Faculty of Mathematics Centre for Education in

Waterloo, Ontario N2L 3G1 Mathematics and Computing

Grade 6 Math Circles

October 13/14, 2015
Tessellations

Tiling the Plane

Do the following activity on a piece of graph paper. Build a pattern that you can repeat
all over the page. Your pattern should use one, two, or three different ‘tiles’ but no more
than that. It will need to cover the page with no holes or overlapping shapes. Think of this
exercises as if you were using tiles to create a pattern for your kitchen counter or a floor.
Your pattern does not have to fill the page with straight edges; it can be a pattern with
bumpy edges that does not fit the page perfectly. The only rule here is that we have no holes
or overlapping between our tiles. Here are two examples, one a square tiling and another
that we will call the up-down arrow tiling:

Another word for tiling is tessellation. After you have created a tessellation, study it: did
you use a weird shape or shapes? Or is your tiling simple and only uses straight lines and

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polygons? Recall that a polygon is a many sided shape, with each side being a straight line.
For example, triangles, trapezoids, and tetragons (quadrilaterals) are all polygons but circles
or any shapes with curved sides are not polygons. As polygons grow more sides or become
more irregular, you may find it difficult to use them as tiles. When given the previous
activity, many will come up with the following tessellations.

Each tiling in the picture on the left uses only reg-

ular polygons. The adjective regular means that
each side in the polygon has the same length and
each angle has the same measure. In fact there are
only three tessellations that use only one regular
polygon as a shape. Can you think of a reason
why this is true, or how we would determine if it is
true or not?

Exercise: What are the names of the following polygons? Which ones are regular?

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Symmetry
Regular polygons are excellent for discussions of symmetry because they have a lot of it.
You may recall from school what symmetries are and which ones are present in shapes
like triangles, squares, or hexagons. Consider the simplest regular polygon, the equilateral
triangle. We can think about the symmetries acting on itself so that, when we perform the
actions, it will appear as the same shape. For example, we can reflect the triangle across the
altitude (in other words, the middle line dividing it in half).
You probably will have seem reflections before. A
reflection is like picking up the shape, flipping it,
and then setting it back down. A reflection al-
ways has a line that it will be reflected across. The
words in the diagram on the left are reflected across
the middle line of symmetry of the triangle. Two
more lines of symmetry exist for the triangle. Each
of these lines reflect the triangle so it appears the
same afterwords.
A triangle also has rotational symmetries as well.
We can rotate it in increments of 120 degrees about
its centre to return it to a similar position. To know
that it is 120 degrees, we can measure the angle
between rotations like in the diagram to the right.
We could also notice that we can rotate the triangle
three times before it has made a full rotation and
is in its original orientation. Since one full rotation
is 360 degrees, each of the three increments make
up 360 ÷ 3 = 120 degrees.

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There are three more types of symmetry we will learn about today.

Translations are simple: they move shapes in the path of a line (or, equivalently, up/down and
left/right some amounts). Glide Reflections are a combination of reflection and translation.
Finally, scaling is a way of taking shapes and making larger or smaller copies.

Using Symmetry as Instructions

There are multiple ways to turn each shaded shape into its grey counterpart using symmetry
(translations, reflections, rotations, glide reflections, and scaling). For each shape, come up
with two different lists of instructions to transform each shape using symmetry.

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Tessellations
We can use symmetry to create more complicated
tessellations First, we need to be able to identify
the symmetries in tessellations. Recall the up-
down arrow tiling; use symmetry to write instruc-
tions so someone else could create this tiling. When
you write your instructions, you do not need to ex-
plain the colouring of the shapes. Colouring does
not change the shapes in a tessellation.

What symmetries do the tessellations below contain?

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Notice that some of these tessellations look like they are made from smaller shapes. For
each tessellation above, if it is possible to build it from a smaller shape, name the shape or
shapes. Can we use a simple tiling to make a more complex one?

Escher Tessellations
To make these strange tessellations, I used regular tessellations and altered them based on
the type of symmetry they possess. What tessellations did I use? How do you think I used
symmetry to alter them?

M.C. Escher was a Dutch artist who could create amazing and surreal art. Often this art
was mathematical in nature. Here is a small part of his famous woodcut Metamorphosis II.

Figure 1: Escher, M.C. Metamorphosis II. 1940. Gallery: Selected Works by M.C. Escher.
www.mcescher.com. Woodcut. September 29, 2015.

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To construct some of his tessellations, Escher chose a regular polygon tiling and subtracted
a shape from its area and then added it to another side. What side he chose depended on
the symmetry he wanted to recreate.

