Symmetry · Lesson 6 of 7
Symmetry in Designs and the World
“Combine reflection and rotation to analyse coloured patterns, buildings, wheels and a strategy game.”
• Classify figures with reflection symmetry, rotational symmetry, both or neither. • Use angle clues to infer possible smallest matching turns. • Explain how sector colours and tiles change a design’s symmetries. • Relate symmetry counts to regular polygons, snowflake patterns and equal spokes. • Construct coloured designs with exact mirror-line conditions. • Explain a half-turn reply strategy for the adjacent-squares game.
Two questions about the same figure
A figure can preserve its appearance under a fold, a turn, both or neither. These are separate tests. A mirror line reverses the sides of the figure across that line; a rotation moves the parts around a centre without flipping them. Learn to ask each question explicitly rather than using the general word symmetric as a complete answer.
Problem
Which familiar shapes illustrate both kinds, just one kind, or neither?
- 1.A nonsquare rectangle has two mirror axes and a 180° matching turn, so it has both reflection and rotational symmetry.
- 2.A general slanting parallelogram, with unequal adjacent sides and no right angles, has a half-turn match but no mirror line. A symmetric trapezium with unequal bases has one mirror line but no smaller matching turn.
- 3.A scalene triangle has neither property. An equilateral triangle has both: three mirror lines and turns of 120°, 240°, 360°. An isosceles triangle with exactly two equal sides has one mirror line but no smaller matching turn.
An equilateral triangle also answers the request for a triangle with at least two mirror lines and at least two matching angles. Exactly two mirror lines are impossible, but at least two includes three. Read words such as exactly, at least and smallest carefully: they express different conditions.
Use an angle clue to recover a repeating step
The angles of a finite-order figure are multiples of its smallest positive matching turn. This lets us work backwards from a list or from a count of earlier matches. First determine what the clue guarantees. Being a matching angle does not automatically make an angle the smallest match.
Problem
A figure’s smallest matching angle is 60°. What other positive matching angles occur through a full turn?
- 1.Repeat the smallest matching turn. Its next multiples are 120°, 180°, 240° and 300°.
- 2.The sixth step is 360°, giving the full turn. There can be no extra match between consecutive 60° steps, because that would lead to a smaller positive matching step.
- 3.The other angles are therefore 120°, 180°, 240°, 300°, 360°. The figure has rotation order 6.
Problem
A figure matches at 60° and has exactly two positive matching angles less than 60°. What is the smallest?
- 1.Let the smallest matching angle be s. The positive matching angles are s, 2s, 3s and so on.
- 2.Since exactly two of these are below 60°, those are s and 2s, while 60° is the next one: 3s.
- 3.Divide 60° into three equal steps. The smallest is 20°, with the earlier matches at 20° and 40°. It gives rotation order 360° ÷ 20° = 18.
Problem
Can 45° or 17° be the smallest matching angle of a finite-order figure?
- 1.A smallest matching turn must repeat a whole number of times to complete 360°. For 45°, 360° ÷ 45° = 8, so eight equal repeats work.
- 2.For 17°, 21 steps give 357°, leaving 3°. Equivalently, 360 divided by 17 is not a whole number.
- 3.Thus 45° is possible, but 17° is not a smallest matching turn. The issue is completing the full turn with an exact whole number of equal repeating positions.
Colour must return to the same positions
A circle divided into 12 equal sectors has a 30° step from one sector to the next. Colouring the sectors can reduce its symmetry because a blue sector must land on blue and a yellow sector on yellow. To produce a particular order, repeat a complete colour arrangement evenly around the circle.
Problem
What finite rotation orders can a 12-equal-sector drawing have when the sector boundaries remain part of the drawing?
- 1.A rotation preserving the boundary lines must shift sectors by a whole number of 30° steps.
- 2.A smallest repeating block must fit into the 12 positions a whole number of times. Block lengths 1, 2, 3, 4, 6 and 12 give orders 12, 6, 4, 3, 2 and 1 respectively.
- 3.The possible orders are therefore 1, 2, 3, 4, 6 and 12. Order 1 means no smaller matching rotation. A single contiguous coloured patch generally gives order 1; the number of coloured sectors alone does not establish the order.
Counting coloured sectors is not the same as counting matching angles. Colours must repeat in a complete cycle. Also, the outside circle may have unlimited symmetry while its divided or coloured interior has only finitely many matching turns.
Read patterns in buildings, polygons and wheels
Symmetry occurs in natural and designed objects, but decide what part is being tested. A building’s simplified outer boundary can have symmetry even when doors, landscaping or a perspective photograph do not. A wheel’s circular rim and its entire spoke pattern also ask different questions.
The Parliament Building example uses its simplified outer boundary. The three matching main directions give three mirror lines and matching turns of 120°, 240° and 360°. Inspecting a sloping photograph as though it were a flat geometrical plan could give a different appearance, so keep the object of the symmetry test clear.
Problem
What happens to the two symmetry counts as regular polygons gain more sides?
- 1.An equilateral triangle has three mirror lines and rotation order 3. A square has four of each, and a regular pentagon has five of each.
- 2.For a regular n-sided polygon, identical sides and equal angles repeat at n positions. It has n mirror axes and rotation order n.
