Chapter 5: Does it Look the Same?

Symmetry Letters:
• (a) Vertical symmetry: A, M, T, V (also H, I, W, Y).
• (b) Horizontal symmetry: B, C, D, E (also H, I, K).
• (c) Both vertical and horizontal: H, I, X, O.

Half-Turn Numbers:
• Among single digits: 0, 1, and 8 look identical after a 1/2 turn.
• Two-digit numbers: 11, 88.
• Three-digit numbers: 101, 111, 181, 808, 888.

Word Rotations (180°):
• (a) ZOOM: On half turn, Z turns into Z, O stays O, M turns into W (reads MOOZ upside down). Does NOT look the same.
• (b) NOON: N turns into N, O stays O. Reading upside down gives N-O-O-N. Looks the same.
• (c) SIS: S turns into S, I stays I. Reading upside down gives S-I-S. Looks the same.
• (d) MOW: M turns into W and W turns into M. Upside down it reads M-O-W. Looks the same.

Quarter Turn Properties:
• (a) Angle: A 1/4 turn is 360° ÷ 4 = 90°.
• (b) Shapes:
- Square: Yes, looks identical after 90°.
- Rectangle: No, a horizontal rectangle becomes vertical after 90°.
- Equilateral Triangle: No, its pointing vertex rotates to the side.
- Circle: Yes, looks identical after any rotation.
Therefore, only Square and Circle look identical after a 1/4 turn.

1/3 Turn Analysis:
• (a) Angle: 360° ÷ 3 = 120°.
• (b) Equilateral triangle: Yes, it looks exactly the same. An equilateral triangle has 3 equal sides and 3 equal angles of 60°. Rotating it by 120° moves each corner to the next vertex position without altering its overall orientation.

1/6 Turn Properties:
• (a) Angle: 360° ÷ 6 = 60°.
• (b) Shape: A Regular Hexagon (or a 6-petalled symmetrical flower / 6-pointed star) looks identical after a 1/6 turn.

Windmill Rotations:
• (a) 1/4 turn: Yes, because rotating by 90° moves each of the 4 identical blades into the position previously occupied by its adjacent blade.
• (b) 1/2 turn: Yes, because a 1/2 turn is two 1/4 turns (180°), so the blades align identically again.

Comparison:
• Line Symmetry: A figure has line symmetry if it can be folded along a straight line such that the two halves fit exactly on top of each other. A rectangle has 2 lines of symmetry (one horizontal, one vertical through the middle of opposite sides).
• Rotational Symmetry: A figure has rotational symmetry if it looks identical to its original position after being turned by less than 360° around its centre. A rectangle has rotational symmetry of order 2 (looks the same on a 1/2 turn / 180°).

Diagonal Reflection Test:
No, the diagonal of a rectangle is NOT a line of symmetry.
When you fold a rectangular paper along its diagonal, the opposite corners do not match or overlap; instead, they stick out in opposite directions. Although the diagonal divides the rectangle into two triangles of equal area, they are not mirror reflections of each other across that diagonal line.

Inkblot Symmetry Steps:
1. Take a clean sheet of paper and fold it in half down the middle to create a sharp centre crease line.
2. Unfold the paper and place a few drops of liquid paint or coloured ink on one side of the fold.
3. Fold the paper back along the crease line and press the two halves gently together with your palm to spread the ink.
4. Open the paper. The paint transfers to both sides equally, producing a beautiful design where the crease line is the exact line of symmetry.

3-Blade Fan Analysis:
• (a) 1/4 turn (90°): No, 90° does not align with the 120° blade intervals.
• (b) 1/3 turn (120°): Yes, each blade rotates exactly into the position of the next blade.
• (c) 1/2 turn (180°): No, 180° points a blade directly between two blades on the opposite side.

Regular Polygon Symmetry Rule (Number of lines = Number of sides):
• (a) Equilateral triangle: 3 lines of symmetry.
• (b) Square: 4 lines of symmetry (2 through midpoints, 2 diagonals).
• (c) Regular pentagon: 5 lines of symmetry.
• (d) Regular hexagon: 6 lines of symmetry.

Vertical Mirror Reflection:
The digits that remain unchanged in a vertical mirror reflection are 0 and 8 (in standard symmetric font). Digits like 1 (with hook), 2, 3, 4, 5, 6, 7, and 9 reverse horizontally.

Order of Symmetry:
• Definition: The number of times a shape looks identical to its original appearance during one complete 360° turn.
• (a) Equilateral triangle: Looks the same at 120°, 240°, and 360°. Order = 3.
• (b) Square: Looks the same at 90°, 180°, 270°, and 360°. Order = 4.

Traffic Signs Analysis:
• (a) 'GIVE WAY' (Equilateral triangle shape): Has 3 lines of symmetry (from each vertex perpendicular to opposite edge).
• (b) 'STOP' (Regular octagon shape): Has 8 lines of symmetry (ignoring text lettering).

Pinwheel Colors Analysis:
• (a) 1/4 turn: No, because a 1/4 turn rotates a red blade into the position of a blue blade.
• (b) 1/2 turn: Yes, because a 1/2 turn (180°) rotates a red blade into the opposite red blade's position, and blue into blue.

Nature Examples:
1. Butterfly: The patterns, shapes, and veins on both wings show striking bilateral (mirror) symmetry.
2. Human face/body: Bilateral vertical symmetry (left eye matches right eye, left arm matches right arm).
3. Flowers & Snowflakes: Radial symmetry (many flower petals have 5-fold or 6-fold symmetry; snowflakes exhibit 6-fold hexagonal symmetry).

Parallelogram Properties:
• (a) Line symmetry: No, a general parallelogram cannot be folded along any line to match its halves.
• (b) Rotational symmetry: Yes, it looks identical after a 1/2 turn (180°). Its order of rotational symmetry is 2.

Circle Symmetry:
• A circle has infinitely many lines of symmetry.
• Any straight line passing through the centre of the circle (i.e. any diameter) is a line of symmetry, as folding along it divides the circle into two identical semicircles.

Mirror Reflection Activity:
• Placing the mirror exactly on the line of symmetry completes the reflection, showing a full, complete, realistic dog face.
• If you tilt or shift the mirror away from the line, the reflected image becomes distorted, stretched, or cut off, and no longer resembles a normal face.

Rotational Symmetry of 4 Ticks:
With only 4 tick marks at 90° intervals, the dial looks identical after every 1/4 turn (90°). Its order of rotational symmetry is 4.

Letter 'Z' Analysis:
• (a) Line symmetry: No, folding 'Z' vertically or horizontally never overlaps its arms.
• (b) Half turn: Yes, rotating 'Z' by 180° produces the exact same letter 'Z'.

Balance for Symmetry:
• The left has 3 circles, right has 2 circles.
• To make it symmetrical, the right side must also have 3 circles.
• Therefore, you must add 1 red circle to the right side at the corresponding mirror position.

Rangoli Design Principles:
• 4-fold and 8-fold radial symmetry creates a harmonious, visually balanced, and aesthetically pleasing geometric design from every viewing angle.
• Using grid dots allows artists to replicate motifs equally in all four quadrants, ensuring mathematical balance and beauty.