Symmetry: Rotational

Rotational Symmetry: Definition

Beyond the static mirror-like quality of reflectional symmetry, figures can also exhibit symmetry related to movement, specifically rotation. This is known as rotational symmetry.

A figure possesses rotational symmetry if, when rotated about a fixed central point by an angle less than a complete turn ($360^\circ$), it looks exactly the same as it did in its original position. In other words, the figure maps onto itself after such a rotation.

To have rotational symmetry, a figure must be able to coincide with its original position after a rotation of some angle $\theta$, where $0^\circ < \theta \leq 360^\circ$. The point around which the rotation occurs is called the centre of rotation.

The illustration depicts a square rotating around its centre. Observe that the square looks identical to its starting position after rotations of $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$. Since it coincides with its original position at angles less than $360^\circ$ (at $90^\circ$, $180^\circ$, and $270^\circ$), the square has rotational symmetry.

If a figure only coincides with its original position after a full $360^\circ$ rotation, it means it has no rotational symmetry other than the trivial rotation which leaves every figure unchanged. This is sometimes stated as having rotational symmetry of order 1.

The concept of rotational symmetry involves understanding the angle of rotation and how many times the figure coincides with its original position during a full turn, which leads to the idea of the 'order' of rotational symmetry, discussed in the next section.

Centre of Rotation, Angle of Rotation, and Order of Rotational Symmetry

To fully understand and describe rotational symmetry, we need to be familiar with three specific terms:

1. Centre of Rotation:

The centre of rotation is the fixed point around which a figure rotates. This point can be inside the figure, outside the figure, or on the figure's boundary. When a figure has rotational symmetry, it means it can be rotated about this specific point and coincide with its original position.

For many symmetrical figures, the centre of rotation is their geometric centre. For example:

• For a square or a rectangle, the centre of rotation is the intersection point of its diagonals.
• For a circle, the centre of rotation is its centre.
• For a regular triangle (equilateral), the centre of rotation is its centroid (the intersection of medians, altitudes, etc.).

In the image, the point where the two diagonals of the rectangle intersect is its centre of rotation.

2. Angle of Rotation:

The angle of rotation is the smallest positive angle (greater than $0^\circ$) through which a figure must be rotated about its centre of rotation so that it fits exactly onto itself. It is the minimum angle required for the figure to look identical to its original state. This angle is always measured in degrees and is strictly between $0^\circ$ and $360^\circ$.

If the figure only matches its original position after a full $360^\circ$ turn, it is considered to have no rotational symmetry in the non-trivial sense, and the angle of rotation would be $360^\circ$. However, often the angle of rotation is referred to as the smallest angle that is *not* $0^\circ$ or $360^\circ$ that makes the figure coincide with itself.

3. Order of Rotational Symmetry:

The order of rotational symmetry is a measure of how many times a figure coincides with its original position during a complete rotation of $360^\circ$ around its centre of rotation. It tells us how many 'fits' or coincidences occur in one full turn.

The order of rotational symmetry is always a positive integer.

There is a direct relationship between the angle of rotation and the order of rotational symmetry. If $\theta$ is the smallest angle of rotation (in degrees), then the order of rotational symmetry is given by the formula:

For example, if the smallest angle of rotation is $90^\circ$, the order is $\frac{360^\circ}{90^\circ} = 4$. This means the figure coincides with its original position 4 times during a $360^\circ$ rotation (at $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$). The order is 4.

If a figure only coincides with itself after a full $360^\circ$ rotation, the smallest angle of rotation is $360^\circ$. The order would be $\frac{360^\circ}{360^\circ} = 1$. An order of 1 indicates that the figure has no rotational symmetry other than the trivial one, as every figure maps onto itself after a $360^\circ$ rotation.

Answer:

An equilateral triangle is a regular polygon with 3 equal sides and 3 equal angles. Its centre of rotation is its geometric centre (centroid).

Imagine rotating the equilateral triangle around its centre. For the triangle to coincide with its original position, each vertex must move to the position previously occupied by another vertex (or itself).

Since there are 3 vertices that are equally spaced around the centre, the triangle will coincide with itself when one vertex moves to the position of the next vertex.

The total angle of rotation for a full turn is $360^\circ$. Since there are 3 such "fits" in a full turn, the smallest angle of rotation will be $360^\circ$ divided by the number of vertices (or sides).

Smallest Angle of Rotation $=$ $\frac{360^\circ}{\text{Number of sides}}$

Smallest Angle of Rotation $=$ $\frac{360^\circ}{3} = 120^\circ$

So, the angle of rotation for an equilateral triangle is $120^\circ$.

