Patterns and tiling

A pattern is a design that repeats. A row of marigolds in a garland, the floor tiles in a temple, the print on a saree , they are all patterns. When shapes fit together with no gaps and no overlaps, we call that tiling or tessellation.

What is this?

Patterns are made by three actions:

• Repeat (translation) , copy a shape and slide it sideways.
• Flip (reflection) , copy a shape as if seen in a mirror.
• Turn (rotation) , copy a shape after rotating it by some angle.

A tile covers the floor when identical shapes fit together without leaving any gap. Squares tile perfectly because four squares meet at a corner and 4 × 90° = 360° , a full turn. Equilateral triangles tile because 6 of them meet at a corner (6 × 60° = 360°). Regular hexagons tile too , 3 hexagons meet at a corner (3 × 120° = 360°).

Circles do not tile , they leave little curved gaps.

Some patterns use two or more shapes together. For example, a kitchen floor may use squares and rectangles together. The trick is that the angles around any point still add up to 360°.

Look closely at a rangoli, a temple wall, or a hand-block printed fabric. The artist has chosen one small shape , the basic unit , and repeated it across the surface.

Worked examples

Example 1. Why do squares tile the floor perfectly?

• Each angle of a square is 90°.
• 4 squares meet at a corner → 4 × 90° = 360° = one full turn.
• No gap, so they tile.

Example 2. Will a regular pentagon (5-sided shape, each angle 108°) tile?

• 3 pentagons at a corner = 3 × 108 = 324°. Less than 360 , gap remains.
• 4 pentagons = 432°. Overlap. So no, regular pentagons don't tile.

Example 3. A pattern repeats every 4 cm. If a strip is 20 cm long, how many full repeats?

• 20 ÷ 4 = 5 repeats.

Try it

1. Will equilateral triangles tile? How many meet at a corner?
2. Will regular hexagons tile? How many at a corner?
3. Name 3 places in your home where you see patterns.
4. A border pattern repeats every 6 cm. How many repeats in 60 cm?
5. Draw a small repeating pattern in your notebook using only squares and triangles.

Hands-on

Square-paper tile. Take a sheet of square ruled paper. Colour every alternate square dark and light. You have made a chessboard tile pattern.

Rangoli mini. Draw a 4×4 grid on paper. Pick one small shape (triangle, half-circle, dot). Fill the grid by repeating the shape with flips and turns. Notice the pattern emerge.