Chapter 14 Symmetry (Concepts)

Line Symmetry

Symmetry is a concept of balance and proportion that we see all around us in nature, art, and architecture. A figure or an object is said to have line symmetry if it can be divided into two identical, mirror-image halves by a straight line. When the figure is folded along this line, one half fits exactly onto the other half, as if they are reflections of each other.

This dividing line is called the line of symmetry or the axis of symmetry. It acts like a mirror placed along the center of the shape.

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Understanding Line Symmetry

Imagine a shape drawn on a piece of paper. If you can fold the paper along a line such that the two parts of the shape on either side of the fold match up perfectly, then that fold line is a line of symmetry. For instance, the wings of a butterfly or a leaf often exhibit line symmetry. If no such fold is possible, the shape has no line of symmetry.

Figures can have different numbers of lines of symmetry: some have only one, some have several, some have none, and some, like a circle, have infinitely many.

Reflection and Symmetry

The concept of line symmetry is directly related to reflection. A figure has line symmetry if one half of the figure is the reflection of the other half. The line of symmetry acts as the mirror line.

For a figure to be symmetrical about a line, every point 'P' on one side of the line must have a corresponding point 'P'' on the other side such that:

1. The line segment connecting P and P' is perpendicular (at a $90^\circ$ angle) to the line of symmetry.
2. The distance of point P from the line of symmetry is exactly equal to the distance of point P' from the line of symmetry.

In other words, the line of symmetry is the perpendicular bisector of the line segment joining any point to its image.

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Examples of Shapes with Line Symmetry

Let's look at some common geometrical shapes and identify their lines of symmetry.

(a) Isosceles Triangle: An isosceles triangle has exactly one line of symmetry. This line connects the vertex between the two equal sides to the midpoint of the base.

(b) Equilateral Triangle: An equilateral triangle has three lines of symmetry. Each line of symmetry connects a vertex to the midpoint of the opposite side.

(c) Square: A square has four lines of symmetry. Two lines join the midpoints of opposite sides, and the other two are the diagonals.

(d) Rectangle: A rectangle (that is not a square) has two lines of symmetry. These lines join the midpoints of opposite sides. The diagonals are NOT lines of symmetry because if you fold along a diagonal, the corners do not match up.

(e) Rhombus: A rhombus has two lines of symmetry. These are its two diagonals.

(f) Circle: A circle has infinitely many lines of symmetry. Any straight line that passes through the centre of the circle (i.e., any diameter) is a line of symmetry.

(g) Regular Hexagon: A regular polygon with 'n' sides has 'n' lines of symmetry. A regular hexagon (n=6) has six lines of symmetry.

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Summary of Lines of Symmetry

Here is a table summarising the number of lines of symmetry for some common shapes:

Shape	Number of Lines of Symmetry
Scalene Triangle	0
Isosceles Triangle	1
Equilateral Triangle	3
Parallelogram (not rectangle or rhombus)	0
Rectangle	2
Rhombus	2
Square	4
Circle	Infinite
Regular Pentagon	5
Regular Hexagon	6

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Line Symmetry in Letters of the Alphabet

Many letters of the English alphabet also exhibit line symmetry. Some have a vertical line of symmetry, some have a horizontal line of symmetry, and some have both. Let's look at the different categories.

Letters with a vertical line of symmetry:

These letters are symmetrical about a vertical line running down their middle. The left half is a mirror image of the right half.

The letters are: A, H, I, M, O, T, U, V, W, X, Y.

Letters with a horizontal line of symmetry:

These letters are symmetrical about a horizontal line running across their middle. The top half is a mirror image of the bottom half.

The letters are: B, C, D, E, H, I, K, O, X.

Letters with both vertical and horizontal lines of symmetry:

These letters are symmetrical about both a vertical and a horizontal line.

The letters are: H, I, O, X.

Letters with no line of symmetry:

These letters cannot be divided into two identical mirror-image halves by any straight line.

The letters are: F, G, J, L, N, P, Q, R, S, Z.

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Rotational Symmetry

A figure is said to have rotational symmetry if it looks exactly the same as its original position after being rotated about a fixed central point by a certain angle, less than $360^\circ$.