The order of rotational symmetry is the number of times the figure coincides with itself during a $360^\circ$ rotation. This is given by the formula:

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

Order of Rotational Symmetry $=$ $\frac{360^\circ}{120^\circ} = 3$

The equilateral triangle coincides with itself at rotations of $120^\circ$, $240^\circ$, and $360^\circ$. Thus, it has an order of rotational symmetry of 3.

Rotational Symmetry in Various Geometric Figures

As discussed, different geometric shapes exhibit varying degrees of rotational symmetry, quantified by their angle of rotation and order of rotational symmetry. These values are determined by the inherent structure and regularity of the figure.

Let's examine the rotational symmetry properties of some common geometric figures:

Let's discuss some of these in more detail:

Regular Polygons:

For any regular polygon with $n$ sides, all sides are equal in length, and all interior angles are equal. This high degree of symmetry results in a straightforward pattern for rotational symmetry. The centre of rotation is the geometric centre of the polygon (the point equidistant from all vertices and sides). The figure will coincide with itself when rotated by an angle that moves each vertex to the position of the next vertex.

The total angle for a full rotation is $360^\circ$. Since there are $n$ identical "sections" or rotations needed to return to the original position, the smallest angle of rotation is $360^\circ$ divided by the number of sides, $n$.

The order of rotational symmetry is the number of times the figure coincides with itself during a $360^\circ$ rotation. Since the angle of rotation is $(\frac{360}{n})^\circ$, the order is:

Thus, a regular polygon with $n$ sides has rotational symmetry of order $n$. Examples:

• Equilateral triangle ($n=3$): Angle $120^\circ$, Order 3.
• Square ($n=4$): Angle $90^\circ$, Order 4.
• Regular pentagon ($n=5$): Angle $72^\circ$, Order 5.
• Regular hexagon ($n=6$): Angle $60^\circ$, Order 6.

Irregular Polygons:

For most irregular polygons, rotational symmetry is absent (order 1). A notable exception among quadrilaterals is the parallelogram. Although not all sides/angles are equal, a parallelogram has $180^\circ$ rotational symmetry about the intersection of its diagonals (its centre). A rotation by $180^\circ$ maps each vertex to its opposite vertex. Thus, general parallelograms (including rectangles and rhombuses, but excluding squares which have higher order) have rotational symmetry of order 2, with an angle of rotation of $180^\circ$. Scalene triangles, general trapeziums, and kites do not possess rotational symmetry (order 1).

Circle:

A circle is the most symmetrical plane figure. Any rotation about its centre leaves the circle unchanged. Since this is true for any angle of rotation, the circle has rotational symmetry for every angle. Consequently, its order of rotational symmetry is infinite.

Visual Examples of Rotational Symmetry

Here are visual representations illustrating rotational symmetry for some figures:

It is possible for a figure to have:

• Both line symmetry and rotational symmetry (e.g., square, equilateral triangle, circle).
• Only line symmetry, but no rotational symmetry (order > 1) (e.g., isosceles triangle, kite).
• Only rotational symmetry (order > 1), but no line symmetry (less common in basic geometric shapes, but possible with specific patterns or designs).
• Neither line symmetry nor rotational symmetry (order > 1) (e.g., scalene triangle, general trapezium).

Analyzing the presence and type of symmetry helps in classifying and understanding the properties of different geometric figures.

Answer:

A rhombus is a parallelogram with all four sides equal in length. It does not necessarily have equal angles (unless it's a square).

The centre of rotation for a rhombus is the intersection point of its diagonals.

If you rotate a rhombus by $90^\circ$ about its centre (unless it's a square), it will not coincide with its original position.

However, if you rotate a rhombus by $180^\circ$ about its centre, each vertex will map to the position of the opposite vertex, and the figure will coincide with its original position.

The smallest angle of rotation for a rhombus (that is not a square) is $180^\circ$.

The order of rotational symmetry is calculated as:

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

Order of Rotational Symmetry $=$ $\frac{360^\circ}{180^\circ} = 2$

A rhombus coincides with its original position after rotations of $180^\circ$ and $360^\circ$. Thus, it has an order of rotational symmetry of 2.

Note: A square is a special type of rhombus. A square has an angle of rotation of $90^\circ$ and an order of rotational symmetry of 4, which is consistent with the regular polygon formula for $n=4$. The general properties of a rhombus (sides equal, diagonals bisect each other perpendicularly) guarantee $180^\circ$ rotational symmetry.