A figure is called symmetrical when a part or parts of it repeat in a definite pattern.

The repetition may be seen through folding/reflection, through rotation, or sometimes through both.

The opening examples—flower, butterfly, rangoli and pinwheel—show definite repeated patterns and are described as symmetrical. The cloud picture does not have such a definite repeating pattern, so it is treated as not symmetrical.

The chapter notes that its repeated parts return to their original positions when the rangoli is rotated through 90° about its centre.

Thus the pattern has rotational repetition in quarter-turns.

The most immediately visible symmetry is reflection or mirror symmetry about a central vertical direction: corresponding features on the left and right are arranged in matching positions.

Real buildings contain many details, so the question is mainly asking us to notice repeated structural organisation rather than claim that every tiny detail is perfectly identical.

If a figure is folded along a line and one half covers the other half exactly, the two parts are called mirror halves.

A line of symmetry is a line that divides a figure into two parts which exactly overlap when the figure is folded along that line.

It is also called an axis of symmetry.

According to the supplied chapter solutions:

• Flower: 6 lines of symmetry.
• Rangoli: 4 lines of symmetry.
• Butterfly: 1 line of symmetry.
• Pinwheel: no line of symmetry.
• Cloud: no line of symmetry.

The pinwheel is important because it can be rotationally symmetric even though it has no reflection line.

The solution page marks lines of symmetry only for the 1st, 2nd and 4th figures.

• Figure 1: one slanting line of symmetry.
• Figure 2 (kite-like figure): one vertical line of symmetry.
• Figure 3: no line of symmetry.
• Figure 4 (L-shaped figure): one diagonal line of symmetry.
• Figure 5: no line of symmetry.

A square has 4 lines of symmetry:

• one vertical,
• one horizontal,
• and the two diagonals.

Folding along any one of these lines makes the two halves overlap exactly.

No. Folding a non-square rectangle along a diagonal does not make the two halves coincide exactly.

A rectangle has two reflection axes through its centre—one horizontal and one vertical—but its diagonals are not reflection axes unless the rectangle is actually a square.

If one side of a figure is carried to the other side by reflection in a line, and the complete figure remains unchanged, the figure has reflection symmetry.

Any figure with one or more lines of symmetry therefore has reflection symmetry.

The left and right corners exchange positions:

No corner remains fixed because none of the four labelled corners lies on the vertical reflection axis.

Points on the mirror line stay fixed, so A and C remain in the same places. The other two vertices exchange positions:

The upper and lower corners exchange positions:

The paint on one half is transferred to the other half while the paper is folded. The crease acts as a mirror line.

Therefore each mark on one side gets a matching reflected mark at the same perpendicular distance on the other side.

The cut appears together with its reflected copy across every fold line involved. The fold acts as a line of symmetry.

If the sheet is folded more than once, one cut may generate 4 or more repeated parts after unfolding.

The supplied solution identifies:

• (a) vertical fold.
• (b) diagonal fold from bottom-left toward top-right.
• (c) horizontal fold.
• (d) the paper was folded along both a vertical and a horizontal line (in either order) before one hole was punched.

For (d), unfolding across two perpendicular folds creates four symmetrically placed holes from a single punch.

For each diagram, reflect the given hole across the dashed symmetry line.

• If there is one symmetry line, add one mirror-image hole.
• If two symmetry lines are present and the hole is not on either line, repeated reflection can produce three additional matching holes.
• The reflected hole must be at the same perpendicular distance from the symmetry line as the original.

The opened shape is obtained by reflecting the cut across the fold line(s).

• (a) produces a horizontally extended, mirror-symmetric decorative opening with matching wavy sides.
• (b) produces a symmetric bow-tie/hourglass-like opening.
• (c) produces the tall symmetric cut-out shown in the supplied solution, with matching top/bottom notches and rectangular openings.
• (d) produces the vertically symmetric I-like yellow shape shown in the solution.

The canvas below shows simplified versions of the opened outcomes.

1. Fold the square sheet horizontally.
2. Fold it again vertically so the centre of the original sheet becomes the closed corner of the folded packet.
3. At that closed corner, make the required small straight-sided cut.
4. Open both folds. The repeated cut forms a square hole at the centre.

1. Fold the sheet horizontally and then vertically.
2. Locate the fully closed corner corresponding to the original centre.
3. Make one slanting straight cut across that corner.
4. Unfold twice. Four copies of the slanting cut form a square rotated by 45°.

The fact that it looks like a ‘diamond’ does not change its mathematical identity: its four sides are equal and its angles are right angles.

The rotated square has 4 lines of symmetry. The eight-point star-like shape has 8 lines of symmetry.

An equilateral triangle has 3 lines of symmetry. Each line passes through one vertex and the midpoint of the opposite side.

