Tessellations Using One Shape

Explore Shapes: Gap or No Gap?

Goal: Why do certain shapes fit together while others leave gaps? Material: a set of physical regular shapes or digital shapes (triangles, squares, pentagons, hexagons, and circles). Links are provided below. Task:

Try to cover a whole flat surface with only one shape. You may not overlap and leave no gaps.
You may use the digital resource below if you do not have physical / concrete regular shapes.
Observe which shapes work easily, and which shape is impossible. Why?

Concept Focus: For a shape to tessellate, the sum of interior angles meeting at a point must be exactly

Digital Resource 1: Regular Shapes

Digital Resource 2: Tessellation Creator

Exploration

By now, you would have noticed that for regular polygons, only equilateral triangle, square, and hexagon work well to create tessellations. Let's explore other kinds of shapes that can tesselate:

What types of irregular polygon can tessellate?
What types of quadrilateral can tessellate?
What types of triangle can tessellate?
What types of non-polygonal shapes can tessellate?

Use a blank graphing calculator to create tessellations that meet the criteria above. Can you build a flowchart to summarise the types of shapes that can be tessellated by using a single type of shape.

Configuration: Naming Tessellations

A tessellation is named by choosing a vertex and counting the number of sides of each shape touching the vertex. These numbers are then listed in order (clockwise / counter-clockwise) starting with the polygon with the least number of sides. Since this topic only cover single shapes, the configuration will have same numbers. Example:

Tessellation consist of 4 squares. Each square has 4 sides. Hence, the configuration is 4.4.4.4

Tessellation consist of 6 triangles. Each triangle has 3 sides. Hence, the configuration is 3.3.3.3.3.3

Tessellation consist of 3 hexagons. Each hexagon has 6 sides. Hence, the configuration is 6.6.6

How to Identify a Tessellation?

In summary,

Identify the repeating unit (motif)

Look for the smallest individual shape or a cluster of shapes that repeats. To be a tessellation, this motif must be able to fill the entire plane without leaving any gaps or overlapping.

Search for Isometries (Transformations)

Translation: Does the pattern slide in a straight line to repeat? Rotation: Does the pattern pivot/ rotate around a point? Reflection/Glide Reflection: Does the pattern flip across an axis / line? A pattern is only a tessellation if these movements can continue infinitely in all directions to cover the surface.

The Vertex Path (The 360° Rule)

Locate a point where the corners of the shapes meet (vertex). Calculate the sum of the interior angles of all shapes meeting at that single point. The sum must be exactly 360°. Note: If the sum is less than 360°, there will be a gap. If it is more than 360°, the shapes will overlap.