The fixed point around which the rotation occurs is called the center of rotation. The minimum angle by which the figure must be rotated to coincide with its original position is called the angle of rotational symmetry or the minimum angle of rotation.

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Understanding Rotational Symmetry

Imagine holding a shape at its centre with a pin and rotating it. If, during this rotation, the shape looks identical to its starting position one or more times before it completes a full circle ($360^\circ$), the shape possesses rotational symmetry.

The amount of turn during rotation is typically measured in degrees. A full turn is $360^\circ$, a half turn is $180^\circ$, and a quarter turn is $90^\circ$.

Order of Rotational Symmetry:

The order of rotational symmetry is the number of times a figure coincides with itself (looks exactly the same) during a complete rotation of $360^\circ$ about its center of rotation. It includes the starting position and the final position (after a $360^\circ$ rotation).

Any figure will look the same after a $360^\circ$ rotation, so every object has rotational symmetry of order at least 1 (this is often called trivial rotational symmetry). We usually talk about rotational symmetry when the order is greater than 1.

Angle of Rotational Symmetry:

The angle of rotational symmetry is the smallest positive angle through which a figure can be rotated about its center of rotation to coincide with its original position. This is also sometimes called the angle of rotation.

The angle of rotational symmetry can be calculated using the order of rotational symmetry:

Angle of Rotational Symmetry $= \frac{360^\circ}{\text{Order of Rotational Symmetry}}$

Conversely, if you know the angle of rotational symmetry ($\theta$), the order of rotational symmetry is $360^\circ / \theta$.

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Examples of Shapes with Rotational Symmetry

Let's examine the rotational symmetry of some common shapes:

• A square: The center of rotation is the intersection of its diagonals. A square looks the same after rotations of $90^\circ, 180^\circ, 270^\circ,$ and $360^\circ$ about its center.
  • The unique positions where it looks the same (including the start) are 4.
  • Order of rotational symmetry = 4.
  • Angle of rotational symmetry $= \frac{360^\circ}{4} = 90^\circ$.
  
• An equilateral triangle: The center of rotation is the centroid (the point where the medians intersect). An equilateral triangle looks the same after rotations of $120^\circ, 240^\circ,$ and $360^\circ$ about its center.
  • The unique positions are 3.
  • Order of rotational symmetry = 3.
  • Angle of rotational symmetry $= \frac{360^\circ}{3} = 120^\circ$.
  
• A rectangle (not a square): The center of rotation is the intersection of its diagonals. A rectangle looks the same after rotations of $180^\circ$ and $360^\circ$ about its center.
  • The unique positions are 2.
  • Order of rotational symmetry = 2.
  • Angle of rotational symmetry $= \frac{360^\circ}{2} = 180^\circ$.
  
• A circle: The center of rotation is the center of the circle. A circle looks the same after rotation by any angle about its center.
  • Order of rotational symmetry = Infinite.
  • Angle of rotational symmetry = Any angle (though we often say the minimum meaningful angle is infinitesimally small, for practical purposes it's best described by its infinite order).
  
• A regular polygon with 'n' sides: The center of rotation is the center of the polygon (where its diagonals or angle bisectors meet). A regular n-sided polygon has rotational symmetry of order 'n'.
  • Order of rotational symmetry = n.
  • Angle of rotational symmetry $= \frac{360^\circ}{n}$.

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Rotational Symmetry in Letters of the Alphabet

Some letters of the English alphabet have rotational symmetry. This means that if you rotate the letter around its center point, it will look exactly the same as it did in its original position before completing a full 360° turn. For the letters that have this property, the angle of rotation is typically $180^\circ$ (a half-turn).

The order of rotational symmetry is the number of times a figure matches its original orientation during a full 360° rotation. The angle of rotational symmetry is the smallest angle of rotation required for the figure to look the same.

The letters that have rotational symmetry of order 2 (meaning they look the same after a $180^\circ$ turn and again after a $360^\circ$ turn) are:

• H, I, N, O, S, X, Z

Letters with Both Line and Rotational Symmetry

The letters H, I, O, and X are special because they have both line symmetry and rotational symmetry. Rotating them by $180^\circ$ makes them look identical.