A regular hexagon has 6 lines of symmetry: three joining opposite vertices and three joining the midpoints of opposite sides.

The supplied solution shows:

• 1st pattern: one vertical line.
• 2nd pattern: one horizontal line.
• 3rd pattern: one vertical line.
• 4th pattern: one horizontal line.

The solution indicates:

• Nested-square figure: 4 lines—vertical, horizontal and two diagonals.
• Elongated symmetric octagon: 2 lines—vertical and horizontal.
• Irregular polygon: 1 horizontal line.
• Four-point curved/star figure: 4 lines—vertical, horizontal and two diagonals.

The supplied solution draws 6 symmetry axes through the centre: a vertical axis, a horizontal axis and four slanting axes matching the six-fold arrangement of the repeated motifs.

Draw an isosceles triangle that is not equilateral. Its only line of symmetry passes from the apex to the midpoint of the unequal side (the base).

Draw an equilateral triangle. It has three reflection axes, one through each vertex and the midpoint of the opposite side.

Draw a scalene triangle, with all three sides of different lengths. No fold can make its two parts match exactly.

No. The supplied solution explicitly states that this is impossible.

Reason: if a triangle had two different reflection axes, the symmetry would force all three sides and all three angles to be equal, producing an equilateral triangle—which has three axes, not two.

Many answers are possible. Valid examples include:

• Exactly 1: a heart-shaped outline with a vertical mirror axis.
• Exactly 2: a non-circular ellipse, with horizontal and vertical axes.
• Exactly 4: a four-petal flower whose petals are identical and placed at 90° intervals.

Reflect every red segment across the blue line.

For a horizontal or vertical blue line, copy each endpoint the same number of grid squares to the opposite side. For a diagonal blue line, rotate the paper mentally if helpful, then reflect each grid point across the diagonal.

The supplied solutions show the missing reflected portions with dashed red segments.

Reflect the given part across the first blue line, then reflect the resulting parts across the second blue line.

Every final point must have matching counterparts required by both axes. When the axes intersect at right angles, a point away from both axes usually appears in four related positions.

Choose the symmetry axis suggested by the existing segments, then add the mirror image of the unmatched boundary using exactly two new line segments.

The official solution uses different convenient axes for different drawings—vertical, horizontal or slanting—and completes each boundary by reflection.

A figure has rotational symmetry if it can be rotated by an angle strictly between 0° and 360° about a fixed point and still look exactly the same.

The fixed point about which a figure is rotated is called the centre of rotation.

An angle through which a figure can be rotated so that it exactly overlaps its original position is called an angle of rotational symmetry, or simply an angle of symmetry.

The windmill matches itself after every quarter turn:

Thus it has 4 angles of symmetry, and its smallest positive angle of symmetry is 90°.

A square overlaps itself after:

Its centre of rotation is the intersection point of its diagonals.

No. The chapter’s definition requires an angle strictly between 0° and 360° that makes the figure overlap itself.

If only a full 360° turn works, the figure is treated as having no rotational symmetry.

Every rotation by one arm-spacing maps each arm onto the next.

Modify opposite arms in matching pairs so that a 180° rotation still pairs every arm correctly, but a 90° rotation no longer does.

Then the only angles of symmetry are:

Although it has three arms, the angular gaps between adjacent arms are not equal. A rotation cannot carry every arm exactly onto another arm.

Therefore only 360° works, which the chapter does not count as rotational symmetry.

The three gaps around the centre must be equal. A full turn is 360°:

Therefore adjacent arms must be separated by 120°. The angles of symmetry are 120°, 240° and 360°.

Use 5 identical arms equally spaced around the centre.

Use 6 identical arms equally spaced around the centre.

Divide the full turn equally among 7 matching positions:

This is not a whole number of degrees.

It has order 4 rotational symmetry.

Since no angle smaller than 360° works, it does not have rotational symmetry under the chapter’s definition.

It has order 2 rotational symmetry.

The supplied solution identifies the following:

• circle with a cross,
• circle divided into three equal radial parts,
• four-bladed curved pinwheel,
• X-shaped pair of crossing lines,
• five-point star.

The triangular figure and the B-like figure are not included because they do not have a non-trivial rotation about the marked point that preserves the whole figure.

Once the smallest successful rotation is known, repeating that same rotation moves the figure through all matching positions until it completes a full turn.

For example, if the smallest angle is 90°, the matching rotations are 90°, 180°, 270° and 360°.

True. A complete turn always returns any figure to its starting position.

However, a figure is said to have rotational symmetry only when it also matches itself at some angle strictly between 0° and 360°.

True. Repeating the smallest rotation an integer number of times must complete exactly 360°.

A circle is exceptionally symmetric:

• Every diameter is a line of reflection symmetry, so there are infinitely many reflection axes.
• Rotating the circle through any angle about its centre leaves it unchanged.

