Mathematics Year 6: Rosettes: rotational symmetry and axes
Mathematics, Year 6 (Portugal): Finding the minimum rotation angle of a rosette, linking it to the number of axes of symmetry, and building a rosette by successive rotations.
A figure that repeats around a centre
A rosette is a figure made of equal copies of one motif, obtained by successive rotations about the same centre. If you turn the rosette so that it lands exactly on itself, that turn is a rotational symmetry. The smallest such turn is the minimum angle, and it equals 360° ÷ n, where n is the number of copies: 10 copies give 36°, 12 copies give 30°. Every other turn that leaves it unchanged is a multiple of it.
The minimum angle counts the copies and tells you how to build
The minimum angle is the key to a rosette for two reasons. First, you can read the number of copies from it: 360° ÷ 45° = 8. Second, it tells you how the rosette was built: the motif was turned by that same angle again and again, copy after copy, until the full 360° closed up. So when you analyse a rosette, measure the minimum angle and you can already explain how it was made.
A case: building a rosette with 5 copies
Motif: a triangle with one vertex at the centre O. You want 5 equal copies around O. Step 1: angle = 360° ÷ 5 = 72°. Step 2: turn the triangle 72° about O (2nd copy), then turn that copy another 72° (3rd), and so on: they end up at 72°, 144°, 216° and 288° from the original. Step 3: another 72° would make 360°, a full turn, landing on the original, so you stop. Turns that leave it unchanged: 72°, 144°, 216°, 288°.
Not every rosette has axes: the pinwheel
The reasonable mistake: thinking the number of axes of symmetry always equals the number of copies. In a rosette with reflection symmetry, the number of axes is n and the minimum angle is 360° ÷ n: 6 axes, 60°; 8 axes, 45°. But a pinwheel with 3 curved blades, all bent the same way, lands on itself every 120° and has no axis at all: a mirror would bend the blades the opposite way. First check whether there are axes; only then count.
Where you see it: windows, logos and programs
Rosettes appear in round windows of old buildings, in tiles, in wheel rims and in logos: find the centre and count the copies to get the minimum angle. A rosette with 12 copies has a minimum angle of 30°. In a block-based program such as Scratch, the recipe is short: repeat 12 times “draw the motif, turn 30°”. A sequence of steps that repeats is an algorithm.
Test what you learned
A rosette has a minimum angle of rotation of 72°. What does that tell you?
- a) Turning it 72° about its centre leaves it looking the same, and no smaller turn does
- b) It has 72 copies of the motif
- c) It has 72 axes of symmetry
- d) Any turn smaller than 72° also leaves it unchanged
Correct answer: a) Turning it 72° about its centre leaves it looking the same, and no smaller turn does — Right: that is what "minimum angle" means. Here 360° ÷ 72° = 5, so the rosette has 5 copies of its motif.
A rosette with reflection symmetry has 9 axes of symmetry. What is its minimum angle of rotation?
- a) 9°
- b) 20°
- c) 45°
- d) 40°
Correct answer: d) 40° — Right: 360° ÷ 9 = 40°. Check: 9 turns of 40° make 360°, a full turn.
A pinwheel has 3 identical curved blades, all bent the same way. Turning it 120° about its centre gives the same picture. How many axes of symmetry does it have?
- a) 3, one for each blade
- b) None, it has no axis of symmetry
- c) 6, one through each blade and one between blades
- d) 120
Correct answer: b) None, it has no axis of symmetry — Right: it lands on itself when turned 120°, but a reflection would bend the blades the other way, so there is no axis.
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