• H: After a 180° turn, the top and bottom parts swap, but the letter remains unchanged.
• I: Similar to H, a 180° rotation leaves the letter I looking the same.
• O: A circle or oval 'O' has infinite lines of symmetry and looks the same after any rotation, but in the context of distinct positions, we consider it to have rotational symmetry of order 2 (or more).
• X: Rotating an X by 180° results in the same shape.

Letters with Only Rotational Symmetry (and No Line Symmetry)

The letters N, S, and Z are interesting because they do not have any lines of symmetry, but they do have rotational symmetry of order 2.

• N: If you rotate 'N' by 180°, it looks exactly the same. However, you cannot fold it along any line to make the two halves match.
• S: A 180° turn makes 'S' look identical to its starting position.
• Z: Like N and S, 'Z' has rotational symmetry of order 2 but no line symmetry.

Letters with No Rotational Symmetry

Most letters do not have rotational symmetry. If you rotate them by 180°, they will appear upside down or as a different character entirely.

For example, let's see what happens when we rotate the letter 'P' by 180°:

As you can see, the rotated shape is not the same as the original letter 'P'. Therefore, 'P' does not have rotational symmetry.

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Symmetry of Some Figures

We have learned about line symmetry and rotational symmetry in the previous sections. Now, let's consolidate our understanding by looking at the symmetry properties of some common geometrical figures and shapes. We will observe whether they have line symmetry, rotational symmetry, both, or neither.

Remember that every figure has at least rotational symmetry of order 1 (after a $360^\circ$ rotation), which is often considered trivial or "no rotational symmetry" in practical terms when the order is greater than 1.

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Combining Line and Rotational Symmetries

Many shapes possess both line symmetry and rotational symmetry.

• A square has multiple lines of symmetry and also exhibits rotational symmetry of a certain order greater than 1.

• An equilateral triangle is another figure that has both line symmetry and rotational symmetry.

• A circle is a classic example with the highest degree of both types of symmetry.

• Regular polygons in general, like a regular hexagon, have both types of symmetry.

However, some figures might have only one type of symmetry or neither (excluding the trivial $360^\circ$ rotational symmetry).

• A rectangle (which is not a square) has line symmetry but its rotational symmetry order is only 2.
• A rhombus (which is not a square) also has line symmetry but its rotational symmetry order is only 2.
• A parallelogram (which is not a rectangle or rhombus) has rotational symmetry of order 2 but no line symmetry.

• A scalene triangle or a general trapezium typically have no line symmetry and no rotational symmetry of order greater than 1.

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Summary of Symmetry Properties for Various Figures

The table below provides a summary of the line symmetry and rotational symmetry properties for several common shapes and figures.

Figure	Line Symmetry	Number of Lines of Symmetry	Rotational Symmetry	Order of Rotational Symmetry	Angle of Rotational Symmetry
Scalene Triangle	No	0	No (Order 1 only)	1	$360^\circ$
Isosceles Triangle	Yes	1	No (Order 1 only)	1	$360^\circ$
Equilateral Triangle	Yes	3	Yes	3	$120^\circ$
Rectangle (not square)	Yes	2	Yes	2	$180^\circ$
Rhombus (not square)	Yes	2	Yes	2	$180^\circ$
Square	Yes	4	Yes	4	$90^\circ$
Parallelogram (not rectangle/rhombus)	No	0	Yes	2	$180^\circ$
Kite (not rhombus)	Yes	1 (along one diagonal)	No (Order 1 only)	1	$360^\circ$
Isosceles Trapezium	Yes	1 (line joining midpoints of parallel sides)	No (Order 1 only)	1	$360^\circ$
Circle	Yes	Infinite	Yes	Infinite	Any Angle
Regular Pentagon	Yes	5	Yes	5	$72^\circ$
Regular Hexagon	Yes	6	Yes	6	$60^\circ$
...Regular n-gon	Yes	n	Yes	n	$\frac{360^\circ}{n}$

This table serves as a quick reference for the symmetry properties discussed in this chapter.

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