Covering the Plane with Rep‑Tiles

• Lesson
6-8
2

Students discover and explore a special kind of tiling of the plane. Rep‑tiles are geometric figures such that n copies can fit together to form a larger, similar figure. Students experiment with various shapes and values of n. Spatial sense is encouraged by the need to visualize and perform transformations with the shapes involved. This lesson was adapted from an article by Linda Fosnaugh and Marvin Harrell, which appeared in the January‑February 1996 edition of Mathematics Teaching in the Middle School.

Activitating Prior Knowledge

Begin the lesson by asking students where they have seen examples of tiling. Some common uses of tilings include the covering of walls, floors, ceilings, streets, sidewalks, and patios.

Background Information about Rep‑Tiles

A tiling is a partitioning of the plane into regions, or tiles, as suggested by Grunbaum and Shephard (1987). It is commonly known that any triangle, quadrilateral, and regular hexagon will tile the plane. Golomb (1964) suggests an unusual type of tile, a rep‑tile, that tiles the plane.

A rep‑tile is a geometric figure whose copies can fit together to form a larger similar figure. Another way that one can think of a rep-tile is as a puzzle piece, where a larger similar figure is the entire puzzle. For example, it is well known that four congruent squares fit together to form another square. The smaller square, the puzzle piece, is a rep‑tile. In addition, four copies of the large square can be fitted together to form an even larger square. By repeating this process infinitely many times with still larger squares, we can tile the plane.

Another example is the triangle. If one makes copies, or replicas, of any triangle, four of these copies can be fitted together to form a larger triangle similar to the original triangle. One could continue this process with the larger triangle and then again with still a larger similar triangle. Again, if this process is continued infinitely many times, the plane will be tiled. (a) Note that four congruent triangles create a similar figure; thus we call this tile a rep‑4 tile. Similarly, as can be seen in (b), four copies of any parallelogram fit together to create a similar parallelogram. Hence, any parallelogram is a rep‑4 tile.

 (a) (b) Two Rep‑4 Tiles: a Triangle and a Parallelogram

In the figure below, additional rep‑tiles are illustrated. The first image (a) shows a rep‑2 tile formed by joining two isosceles right triangles, whereas (b) illustrates a rep‑3 tile in which three 30o-60o-90o triangles are fitted together to form a larger 30o-60o-90o triangle. The sphinx shape (c) is made up of six equilateral triangles and is a rep‑4 tile. Nevada (d) is formed by attaching an isosceles right triangle to a square and repeating this shape four times, and in figure (e) we have a right triangle whose sides measure 1, 2, and the square root of 5. As one can see from the two figures above and the five figures directly below, although simple examples of rep‑tiles exist, one quickly finds oneself working with more complex geometric figures.

 (a) Rep‑2 tile (b) Rep‑3 tile (c) Rep‑4 tile (d) Rep‑4 tile (e) Rep‑5 tile Additional Examples of Rep‑Tiles

It should be noted that a rep‑n tile exists for any natural number n > 1. In other words, for any natural number n > 1, a tile exists in which n copies can be fitted together to create a larger similar figure.

It is interesting to note that the rep‑4 triangle previously discussed is also a rep‑9 triangle. We can show this fact by simply adding a row of five small congruent triangles to the base of the large triangle found in the initial figures, and also (a) below. One interesting fact about rep‑tiles is that each of our rep‑4 tiles is also a rep‑9 tile. Conversely, each of the rep‑9 tiles is also a rep‑4 tile. The second and third images (b) and (c) below also illustrate this fact.

 (a) (b) (c) Rep‑9 Tiles that were shown earlier to be Rep‑4 Tiles

Rep‑tiles not only present an alternative way to tile the plane but also help students acquire spatial sense.

Introducing the Activity

After the class reviews such concepts as tilings (tessellations), congruent figures, and similar figures, the teacher can introduce rep‑tiles using squares, equilateral triangles, and parallelograms. For example, pattern blocks on an overhead projector can be used to illustrate rep‑tiles. Otherwise, cutouts from colored transparency sheets or cardboard figures work just as well. After having the students form small groups, the teacher can then distribute copies of the Rep‑Tiles Activity Sheet.

After students have worked through the activity sheet, a class discussion summarizing their findings is beneficial. Students should realize that any of the similar figures, no matter their size, can be used to tile the plane.

Conclusion

With the introduction of rep‑tiles in a classroom setting, students can visualize and represent geometric figures with special attention to developing spatial sense as well as explore transformation of geometric figures. Creating a tiling pattern requires such mental imagery as visualizing the possible rotations and placements of a tile in the tiling pattern, thus further developing spatial sense. In addition, the construction of a tiling with rep‑tiles allows students to see how congruent copies of a polygon can be fitted together to form a larger similar copy of the polygon. After discovering the pattern, the students may see that each tile can be dissected into smaller, similar copies of itself, thus discovering how any of the similar figures can be used to generate an entire tiling.

Assessment Options

1. Students may be assessed by their responses to Items 1 - 8 on the Rep‑Tiles Activity Sheet.
2. For an individual project, students can pick their favorite rep‑tile and create a poster illustrating a tiling of the plane. Some of the more interesting posters are those in which students color their tile as a reptile. This project not only reinforces the notion of rep‑tiles but also aids students in making a connection between mathematics and certain principles of art.

Extensions

1. Assign items 9 and 10 on the Rep‑Tiles Activity Sheet to students who may finish early or need an extra challenge.
2. Students may also wish to experiment with creating tilings on the computer. The figures below provide examples of computer-generated tilings.
 (a) (b) (c) Computer-Generated Tilings

Questions for Students

1. What geometric concept can rep-tiles be compared to?

[Tessellations.]

2. What are rep-tiles?

[A rep‑tile is a geometric figure whose copies can fit together to form a larger similar figure. Another way that one can think of a rep-tile is as a puzzle piece, where a larger similar figure is the entire puzzle.]

3. What polygons did you use to create your rep-tiles?

[Students may suggest any of the following polygons: isosceles right triangles, 30o-60o-90o triangles, equilateral triangles, parallelograms, trapezoids, squares, and so on.]

Learning Objectives

Students will:

• Precisely describe, classify, and understand relationships among types of two-dimensional objects using their defining properties.
• Examine the similarity of objects using transformations.
• Identify and create rep-tiles using familiar polygons.

NCTM Standards and Expectations

• Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.
• Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.