The flower looks the same from many different angles. What about the butterfly? No doubt, the colours are very attractive. But what else about the butterfly appeals to you?
Answer
What appeals to us in the butterfly is that its left half and right half are exactly alike.
Fold the butterfly along its body and the two wings cover each other exactly.
If we fold the picture along the line through the body, the left wing falls exactly on the right wing — the same spots, the same shape, the same size. These are mirror halves.
Why it is different from the flower: the flower repeats when you turn it (rotation), while the butterfly repeats when you flip it (reflection). The butterfly has exactly one line of symmetry, and no rotational symmetry.
Q2.
In these pictures, it appears that some parts of the figure are repeated and these repetitions seem to occur in a definite pattern. Can you see what repeats in the beautiful rangoli figure?
Answer
Yes. In the rangoli, one quarter of the design repeats four times around the centre.
The red petals, the green leaves and the small blue drops all come back onto themselves when the rangoli is turned by 90° about its centre.
Turning again by 90° (that is 180°), again (270°) and once more (360°) also brings it back.
Angles of symmetry of the rangoli = 90°, 180°, 270°, 360°
The rangoli also has 4 lines of symmetry — one vertical, one horizontal and the two slanting lines through the centre.
Try This: Draw only one quarter of a rangoli on a square of paper. Fold the paper twice, cut, and open it — the full rangoli appears by itself.
Q3.
What about the pinwheel? Can you spot which pattern is repeating? Hint: Look at the hexagon first. Now, can you say what figure repeats along each side of the hexagon? What is the shape of the figure that is stuck to each side? Do you recognise it? How do these shapes move as you move along the boundary of the hexagon?
Answer
Start from the middle. The centre of the pinwheel is a regular hexagon (six equal sides), and a rhombus — made of two equilateral triangles joined along a side — is stuck to each of its six sides.
A regular hexagon with a rhombus (two equilateral triangles) stuck to each of its six sides, every rhombus tilted the same way round.
How the shapes move: as you walk round the boundary of the hexagon, each rhombus is the previous one turned by 60° about the centre. They all lean the same way, like the blades of a fan.
Angles of symmetry of the pinwheel = 60°, 120°, 180°, 240°, 300°, 360° (6 angles)
Important: because every blade leans the same way, a mirror would tip them the other way. So the pinwheel has no line of symmetry at all, yet it is beautifully symmetric — this is rotational symmetry, studied in Section 9.2.
Q4.
What are the symmetries that you see in these beautiful structures? (Taj Mahal and Gopuram)
Answer
Taj Mahal. Looking at it from the front:
There is one vertical line of symmetry right through the middle of the dome and the main archway. The left half and the right half — arches, small domes, and the four minarets — are exact mirror halves.
The water channel in front adds a second, horizontal mirror: the reflection in the water is an upside-down copy of the building.
Seen from directly above, the whole monument even has 4 lines of symmetry and turns onto itself every 90°.
Gopuram (temple gateway tower).
It also has one vertical line of symmetry — the left and right sides of every storey match exactly.
Going up, the same row of carved figures repeats storey after storey, each one a little smaller. This is a repeating pattern (translation), which is why the tower looks so orderly.
Math Talk: Look at photographs of the Qutub Minar, the Konark wheel, the Charminar and a temple kolam. For each one ask two questions — “Can I fold it?” and “Can I turn it?”