Starting with a single equilateral triangle, we can use rotations to get a bigger equilateral triangle.
C
Starting with a single square, pentagon, or hexagon, can we use rotations, translations, and reflections to fill the whole plane with no gaps or overlaps?
A
See solution.
B
See solution.
C
Square and hexagon
Practice makes perfect
A tessellation is a design in a plane that uses congruent figures to cover the entire plane with no overlaps or gaps. We are given an example of a tessellation using an equilateral triangle.
We want to explain how this tessellation is formed using reflections. Let's start with a single equilateral triangle.
Next, we will reflect the triangle three times, once over each side.
Note that the three new triangles, together with the original triangle, form a bigger equilateral triangle.
Let's reflect this bigger triangle three times, just like we reflected the original triangle.
Again, the images of the reflections, along with the preimage, form a bigger triangle. We can reflect this new triangle three times, creating another, even bigger triangle. We will repeat this process infinitely many times to cover the entire plane.
This time, we want to explain how the tessellation of an equilateral triangle can be formed using rotations. Once again, we start with a single triangle.
We will rotate the triangle 60^(∘) about each vertex.
This created another, bigger equilateral triangle. We can now rotate this triangle by 60^(∘) about each vertex, which will create an even bigger equilateral triangle. We can cover the entire plane by repeating this infinitely many times.
We are asked to determine which of the given regular polygons can be tessellated using a series of transformations. Let's consider each of the figures separately, starting with a square. We start with a single square.
Now, we rotate the square around one of the vertices three times. First, we rotate it 90^(∘), then 180^(∘), and, finally, 270^(∘).
This results in another, bigger square. We can then rotate this new square three times, creating an even bigger square. We repeat this an infinite number of times to fill the entire plane.
Next, we will consider the regular pentagon. Once again, we start with a single pentagon.
For the pentagons to cover the whole plane, each vertex of a pentagon must touch either a vertex of another pentagon or a side of another pentagon. Consider the first situation.
Here, the double vertex cannot be on a side of another pentagon because then there would be an overlap.
Then, a vertex of another polygon must touch the double vertex. This creates a space that cannot be filled by another pentagon without creating an overlap or a gap.
This is because the measure of every internal angle of a regular pentagon is 108^(∘), and it is not possible to create a full angle using 108^(∘) angles because 360^(∘) is not divisible by 108^(∘). Now, let's consider the situation where a vertex of the pentagon touches a side of another pentagon.
This also creates a space that cannot be filled by a regular pentagon without creating an overlap. Therefore, a regular pentagon cannot be tessellated using a series of transformations. Now, let's consider the regular hexagon.
