Symmetry · Lesson 3 of 7
Finding and Completing Lines of Symmetry
“Count reflection axes and build complete figures on square and dot grids.”
• Find every mirror line of regular and composite figures. • Distinguish triangles with zero, one and three mirror lines. • Explain why a triangle cannot have exactly two reflection axes. • Complete a drawing across a vertical, horizontal or diagonal axis. • Use two given axes to construct a complete mirror design. • Check symmetry of a whole repeated pattern rather than one motif.
Search systematically for every axis
A drawing can pass one fold test and still have other mirror lines. To find them all, look at the arrangement of sides, corners and repeated parts. For a regular shape, the equal sides and equal angles limit which corners can match. For a composite shape, every small part must agree under the same reflection.
A regular hexagon has six equal sides and six equal angles. Three axes join opposite vertices, and three join the midpoints of opposite sides. These are different lines, even though they all pass through the centre. An eight-point star made from eight identical evenly arranged points can have eight axes. A star with one point made longer may have fewer, so counting pointed tips is not a substitute for the fold test.
Problem
Why does an undecorated regular hexagon have six lines of symmetry?
- 1.Start with one line joining a vertex to the opposite vertex. Reflection pairs the corners to its left and right and matches their connecting sides.
- 2.There are three different opposite-vertex pairs, giving three such lines.
- 3.Next join the midpoints of opposite sides. Each such line also pairs the remaining corners and sides. There are three of these, making six distinct axes altogether.
For an overlapping chain of identical diamonds, do not count the axes of each diamond separately and add them. A line through the middle of one diamond may send the other diamonds to empty space. An outer square surrounding a smaller square also requires checking their common centre and orientation. Moving the small square slightly away from the centre can destroy axes of the combined drawing.
Problem
Why must you check more than one small flower in a kolam?
- 1.A single six-petal motif may have several mirror lines, but the complete design contains other motifs in specific positions.
- 2.Reflect the centres of the surrounding motifs as well as the shapes inside them. A proposed axis fails if one motif is sent to a position with no matching motif.
- 3.In the displayed arrangement, the six outer centres and their identical motifs pair across each orange line. Those six lines are axes of the complete pattern. A changed outer motif would need a fresh check.
Triangles cannot have every possible axis count
A triangle’s corners make it a good setting for a reasoning question. The three useful cases are a scalene triangle, an isosceles triangle with unequal base length, and an equilateral triangle. Their symmetry counts depend on the side relationships, not on whether the drawing happens to stand upright.
A triangle whose three side lengths are different. It has no line of reflection symmetry.
A triangle with at least two equal sides. When exactly two sides are equal, it has one reflection axis. The equilateral special case has three.
A triangle with all three sides equal. Its three angles are equal too, and it has three mirror lines, each joining a vertex to the midpoint of the opposite side.
Problem
Can a triangle have exactly two lines of symmetry?
- 1.A mirror line of a triangle passes through a vertex and exchanges the other two vertices. The sides joining the fixed vertex to those two vertices must therefore be equal.
- 2.Suppose a second, different vertex also supplies a mirror axis. Its reflection requires another pair of side lengths to be equal.
- 3.Together the two equalities force all three sides to be equal. The triangle is equilateral and automatically has the third axis. Exactly two is therefore impossible.
Curves can obey the same mirror rules
A mirror line applies to curved boundaries just as it does to straight sides. A semicircle has one axis through the midpoint of its straight edge. A nonsquare ellipse has two, along its longer and shorter directions. A balanced four-lobed outline can have four, but the folds must match the lobes and the inward curves too.
When asked to make a figure with exactly one, two or four axes, begin with the desired reflected arrangement and then test for accidental extra lines. For example, a circle would not be a suitable answer for exactly four: it has many more. In an ellipse, making the two dimensions equal turns it into a circle and changes the symmetry count completely.
Complete a drawing across one given line
A square grid supplies a way to compare distances. For a vertical axis, count horizontal steps to the line and place the partner the same number of steps across it. For a horizontal axis, use vertical steps. A sloping axis needs the same perpendicular matching idea; rotating the paper can make it easier to see.
Problem
A shape has corners 1, 2 and 3 squares left of a vertical axis at three different heights. How do you complete it?
- 1.At each corner’s own height, place a new corner the same number of squares to the right of the axis: 1, 2 and 3 respectively.
- 2.A corner already on the axis needs no new partner. Keep its position.
- 3.Join the new corners in the order corresponding to the original edges. Then check each entire segment, not just the new corner positions, against the original boundary.
Across a diagonal of the grid, horizontal and vertical changes exchange roles. For the blue rising diagonal in C and F, a corner above-left of the line needs a corresponding corner below-right. Check the short horizontal and vertical pieces as well as sloping edges: reflection can turn a horizontal edge into a vertical one. This is why copying the same staircase orientation can fail.
Do not draw a matching-looking shape somewhere on the other side. Every point needs the correct position relative to the specified axis. Equal distances measured along an arbitrary slant are insufficient; compare straight across the mirror line.
Two axes impose two checks
Two specified mirror lines are simultaneous conditions. Completing a figure across only one of them may leave the other condition unsatisfied. In the displayed grids the axes meet at right angles, so a part in one region supplies matching parts in the other three regions.
Problem
A point is 2 squares left of a vertical axis and 1 square above a horizontal axis. Which partners are required?
- 1.Reflect across the vertical line first: the partner is 2 squares right and 1 square above the horizontal line.
- 2.Reflect the original point across the horizontal line: the partner is 2 squares left and 1 square below.
- 3.The fourth point is 2 squares right and 1 square below. Checking all four positions shows that either reflection preserves the set. The same construction applies to every edge of a given part.
Dot-grid completion allows choices. You may find more than one suitable axis or more than one valid completion. State the axis you are using, add the required missing edges and test the entire resulting boundary. A correct explanation matters more than whether your drawing resembles someone else’s answer.
Quiz
How many mirror lines does an undecorated regular hexagon have?
Which triangle has exactly one line of symmetry?
Why can a triangle not have exactly two mirror axes?
A point is on the given mirror line. When completing a drawing, where is its partner?
When a design must have two specified mirror lines, what must be checked?
Which curved figure has exactly two mirror axes?
Three opposite-vertex axes and three opposite-side-midpoint axes give six.
Practice Problems
- Explain the six mirror lines of a regular hexagon and the eight of the regular eight-point star in the diagram.
- Draw triangles with zero, one and three mirror lines. Give the side relationships that justify each count.
- Explain the impossibility of a triangle with exactly two axes without relying only on a list of familiar triangle names.
- Complete all six one-axis grid drawings. Explain what changes when the axis is diagonal.
- Complete all six two-axis grid drawings and verify each blue line separately.
- For each partial dot-grid drawing, add two segments to make a closed symmetric shape. Show at least one valid axis.
- Draw figures with curved boundaries having exactly one, two and four axes. Test for additional lines.
- In the kolam diagram, change exactly one outer motif. Explain which proposed axes now fail and why.
- Draw three identical diamonds in a horizontal chain. Compare the axes of one diamond with those of the entire chain.
The possible reflection counts are zero, one and three. Two distinct vertex axes force two pairs of sides equal, so all three sides become equal and a third axis follows.
Key Takeaways
• Count axes of the entire figure, including positions of repeated motifs. • An equilateral triangle has three axes, an exactly-two-equal-side triangle one, and a scalene triangle none. • A triangle cannot have exactly two axes because the first two force the third. • Grid reflection pairs points at equal perpendicular distances from the given line. • A drawing with two required axes must satisfy both reflection conditions. • Curved boundaries follow the same exact matching test as straight edges.