Simplicity: few rules, big result
A shape or a pattern is simple if a short rule can make all of it. A circle is every point at the same distance from one centre. A rangoli with 8 petals is one petal copied 8 times. The shorter the rule, the more beautiful many people find the result.
Mathematicians love this too. A very short statement, like "the angles of a triangle add up to 180°", can explain an endless number of triangles.
Symmetry: mirror and rotation
Symmetry means a shape looks the same after you move it in a certain way.
- Mirror (line) symmetry: fold along a line and both halves match. A butterfly and the letter A have it. A square has 4 mirror lines, a circle has endless ones.
- Rotation symmetry: turn the shape about its centre by less than a full turn and it looks the same. The number of times it fits in one full turn is its order n. The smallest turn is 360° ÷ n. A starfish has order 5, so its turn is 72°. A snowflake has order 6, so its turn is 60°.
A shape can have only one kind. The letter S has rotation symmetry of order 2 but no mirror line. The letter E has a mirror line but no rotation symmetry.
Periodicity: patterns that repeat
A pattern is periodic if it repeats after a fixed distance or time. That distance is the period. A border of "flower, leaf, leaf, flower, leaf, leaf..." has a period of 3 items. A heartbeat, day and night, and the swing of a pendulum are periodic in time.
The most important periodic curve is the sine wave. One hump up and one hump down make one full period. If 3 full waves fit in 12 cm, the period is 12 ÷ 3 = 4 cm.
To find the item at any place in a repeating pattern, divide the place number by the period and look at the remainder.
Harmony: simple ratios and balance
Harmony means the parts fit together well. In maths this often shows up as a simple ratio. Two strings whose lengths are in the ratio 1 : 2 sound the same note an octave apart. A ratio of 2 : 3 sounds sweet too. Their waves line up again and again, so the ear finds the sound calm.
The same idea works for sight. The golden ratio, about 1.618 : 1, is a rectangle shape many artists find balanced. You can read more in The golden ratio and Mathematics in music.
Try it: fold and cut
Fold a square paper in half, cut a shape on the fold, and open it. You get mirror symmetry. Now fold it in half twice, and then once more diagonally. Cut and open: you get rotation symmetry. Count the order and check that the turn angle is 360° ÷ order.
Key formulas and definitions
- Angle of rotation = 360° ÷ n (n = order of rotational symmetry)
- A regular polygon with n sides has n mirror lines and rotation order n
- Period = length of one full repeat
- Item number in a repeating pattern: use the remainder of (place ÷ period)
- Simple ratios such as 1 : 2 and 2 : 3 give harmony
Worked examples
1. A regular hexagon: how many mirror lines, what is the order of rotation, and what is the smallest turn?
A regular hexagon has 6 sides, so 6 mirror lines and order 6. Smallest turn = 360° ÷ 6 = 60°.
2. Describe the symmetry of the letter H.
It has 2 mirror lines (one up-down, one left-right). It looks the same after a half turn (180°), so its rotation order is 2.
3. A sari border repeats "flower, leaf, leaf". What is the 20th item?
The period is 3. 20 ÷ 3 = 6 remainder 2. The 2nd item of the group is a leaf, so the 20th item is a leaf.
4. A wave has 5 full cycles in 20 cm. Find its period.
Period = 20 ÷ 5 = 4 cm.
5. A wave has 2 cycles and another has 3 cycles over the same length. How many times does the pair line up and begin again over this length?
They line up once at the start and once at the end. The two numbers 2 and 3 have no common factor except 1, so the joint pattern repeats once in this length. A small pair of numbers means a short, neat repeat.
6. A rangoli has rotation order 12. A friend turns it by 45°. Does it look the same?
The smallest matching turn is 360° ÷ 12 = 30°. A turn of 45° is not a multiple of 30°, so it does not look the same.
Common mistakes
- Saying a rectangle has 4 mirror lines. A rectangle has only 2 (a square has 4).
- Mixing up the order and the angle. The order n is how many times the shape fits in a full turn; the angle is 360° ÷ n.
- Counting a parallelogram as having mirror lines. It has none (only rotation order 2).
- Thinking "periodic" means "random". A periodic pattern repeats exactly after the period.