This unit examines tessellations, that is, ways of covering the plane with copies of the same shape so that there are no gaps or overlaps. Students will investigate what properties tessellating shapes must have in order to cover the plane with no gaps and with no overlapping. The tessellations investigated involve both non-regular and regular polygons.
- demonstrate why a given tessellation will cover the plane
- create regular tessellations
Tessellations are all over the place but especially in the kitchen and bathroom on tiles and lino. Occasionally you can see them in the living room as the basis of the pattern on carpets and in parquetry wooden floors. Tessellations are a neat and symmetric form of decoration. They also provide a nice application of some of the basic properties of polygons.
They also have other, practical uses. Brick walls are made of the same shaped brick repeatedly laid in rows. Bees use a basic hexagonal shape to manufacture their honeycombs. The brick wall provides a tessellation with rectangles and the honeycomb is a tessellation of regular hexagons. These tessellations provide a strong structure for their two different purposes.
The key features of tessellations are that there are no gaps or overlaps. The same figure (or group of figures) come together to completely cover a wall or floor or some other plane. This requires the vertices to fit together. This can be done in two ways. Either the corners of the basic shape all fit together to make 360° , or the corners of some basic shapes fit together along the side of another to again make 360° . So a precursor to this unit is a lesson or series of lessons that give the class a sound knowledge of angles in degrees. You might use Measuring Angles, Level 3 for this purpose.
You might find it useful too to have done the Problem Solving lesson Copycats, Geometry, Level 3 or it could used as part of the Exploring stage of this unit.
Following on from this unit is Fitness, Level 4. This goes on to looking at both regular and non-regular tessellations. In the regular case it shows that regular tessellations can be made only with equilateral triangles, squares and regular hexagons. Semi-regular tessellations involve two or more regular polygons. Fitness investigates the possible non-regular tessellations.
Related to the idea of tessellations is that of Escher drawings. There is a unit on that at Level 4, Tessellating Art, though some of the concepts there would be accessible to students at Level 3.
Copymaster 1 (picture showing brick wall)
Copymaster 2 (a sheet of 2 x 1 bricks)
Copymaster 3 (a sheet of regular polygon shapes from equilateral triangles up to dodecagons)
Copymaster 4 (a sheet of 4-sided figures: two different rectangles, two different parallelograms one which tessellates and one which does not
School Journal Part 4, Number 1, 1993, Tessellations
Copymaster 5 (a sheet of three types of pentagons, two of which tessellate and one which does not
regular polygonal shapes
scissors, paper, rulers, pencils, protractors, Lego
- Talk to the students about buildings and how they are constructed.
What material is used to construct buildings? (wood, bricks, mortar, sand, cement, etc.)
How do builders put a house together?
How do they build walls?
- Get the students to describe the brick pattern on walls. Show them Copymaster 1. Talk about the features of the pattern. (There are no gaps; the bricks are in rows; each row is slightly displaced from the row below.)
Why do you want to have no gaps in a wall?
(You might note that some walls do have gaps. Why?)
- Are there any other ways of using bricks to build a wall?
- Get the students to produce their own (2-dimensional) bricks. The bricks should be rectangles that are twice as long as they are wide. This can be done using graph paper and then tracing round one to produce many. They could also draw a number of bricks on graph paper and use the photocopier to produce 20 or so. Then cut up the photocopied bricks.
(We have provided Copymaster 2 to photocopy if you would prefer students not to make their own bricks.)
- Ask the students to produce a wall using the bricks that they have just produced. They may copy the pattern in 1. But they may also arrange them in any pattern they like. However, if they think up their own patterns then there are two important rules that have to be obeyed: (i) there are to be no gaps and (ii) the pattern must be capable of being continued indefinitely (to cover a very big wall!).
- Go round and work with the students. Once they have one brick pattern, see if they can find others. (The bricks don’t always have to be ‘horizontal’. The pattern doesn’t have to be practical. It’s OK if the pattern won’t give a very strong wall.)
- Bring the class together and let the students discuss the patterns that they have made. The students should be able to justify that the patterns have no gaps and that they can go on for ever in all directions. Help them with their explanations if necessary, as these two rules are a key part of this unit.
- Then tell them that they have been making tessellations. A tessellation is a way of using a fixed shape to cover the whole of a flat surface (a plane). Tessellations have two important properties: (i) they have no gaps (all of the plane is covered) and (ii) they go on for ever (no matter where you go in the plane the shapes will still be covering the part of the plane that you can see. Sometimes we call a tessellation a tiling.
Session 1: Regular Polygons
In this session the students will explore tessellations by regular polygons. If you haven’t got any solid regular polygons handy then they can be cut using the pattern in Copymaster 3.
- Ask the students to recap the work of the last lesson.
What did we do in the last lesson?
What shapes did we use?
What word did we learn? (tessellation or maybe tiling)
How many rules does a tessellation have? What are they?
- Today we are going to look at special types of shapes called regular polygons. (If necessary remind them what it means for a polygon to be regular – all sides equal, all interior angles equal.)
- What we want to do is to see which of these shapes tessellate and which don’t. We’ll then construct this table together.
- Let the students tackle the problem in any way that they like. As you go round their groups check that they understand that there are to be no gaps and that the patterns must continue forever in all directions. Check too, that if they can’t find a tessellation, they can explain why one doesn’t exist. (It isn’t good enough to say that they have tried for 5 minutes and they can’t find one. They need to be able to justify why there have to be gaps or why the pattern can’t go on for ever. It should usually be the first reason. This is fortunate, as it is easier to justify.)
