Symmetry · Lesson 5 of 7
Radial Patterns and Rotation Order
“Build equal-arm patterns and connect their smallest matching turns to a full rotation.”
• Explain why identical radial arms need equal angular spacing for a one-arm matching turn. • Derive the 120° spacing of three equally arranged arms. • Find matching angles and rotation order for finite repeating patterns. • Use the smallest matching angle to calculate other matching turns. • Distinguish the circle’s unlimited matching turns from finite rotation orders.
Arms arranged around a centre
A radial arm is a part that extends out from a common central point, like a spoke or a fan blade. If several arms have identical shapes and equal spacing, turning one arm into its neighbour’s position can move every other arm into a matching position too. The gaps between neighbouring directions are angles at the centre.
Four identical straight arms separated by 90° produce a quarter-turn match. Merely having four arms is not enough. Unequal lengths, unequal thicknesses or unequal gaps can prevent the match. When asking whether the spacing alone may change while four identical arms still give a one-arm match, all four gaps must remain equal, so they remain 90°.
Problem
How could a four-arm figure be changed so that it matches at 180° but not at 90°?
- 1.Keep opposite parts as identical half-turn partners, but make the two pairs different from each other. For instance, alter the ends of one opposite pair.
- 2.A 180° turn still takes each part onto its matching opposite part.
- 3.A 90° turn would exchange the different pairs and fails. The only positive matching angles through a full turn are now 180° and 360°. The displayed bent-end version illustrates the idea.
Three arms do not automatically mean three matches
Suppose three identical arms point in directions with unequal gaps. Turning one arm onto the next does not necessarily carry the third into any arm position. The whole figure has to match at once. A tracing overlay is a useful test because it exposes a gap that your eye may overlook.
To make a turn cycle three arms through the three positions, the first gap must become the second, the second must become the third and the third must become the first. Their angle sizes must therefore agree. Call the gaps A, B and C. The matching requirement gives A = B = C, while together they complete the full 360° around the centre.
Problem
What are the matching angles when the three equal arms are equally spaced?
- 1.Divide the full turn into three equal gaps: 360° ÷ 3 = 120°. Each adjacent pair of directions is therefore 120° apart.
- 2.A 120° rotation cycles all three identical arms into the next positions. A second such step gives a total of 240° and also matches.
- 3.A third step gives 360° and returns the arms to their original positions. The complete positive list is 120°, 240°, 360°. A turn smaller than 120° cannot take any arm to the next identical arm position.
Draw a three-arm figure with equal 120° gaps and another with unequal gaps. Trace and cut out each one. Keep the original drawings fixed and test the cut-outs at 120° and 240°. Temporary labels A, B, C may help track the arms; do not treat those tracking labels as permanent decoration when testing the plain arm outlines.
Count the matches in one full turn
The number of matching positions in a full turn is called the order of rotational symmetry. Count the positive matching angles up to and including 360°, rather than counting both 0° and 360° as different returns. A figure with only the full-turn match has order 1 and has no nontrivial rotational symmetry.
For a figure with finitely many matching turns about a chosen centre, the number of positive matching angles greater than 0° and at most 360°. The full-turn position is counted once.
If the smallest matching turn is s degrees, repeating it produces the next matching positions: s, 2s, 3s, and so on until the full turn. Why are there no other angles between consecutive steps? Any extra match between them could be compared with the earlier matching position to give an even smaller positive matching turn, contradicting the choice of s as the smallest.
| Order | Smallest positive match | All positive matching angles through 360° |
|---|---|---|
| 2 | 180° | 180°, 360° |
| 3 | 120° | 120°, 240°, 360° |
| 4 | 90° | 90°, 180°, 270°, 360° |
| 5 | 72° | 72°, 144°, 216°, 288°, 360° |
| 6 | 60° | 60°, 120°, 180°, 240°, 300°, 360° |
Problem
How do you draw identical-arm designs with orders 5 and 6?
- 1.For five arms, use five equal gaps. Each gap is 360° ÷ 5 = 72°. Copy the same arm at these equally spaced directions.
- 2.The matching turns are multiples of 72° through the fifth multiple, 360°: 72°, 144°, 216°, 288°, 360°.
- 3.For six arms, each gap is 360° ÷ 6 = 60°. The matching turns are 60°, 120°, 180°, 240°, 300°, 360°. Check the arm outlines and any decoration at every proposed turn.
Problem
What is the smallest matching turn of seven identical equally spaced arms?
- 1.There are seven equal gaps around a full turn, so each gap is 360° ÷ 7.
- 2.Seven goes into 360 fifty-one times, using 357°. The remaining 3° is divided equally among the seven gaps.
- 3.Thus the smallest angle is 51 3/7 degrees. It is not a whole number of degrees, but seven such turns still complete exactly 360°. Rounding to 51° would give the wrong full-turn total.
The number of arms, sides or points gives an order only when the complete arrangement actually repeats. Check identical shapes and equal angular spacing. Also, 0° and 360° represent the same starting position; counting both would overcount the order.
The circle is different
A plain circle has no special petal or corner that must reach the next position. Any rotation about its centre leaves the boundary at the same distance from that centre in every direction. It matches after 1°, half a degree or any smaller positive angle. There is therefore no smallest positive matching angle.
Every diameter is also a mirror line of a plain circle. The number of mirror lines and matching rotation angles is infinite. Do not assign a plain circle order 360 because you happened to check whole-number degrees: angles between those degrees match as well. The finite-order formula cannot be used by choosing a smallest positive angle that the circle does not have.
A fan, a flower or a spoked wheel may resemble a circle overall but has distinguished parts. Its order depends on the actual number and arrangement of identical blades, petals or spokes. This connects the unlimited symmetry of the plain circle with the finite repeating patterns of real objects.
Quiz
Why must three identical arms have equal gaps to match by a one-arm turn?
What is the smallest matching turn of six identical equally spaced arms?
Which list gives every positive matching angle for an order-3 figure through one full turn?
What is 360° ÷ 7 as an exact mixed number?
Why does a plain circle have no smallest positive matching angle?
A figure has smallest matching angle 72°. What is its rotation order?
A rotation that cycles the arms also cycles their gaps, so those gaps must have equal sizes.
Practice Problems
- Draw four equal radial arms and change their ends so that a half-turn matches but a quarter-turn fails. Explain each test.
- Draw three equal arms with unequal gaps. Explain why at least one arm fails to align after a proposed one-arm rotation.
- Derive the 120° gap of a three-arm pattern from the full turn rather than recalling the value.
- Make five-arm and six-arm designs, and list all their matching angles through a full turn.
- Find the exact smallest matching turn of seven equal arms. Explain why rounding would prevent an exact seven-step return.
- Find the orders and angle lists for the six figures in the gallery. Justify the smallest angle in each case.
- Decide whether the following statements are true: every figure has a 360° matching turn; a whole-number smallest matching angle is a factor of 360. Explain both.
- Compare a plain circle, a circle with one spoke and a wheel with eight identical equally spaced spokes.
- Explain why all matching angles of a finite-order figure are multiples of its smallest positive match.
In reading order, the orders are 2, 4, 6, 3, 4, 5. Their smallest turns are respectively 180°, 90°, 60°, 120°, 90°, 72°. Continue in equal steps through 360°.
Key Takeaways
• Identical arms need an arrangement that preserves both their shapes and the gaps. • Three equal gaps around a centre are each 120°. • Finite rotation order counts positive matching turns through 360°. • Matching angles occur in multiples of the smallest matching turn. • For finite order n, the smallest matching turn is 360° divided by n. • A plain circle has infinitely many symmetries and no smallest positive matching turn.