.
mB''''AB = 144.00° mA''''AB = 140.00° mBAB' = 135.00° mOBA = 90.00° mBAB' = 128.60° mNBA = 120.00° mBAK = 108.00° mA'AB = 60.00° A' B'''' A'''' B' B' N K L M O B A B A A B A B A B A B A B A B mB''''AB = 144.00° mA''''AB = 140.00° mBAB' = 135.00° mOBA = 90.00° mBAB' = 128.60° mNBA = 120.00° mBAK = 108.00° mA'AB = 60.00° A' B'''' A'''' B' B' N K L M O B A B A A B A B A B A B A B A B mB''''AB = 144.00° mA''''AB = 140.00° mBAB' = 135.00° mOBA = 90.00° mBAB' = 128.60° mNBA = 120.00° mBAK = 108.00° mA'AB = 60.00° A' B'''' A'''' B' B' N K L M O B A B A A B A B A B A B A B A B
First, we will create a parallelogram like the one pictured below.
We can hide the dashed parallel lines to make our parallelogram easier to see. Using the line segment tool, construct one of the sides of the parallelogram to look like this irregular polygon:
Mark a vector along the right side of the parallelogram from the bottom of the shape to the top as shown by the arrows. Translate the irregular polygon along this vector:
Repeat the previous two steps for the remaining sides of the parallelogram, creating the figure below:
Note which points represent the vertices of the original parallelogram we drew. It may help to make these another color, because we will be using them as the endpoints of the vectors needed to create the tessellation. Hide the original sides of the parallelogram and create the interior of the remaining
irregular polygon. Drag the vertices and explore the different sizes and shapes that can be created by it. Notice that the two images and their pre-images remain translations of each other.
Now we are going to use this polygon to make our tessellation through transformations. First, let’s translate this image to the right. We will use what was the top of our parallelogram as the translation vector:
Continue translating the polygon using the original sides of the parallelogram to make a tessellation. You may continue your tessellation as long as you please, creating one that covers the plane:
After completing this activity, we have the knowledge to explore other polygons and whether or not they will tessellate. While we already formed generalizations about equilateral triangles, we could look at isosceles and scalene triangles. Do you think any triangle has the possibility to tessellate the plane? Why? What about trapezoids or rhombi? A concave polygon? The possibilities are endless, and with the help of a DGE, they are able to be explored.
IV. Potential Use in Classroom
Tessellations, and the techniques used to create them, are an integral part of the curriculum for the future of math students. It is important to at least introduce them to the topic early in their academic careers, as several topics in higher math classes, such as algebra and calculus, are based upon them.
This topic is first introduced in the 2010 Alabama Mathematics Course of Study under Geometry for Grade 8:
Understand congruence and similarity using physical models, transparencies, or geometry software.
16. Verify experimentally the properties of rotations, reflections, and translations: [8-G1] 17. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. [8-G2]
18. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. [8-G3]
19. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. [8-G4]
Because tessellations are created using rotations, reflections, and translations, any of these objectives could apply to its use in the classroom. For example, objective [8-G4] can be extended to where a student is given a tessellation of multiple similar two-dimensional figures in a DGE, and asked to
describe what sequences could be used to create it. After making his assumptions, the student could then be asked to perform his guesses, and see if they hold and match the given tessellation.
In the geometry conceptual category of the high school mathematics curriculum, the importance of knowing transformations and their properties is clearly written:
“The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles, and therefore shapes generally. Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes—as when the reflective symmetry of an isosceles triangle assures that its base angles are congruent.”
Here, it is easy to see how complete understanding of transformations is essential in a high school geometry classroom, because it can help students understand further concepts such as
congruence and symmetry. Further in the course of study under the same category, it states, “Dynamic geometry environments provide students with experimental and modeling tools. These tools allow them to investigate geometric phenomena…”. Using a DGE such as Geometer’s Sketchpad, students can develop a more thorough understanding of transformations and, as a result, tessellations, because they are not inhibited as they are with a pencil and paper.
V. References
"Alabama Course of Study for Mathematics (2010)."Alabama Mathematics Course of Study. Alabama Department of Education, 2010. Web. 17 Sep 2012.