Here is a beetle tessellation I created earlier. I chose

a beetle because I could imagine its pincers connecting
nicely with its abdomen. I chose a square tiling be-
cause its symmetry is fairly simple. Since creating the
pincers would also create the back end of the beetle,
I knew I needed to use the translative symmetry of a
square tiling so I could fit each beetle into the backside
of the other. In a square tessellation, only the right
side touches the left and only the top touches the bot-
tom. The same must be true for my beetle. Therefore,
anything I subtracted from the left needed to be added
to the right, or vice versa. Similarly, pieces from the
bottom should be moved to the top, or vice versa.

I then needed to draw the legs and that took a bit of

playing around to understand but eventually I settled
on the square you see at the top. Using your imagina-
tion you can see the pincers and the legs. The arrows
tell us the direction in which I move the pieces. For this
square, I only cut from the bottom or top and moved
to the other side, there are no left and right cuts. The
second picture is what the square looks like with pieces
moved and translated four times.

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Exercises

1. Name as many types of symmetries you can find in the following objects, or combina-
tions of symmetries:

2. Have a look at the room you are in and the objects it contains. Search the room for
symmetry, what do you find?

(a) List three objects that have symmetry. What type of symmetry do they have?

(b) Is there be a reason the items you found are symmetrical?

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3. Perform these operations on each of their shapes. Everything should fit.

1. Rotate 90 degrees clockwise about the grey dot. Translate 2 units to the right.

2. Translate 1 unit to the right. Rotate 90 degrees clockwise about the grey dot.

3. Reflect across the dashed line.

4. Scale the hexagon so that the new one has half the height. Translate 4 units right.

MATH. Reflect MATH across the dashed line below it.

6. Rotate 90 degrees counter-clockwise about the center of the shape. Translate

upward until the shape does not intersect the original.

7. Rotate 180 degrees about the base of the 7.

4. Use only rotations to turn the black shape into the grey one. You may use different
points of rotation. Draw each of your steps inbetween.

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5. A King and Queen have four children who will eventually inherit their lands. They
want to split up their land into equal portions so that their children will not fight for
more land. Find a shape that will tessellate their land perfectly into four territories.
Note: territories must be together and cannot be separated by other land; they also
must be the same shape.

Can you take the shape in your solution to the King and Queen problem and find a
tessellation? Use graph paper.

6. Here is a question from the 2006 Grade 7 Gauss contest:

The letter P is written in a 2 × 2 grid of squares as shown:

A combination of rotations about the centre of the grid and reflections in the two lines

through the centre achieves the result:

When the same combination of rotations and reflections is applied to , the result
is which of the following:

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7. Sydney is decorating her bathroom floor. She wants to use two types of tiles to make
her floor. Her choices are equilateral triangles, squares, regular hexagons, and regular
octagons.

(a) What are all of the combinations that will work to tile her floor? Remember, a
good bathroom floor has no gaps between the tiles.

(b) * Later, Sydney changes her mind and wants to use three tiles. Now there is only
one possible combination of tiles, what is it?

8. A vertex is a single point and often a meeting place for lines or corners. We can describe
tessellations made from regular polygons by how many fit around at least one vertex.
For example, here are three tessellations whose pieces are placed around a single vertex
(the vertex in white).

(a) How would you describe the three tessellations above using the vertex?

(b) * A very short way to label the above three tessellations is as follows:

4.82 3.122 (3.6)2

Can you figure out what these short labels mean?

(c) Find a vertex on each of the three regular polygonal tessellations (see page 2).
Use it to either describe them in words or give them a short label.

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 9. Here are three more of Escher’s creations. Determine what regular polygon tessellation
Escher began with to make these pieces. Did he choose rotational, reflective, or transla-
tive symmetry for each? Hint: Try drawing squares or triangles over the tessellations
and see which fits best then decided on the type of symmetry.

Figure 2: From left to right: Escher, M.C. Crab (1941), Winged Lion (1945), Bird (1941).
Gallery: Selected Works by M.C. Escher. www.mcescher.com. Pencil, Ink, Watercolour.
September 30, 2015.

10. (Practice makes perfect!) Make your own Escher Tessellation. I recommend starting
with either a square or triangle tessellation and choosing either translative or rotational
symmetry to start. If you get a handle on those and want to create more, try other
tessellations and other symmetries. YouTube is a great resource for more instruction.

11. * Can you name the reason why there are only three regular polygon tessellations of
the plane? You do not have to prove your reason is correct.

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