- 3.The sequence for triangle through decagon is 3, 4, 5, 6, 7, 8, 9, 10 for both counts. Its smallest matching turn is 360° divided by the number of sides.
| Regular polygon | Mirror lines | Rotation order | Smallest matching turn |
|---|---|---|---|
| Equilateral triangle | 3 | 3 | 120° |
| Square | 4 | 4 | 90° |
| Regular pentagon | 5 | 5 | 72° |
| Regular hexagon | 6 | 6 | 60° |
| Regular heptagon | 7 | 7 | 51 3/7 degrees |
| Regular octagon | 8 | 8 | 45° |
| Regular nonagon | 9 | 9 | 40° |
| Regular decagon | 10 | 10 | 36° |
The Koch snowflake sequence begins with an equilateral triangle, then replaces the middle third of each side with the two sides of a small outward equilateral bump. Every side receives the same change. In the usual sequence, the initial triangle has three mirror lines and order 3; the first snowflake outline and later equally constructed outlines have six mirror lines and order 6. More small edges do not keep increasing the order, because those edges remain organised in the same six-direction arrangement.
Problem
How many mirror lines and matching angles does an idealised Ashoka Chakra with 24 identical equally spaced spokes have?
- 1.The 24 spokes give 24 equal gaps, each 360° ÷ 24 = 15°. A turn of 15° moves each spoke onto its next identical neighbour.
- 2.There are 24 matching positive angles: 15°, 30°, 45° and the following multiples of 15° through 360°. Thus the rotation order is 24.
- 3.Twelve mirror lines pass through opposite spoke pairs and twelve pass midway between opposite gaps. They give 24 mirror lines. The rim alone would have infinitely many, but the spokes limit the entire design.
Build a design with coloured tiles
Each square tile below is divided diagonally into blue and yellow parts. Reflection can reverse the diagonal direction and exchange where the blue part lies. Matching the outside square boundaries alone is insufficient. To complete a design across a specified line, reflect the colours inside each tile as well as the tile positions.
Problem
How can you make a 16-tile square design with exactly one or exactly two mirror lines?
- 1.For one vertical axis, choose the left eight tiles and reflect them to the right. Avoid arranging the top and bottom as reflected copies of each other.
- 2.Check the horizontal and the two diagonal axes of the outside square. They must fail because of the tile colours if exactly one axis is required.
- 3.For two perpendicular axes, choose the upper-left four tiles and reflect them across both centre lines. Verify that the diagonal folds still fail. The illustrated designs demonstrate these exact counts; the outside square’s four axes are only candidates.
Symmetry can supply a strategy
Draw a 6-by-6 board. On each turn a player covers two adjacent unused squares, represented by a horizontal or vertical segment between their centres. No square may be used twice. A player who cannot make a legal placement loses. Think of a placement and its image under a half-turn about the board centre.
Problem
How can the second player use symmetry in the 6-by-6 adjacent-squares game?
- 1.After each first-player move, place its image under a 180° turn about the board centre. This sends row r, column c to row 7 − r, column 7 − c, so adjacent squares stay adjacent.
- 2.After each reply, the occupied squares occur in half-turn pairs. If the next proposed first-player move uses free squares, their half-turned partners are also free in that paired position.
- 3.A two-square placement on this even-by-even board never coincides with or overlaps its own half-turned placement: the centre lies between four squares, and no adjacent pair is its own opposite pair. The reply therefore remains legal.
- 4.The board is finite. Whenever the first player can move, the second can reply. Eventually it is the first player’s turn with no legal placement, so the second player wins by maintaining this pairing.
Quiz
Which quadrilateral generally has rotational symmetry but no mirror line?
A 60° match has exactly two positive matches below it. What is the smallest match?
Which angle can be a smallest positive match for a finite-order figure?
What is essential for a 12-sector colour design to have order 3?
What is the smallest matching turn of 24 equal spokes?
Why can the second player use a half-turn reply in the 6-by-6 game?
The general parallelogram has a half-turn match, while the other named shapes have reflection axes.
Practice Problems
- Draw a triangle and a quadrilateral with both kinds of symmetry. Give their mirror-line counts and matching angles.
- Draw an isosceles triangle with exactly one axis, a general parallelogram and a symmetric trapezium. Explain the two symmetry tests for each.
- A figure has smallest matching angle 60°. List the other angles. Then solve the different clue in which 60° has exactly two positive matches below it.
- Explain why 45° is possible as a smallest matching turn but 17° is not.
- Colour a 12-sector circle for exact orders 2, 3, 4 and 6. State the repeating block and check whether a smaller shift fails.
- Compare the symmetry of a plain circle, a spoke pattern and a coloured sector drawing.
- Extend the regular-polygon table to a regular 12-sided polygon. Give both counts and the smallest angle.
- Explain why the snowflake sequence can add many edges without increasing the symmetry count at every stage.
- Analyse the simplified three-direction building outline and the 24-spoke Chakra drawing. State which details the test includes.
- Complete the partial tile board across both red lines, then check that neither diagonal is an extra mirror line.
- Create your own 16-tile designs with exactly one and exactly two mirror lines. Explain how the colour choices rule out other axes.
- Play the 6-by-6 game and record each move’s half-turn partner. Explain why the reply is legal even for moves next to the four central cells.
A smallest 60° gives order 6 and other matches 120°, 180°, 240°, 300°, 360°. Two positive matches below a 60° match instead give smallest angle 20° and order 18.
Key Takeaways
• Reflection and rotation must be tested separately on the whole figure. • Smallest matching turns divide a full turn into a whole number of repeated positions. • Sector colours and tile orientations are part of a decorated design. • A regular n-sided polygon has n mirror lines and rotation order n. • Idealised building boundaries and equal-spoke wheels have testable finite symmetries. • A half-turn can pair legal moves and maintain a winning reply strategy on an even square board.