Use a colouring pattern that repeats every 120°—that is, every 4 sectors.

Then the figure matches at 120°, 240° and 360°, giving 3 angles of symmetry.

Use a colouring pattern that repeats every 90°—that is, every 3 sectors.

Then the figure matches at 90°, 180°, 270° and 360°, giving 4 angles of symmetry.

The supplied solution explicitly gives 12 angles of symmetry as possible—for example, with a colouring that repeats sector by sector.

The supplied solution gives two four-fold examples, each having:

Other valid examples include an equilateral triangle or a regular hexagon, provided the question’s exclusions are respected.

Use an equilateral triangle.

It has 3 lines of symmetry and rotational angles 120°, 240° and 360°.

Use an isosceles triangle that is not equilateral. It has one reflection axis through the apex and base midpoint, but no angle between 0° and 360° maps it onto itself.

Use a general parallelogram that is neither a rectangle nor a rhombus.

It has no line of symmetry, but a 180° rotation maps it onto itself.

Use a kite that is not a rhombus. It has one line of reflection symmetry, but a 180° turn does not generally map it onto itself.

They are successive multiples of 60°.

The symmetry angles are multiples of the smallest angle. If there are exactly two positive symmetry angles below 60°, they can be 20° and 40°.

Yes.

So eight equal rotational steps complete a full turn.

No, under the chapter’s natural-number-degree rule.

Therefore repeated 17° rotations cannot land exactly on 360° after a whole number of steps.

According to the supplied solution:

• Reflection symmetry: 3 lines of symmetry.
• Rotational symmetry: 120°, 240° and 360°.

Thus its rotational order is 3.

The sequence is therefore 3, 4, 5, 6, 7, 8, 9, 10, ….

A regular n-gon has rotational order n, because each rotation by 360°/n moves every vertex to the next vertex.

So from regular triangle to regular decagon, the numbers of angles of symmetry are:

This follows the same counting-number sequence beginning at 3.

The supplied solution lists:

The initial equilateral triangle has 3. Once the symmetric ‘bumps’ are added, the later snowflake stages have six-fold symmetry.

According to the supplied solution:

The 24 equally spaced spokes create repeated radial structure around the centre.

Reflect the completed upper-left 4×4 tile block first across the vertical red line and then across the horizontal red line.

This fills the remaining three quadrants so that the two red centre lines become symmetry axes.

This is an open construction task. For exactly one line, design the left half and mirror it to the right while deliberately avoiding horizontal or diagonal repetition.

For exactly two lines, design one quarter and reflect it across perpendicular horizontal and vertical axes, while choosing tile orientations that do not accidentally create diagonal axes.

The chapter asks this as a strategy problem but the supplied solution pages do not provide an answer. A strong symmetry strategy is:

The second player mirrors every move by a 180° rotation about the centre of the board.

Because the 6×6 board has central rotational symmetry and a legal domino-like line has a distinct 180° partner, whenever the first player can move, the mirrored move is also available. The second player therefore responds in pairs and aims to make the final move.

It has 2 lines of symmetry: horizontal and vertical through the centre. Its diagonals are not symmetry axes because the rectangle is not a square.

No, because 360 is not divisible exactly by 28.

Two—the two vertices lying on the chosen diagonal.

Three grid units to the right of the line at the same height.

It remains fixed because its perpendicular distance from the mirror line is zero.

No. No rotation smaller than 360° maps three unequal sides onto themselves.

Yes. A 180° rotation maps it onto itself, so its order is 2.

Two—its diagonals are reflection axes.

A general parallelogram has rotational order 2 but no reflection axis. A rectangle has rotational order 2 and two reflection axes.

No. A general kite has no non-trivial rotational symmetry.

No. If 40° were the smallest angle, every symmetry angle would be a multiple of 40°, but 100° is not.

The repeated rotational match is broken; normally only the full 360° turn remains.

Rotation preserves the direction in which the blades spiral, but reflection reverses that handedness, so the reflected pinwheel does not match the original.

Infinitely many, because every diameter is an axis.

No. Every positive rotation angle works, so there is no least positive one.

It must have at least a 180° rotational symmetry about the intersection of the two perpendicular axes.

Not for an ordinary finite plane figure with the same centre: order 4 rotation would rotate that single axis into additional symmetry axes. So the combination is not possible in the usual setting.

The colouring repeats every two arms, so a 120° rotation can preserve it. The rotational order can reduce from 6 to 3.

Rotations by successive multiples of 360°/n move each vertex to the position of another vertex. After n such steps, the figure completes 360°.

360° is always an angle that returns a figure to itself. But the chapter says a figure has rotational symmetry only if at least one successful angle lies strictly between 0° and 360°.