- Those groups who finish early could be asked to see if they could find some different tessellations using the same shapes.
- Bring the class together to discuss their results. After each group has reported and justified their claims add them to the table. A completed form of the table is given below. The angles talked about in the third column are the interior angles of the polygons. The students can find these by directly measuring the angles concerned.
show pattern; 6 x 60° = 360° so they fit.
show pattern; 4 x 90° = 360° so they fit.
no multiple of 108° makes 360° nor do multiples of 108° plus 180° .
show pattern; 3 x 120° = 360° so they fit.
no multiple of 128.57° makes 360° nor do multiples of 128.57° plus 180° .
no multiple of 135° makes 360° nor do multiples of 135° plus 180° .
no multiple of 140° makes 360° nor do multiples of 140° plus 180° .
no multiple of 144° makes 360° nor do multiples of 144° plus 180° .
no multiple of 147.27° makes 360° nor do multiples of 147.27° plus 180° .
no multiple of 150° makes 360° nor do multiples of 150° plus 180° .
- Ask the class if they have a conjecture about tilings by regular polygons.
Session 2: Quadrilaterals
In this session the students try to extend their results about regular polygons to more general polygons.
- Get the class to recall what happened in the last session. The key points are (a) what is a tessellation; (b) tessellations were found for only three types of regular polygons; and (c) which types?
- Can you tell me which shapes we now know will tessellate? (The three regular polygons and the rectangle – from the brick patterns.) We know that two sorts of 4-sided figures will tessellate. What other sorts of quadrilaterals will tessellate?
- Provide the students with various quadrilaterals (see Copymaster 4) and let them experiment. A table can again be constructed like the one in Session 1.
- If possible give the students the opportunity to experiment for themselves with other quadrilaterals. (For instance the students who finish quickly and can justify what they have done should go on to inventing quadrilateral tilings of their own.) Remind them that there are two things that can be varied – the lengths of the sides and the sizes of the interior angles. Suggest that they vary the interior angles first.
- Let the students report back on what they found. What conjecture do they have about tessellations by quadrilaterals? Can they justify that conjecture?
The students will discover that any quadrilateral will tessellate. Because the interior angles of any quadrilateral add up to 360° , we need to put four figures together at a point so that each one of the four (possibly) different angles is used. In this pattern, sides of equal length also have to fit together. The pattern below shows how to do this.
So we have shown that we can fit four of these polygons together at a vertex without gaps. How can we be sure that we can continue the pattern indefinitely? If you look you can see that the pattern in the figure is made up of a wiggley strip of quadrilaterals. In this strip, one quadrilateral is placed one way and then it’s placed another. These wiggley strips can put side by side for ever. So what you see in the drawing is what you would see anywhere in the plane.
Notice that it doesn’t matter whether the quadrilateral is convex (no interior angles bigger than 180° ), as above, or concave, as below.
Session 3: Triangles.
If all quadrilaterals tessellate is the same true for all triangles? Repeat the last session but this time use triangles.
This is easier to establish than the fact that all quadrilaterals tessellate. Since the interior angles of a triangle add up to 180° , we again need to make sure that each angle is represented at a point. Here they make a straight line. This means that we can put the triangles together to make a row. We can fit two such rows together. In fact we can fit as many rows together as we like until the entire plane is covered. This is shown in the diagram below.
Session 4: Pentagons
Review the facts: all triangles tessellate; and all quadrilaterals tessellate. What do we know about regular polygons? (They don’t tessellate.) Does that mean that no pentagon tessellates? Ask the students to experiment using the pentagons from Copymaster 5 and by inventing their own pentagonal shapes. What results can you find?
If the students can’t find any pentagonal tessellations, you might remind them how they constructed the triangular and quadrilateral tessellations. Rows were very important. So it might be an idea to try to use rows in some way to try to form pentagonal tilings.
The situation with pentagons is more complicated than with either triangles or quadrilaterals. Some pentagons do tessellate and some do not. From the evidence of the regular pentagon it is unlikely that all the interior angles of a pentagons could cluster around a similar point in a similar way. That means that there have to be at least two types of points where pentagons come together. So we have to distribute the sizes of the interior angles so that this can happen. This enables us to get tilings like the one below.
- Discuss with the class the results of the last few days.
What have you discovered?
What is a tessellation?
What polygons do you know that tessellate?
What polygons definitely do not tessellate?
- Now we know that all triangles tessellate, that all quadrilaterals tessellate and that regular hexagons tessellate.
What questions do you think that we ought to ask now? (Do you think that all hexagons tessellate? What do you think? Why?)
- Experiment for yourselves. Make your own hexagonal shapes and come up with a conjecture.
- After they have tried this for a while get them to report back on their conclusions.
- Revisit their conjecture about regular tessellations. How could you prove this?
Clearly there are other hexagons apart from regular hexagons that do tessellate. We show one example below.
But if we can arrange for the angles of a hexagon to be such that no combination of them will add up to 360° (or 180° ), then it won’t be possible for that hexagon to tessellate.
The conjecture, the only regular polygons that tessellate are the equilateral triangle, the square and the hexagon, is proved in Fitness, Level 4.