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

1 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?

2 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.

3 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.

4 Answers may vary. Here are some solutions.

1. Translate 7 units right. Rotate about bottom left corner 90 degrees counter-clockwise.

2. Rotate 135 degrees counter-clockwise.

3. Scale to twice the height and width. Translate 4 units to the right.

4. Reflect across a horizontal line exactly 1.5 units up from the bottom of the grid. Translate 4 units to the right.

5. Reflect across the bottom right side of the hexagon (the side that touches the new hexagon).

6. Rotate 90 degrees clockwise about the bottom left bulge of the shape. Reflect across a vertical line from the top of the shape to the bottom (just flips the shape in place). Translate 1 unit right and 3 units up.

7. Reflect across the line (not a grid line, but half-way between grid lines) that is exactly half-way between the bases of the sevens.

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.

5 What symmetries do the tessellations below contain?

The types of symmetry present in each tessellation are:

1. Translative, reflective, rotational, glide reflective. No scaling.

2. Translative, rotational, glide reflective. No reflection, no scaling.

3. Reflection and translation.

4. Scaling.

5. Translative, reflective, glide reflective.

6 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.

7 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.

8 In the third image you see on the previous page, I have added a bit of detail to make the beetle more realistic. One should be very careful bending the straight lines in our tessellation because it may be that our finished creation does not fit together afterwords! I used symmetry to accomplish this task as well. The only pieces that could affect how the beetles fit together are perimeter lines of the square. Fortunately, since I cut pieces only from the top and bottom perimeter, I do not need to worry about them having poor connections (since they do not touch another beetle). So the left and right perimeter that make up the hind legs are what concern us. Since we are using translations, I am required to draw exactly the same line on the left side of the square as the right. This ensured that any pimples on that line would be dimples on the right line.

Think about how the lines would change if I wanted rotational or reflective symmetry on a square. Does this identical line rule always work?

Any of the lines that are inside the square, I can simplify my identical lines rule by just changing the line before I cut out the pieces. This way, my cut will produce two lines that fit together perfectly.

9 Exercises

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

Here is one type of symmetry for each object:

(a) Reflective.

(b) Rotative.

(c) Rotative.

(d) Reflective.

10 (e) Scaling.

(f) Glide reflective.

(g) Rotative.

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?

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.

11 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. Solutions may vary.

12 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.

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:

The solution is (b). The symmetry instructions are: rotate 90 degrees counter-clockwise and then reflect across the horizontal grid line.

13 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.

The pairs are: (Squares, Octagons), (Triangles, Squares), and (Triangles, Hexagons).

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

The triplet is (triangles, squares, hexagons).

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?

Two octagons, one square about a vertex. Two dodecagons, one triangle about a vertex. Two triangles, two hexagons about a vertex.

14 (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?

Each number stands for a shape with that many sides. The exponent tells us how many of each number (shape with that number of sides) we have. The period separates shapes. The brackets in the rightmost tessellation indicate we have two of both of the shapes, since the exponent is outside of the brackets.

(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.

Six triangles about a vertex or 36. Four squares about a vertex or 44. Three hexagons about a vertex or 63.

15 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.

Crab is made from squares and uses translative and reflective symmetry. Winged Lion is made from squares and uses reflective symmetry. Bird is made from triangles and uses rotational symmetry.

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.

If we place polygons around a single vertex (see Question 4 ), only triangles, squares, and hexagons have internal angles that have 360 as a multiple.

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