At-Home STEM Activities: Tessellations—Exploration and M.C. Escher-Inspired Drawing
A regular polygon is a shape where each side is the same length and each angle is the same size. Here are a few examples:
We can study the way regular polygons interact with each other, and one way they can do so is through tessellations. A tessellation, also called a tiling, is a way to cover a surface with a repeating pattern of flat shapes such that there are no overlaps or gaps. A good example of a tessellation is actual tile, like what you would find on a bathroom floor.
A regular tessellation is one made using only one regular polygon. A semi-regular tessellation uses two or more regular polygons. Triangles and squares, for example, form regular tessellations and octagons and squares for a semi-regular tessellation.
Regular tessellation of triangles
Regular tessellation of squares
semi-regular tessellation of octagons and squares
A tessellation can be described by the shapes that meet at each vertex, or a corner point. In a tessellation, the shapes that appear at every vertex follow the same pattern of shapes.
each vertex has 6 triangles around it, so we can call this tessellation 3-3-3-3-3-3
each vertex has 1 square and 2 octagons, so we can call this tessellation 4-8-8
Let’s explore regular and semi-regular tessellations a little more. For this activity, you’ll need:
Scissors
The design from the button below, printed on cardstock
Note: You’ll be able to find the tessellations with only one copy of this print-out. For younger learners, it may be easier with more copies of each shape, so we recommend printing two copies.Colored pencils or crayons (optional)
If you want, color the shapes with crayons or colored pencils, then cut out the regular polygons.
We know that triangles and squares form regular tessellations. What about pentagons, hexagons, and octagons?
We know that octagons and squares form a semi-regular tessellation. What other combinations of the regular polygons that we have can form a semi-regular tessellation?
Try to find all the shapes that form a regular tessellation and all the combination of shapes that form a semi-regular tessellation. When you think you’ve found it, scroll down to see the answer.
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All regular tessellations:
triangles (3-3-3-3-3-3)
squares (4-4-4-4)
Hexagons (6-6-6)
All semi-regular tessellations:
Triangles and squares (3-3-3-4-4)
Triangles and hexagons (3-6-3-6)
Triangles and Squares (3-3-4-3-4)
Triangles, Squares, and Hexagons (3-4-6-4)
Triangles and hexagons (3-3-3-3-6)
Squares and Octagons (4-8-8)
These are all the tessellations we can form with the shapes we have. Notice that no tessellations involve the regular pentagon. In fact, there are no other regular tessellations and only two other semi-regular tessellations: 3-12-12 and 4-6-12. (A 12-sided polygon is called a dodecagon)
Let’s think about why that is.
We already talked about tessellations being classified by the pattern of shapes that appear around each vertex within the tiling. Let’s take a closer look at those shapes, and, in particular, the size of the angles of each shape:
This includes the two tessellations with the dodecagons
Now add up the size of the angles for each tessellation. What do you notice?
All the angles add up to 360°! And so we see that to tessellate, the shapes involved need angles that add up to 360°. That’s why only certain regular polygons form tessellation and why there is a finite number of tessellations that can be formed by regular polygons—only regular triangles, squares, hexagons, octagons, and dodecagons have interior angles that can combine to add up to 360°.
Can irregular polygons form tessellations?
Yes! Consider the following examples:
Rectangles
l-shapes and rectangles
Different pentagons
Rhombuses
Tessellations show up in art and architecture throughout the world. The Alhambra palace in Grenada, Spain is almost entirely decorated with geometric, Arabic tiling.
image via alhambradegranada.org
image via alhambradegranada.org
Another place in art to see tessellations is in the work of M.C. Escher. Escher was actually inspired by the geometric figures covering the walls and floors of the Alhambra. He created mathematical woodcuts, sketches, and lithographs that utilized tessellations and symmetry. Today, M.C. Escher is one of the world’s most famous graphic artists.
via mcescher.com
via mcescher.com
via mcescher.com
Let’s create our own tessellation drawings like M.C. Escher!
M.C. Escher-Inspired Drawings
Materials:
Thin cardboard (like from a cereal box) cut into a rectangle
Scissors
Tape
Pencil
Colored pencils, crayons, markers or other coloring material of your choice
Instructions:
1. Draw a design on either the top or bottom and either the left or right of the rectangle.
2. Cut out the designs and move them to the opposite side of the rectangle. Tape together.
3. Trace your stencil on the piece of paper to tessellate your design.
4. Color your tessellation any way you’d like, with your favorite colors or with funky patterns. Maybe the shapes remind you of your favorite animal—we thought ours looked like chickens—make your tessellation look like that!
