What are you going to learn?

to use the principle of tessellation to obtain a good design

to identify a basic pattern of a tessellation (if any) from a particular design

Key term: regular tessellation

semi regular tessellation

non-regular tessellation

naming tessellation

1. Definition of Tessellation

Look at the picture on the right.

Where can we find such an object?

What kind of a geometrical shape forms this design?

How many geometrical shapes form this design?

If we notice our surrounding, many objects are formed by repeating a shape. A honeycomb is one of the examples.

Is there a hole amongst the shapes? Is there an overlapping shape?

Such a shape replicating a honeycomb is called a tessellation.

The following figures exemplify particular shapes which can be constructed into regular tessellations. There are only three regular tessellations:

Example 1

A tessellation is a shape or tile that repeats to fill a surface without any gaps or overlaps.

1.1.44

Discuss with your classmates whether a combination of 2 or 3 shapes in the first example can form a tessellation.

Notice that each vertex of patterns constructing a tessellation has a common vertex, 6 triangles forming a hexagon and 4 squares forming a new square. Are there other regular-n-shapes?

To answer the above question, complete the following table:

Regular Shapes

The size of the angle

Factor of 360?

TriangleSquarePentagonHexagon

60......................................................

Yes...............................................................

one of triangles

one of squares

one of hexagons

Problem 1

A regular tessellation is a tessellation that is constructed from a regular-n-shape.

...

Can a regular pentagon form a tessellation? What about with other regular-n-shapes? What is the relation of the tessellation contruction to the results of

the factor in the table?

Although not all regular-n-shapes can form a tessellation, we can combine some regular polygon tessellations to construct a tessellation which will be discussed in the next section.

2. Naming Tessellations

A tesellation of squares is named 4.4.4.4. Choose a vertex, and then look at one of the polygons that touches that vertex. How many sides does it have?

What is the name of the following tessellation?

Notice that this tesellation is constructed of squares. Pick up one of the squares. How many sides does it have? Since it is a square, it has four sides. That is where the first square comes from. Now keep going around the vertex in either direction, finding the number of sides of the polygon until you get back to the polygon you started with. Since there are four polygons, and each has four sides, its name is 4.4.4.4.

Not all regular-n-shapes form a tessellation.

Example 2

Explain how we name the following tessellations.

3. Semi-regular Tessellations Up to this point, the tessellations are constructed only of one regular n-shape. Now, we can also use a variety of regular polygonal tessellations, known as semi-regular tessellations.

A semi-regular tessellation has two properties, namely: a. It is constructed of regular polygons. b. The arrangement of polygons at every vertex is identical.

Look at these two tessellations below.

Example (a) is a combined tessellation of triangles and squares, whereas example b) is a tessellation resulting from the combination of octagons and squares.

a)

b)

Problem 2

Example 3

How to name semi-regular tessellations follows ways which have been used for the regular tessellations. The following are some examples and how to name them.

Determine the names of the following semi-regular tessellations.

4. Non-regular Tessellations

Many tessellations can be made by combining regular and non-regular

polygons. In its later development, tessellations can be made by using

special shapes, which have no common points, and by using shapes with

curved lines. Some examples are given below.

Tessellations can be regular, semi-regular, and non-regular.

Example 5

Problem 3

Example 4

A tessellation can be formed from a simple shape. Some tessellations are

formed by combining two or more shapes. Many beautiful designs can be

made by using both regular and non-regular tessellations.

1. In your own words, write the steps in naming a tessellation. Use those

steps to name a tessellation.

2. Name the following tessellations.

3. Can a regular pentagon form a tessellation? Prove it.

4. Check if the folowing combinations can form tessellations.

5. Draw tessellations whose names are as follows. Colour them with your

favourite colour.

a. 3.3.3.3.3.3

b. 4.4.4.4

c. 6.6.6

d. 3.3.3.4.4

e. 3.3.4.3.3

f. 3.4.6.4

g. 3.6.3.6

h. 4.8.8

i. 4.6.12

j. 3.3.3.3.6

k. 3.12.12

6. Identify the basic shape that forms the following tessellations.

a) b)

c) d)

e) f) g)

7. Use the following shapes to make tessellations.

8. Find different kinds of tessellations, both natural and artificial ones, in our surroundings.

Identify the basic shapes and colour them.

9. A unique tessellation can be made by changing the shape. A rectangle, for example, can

be modified but the alteration must be balanced. If there is an added part, there must be a

reduced part at the opposite.

Example:

Copy and complete the following rectangles. Draw the final shapes. Then show the

tessellation shapes.

10. Create a tessellation of your own and then colour it.