An example related to rotational symmetry is a shape like a rectangle, which possesses this property, and such shapes often form parts of patterns or are foundational to understanding patterns with this type of symmetry.
Understanding Rotational Symmetry
Rotational symmetry is a fundamental concept in geometry and design. It describes how a shape or pattern appears when rotated around a central point.
As defined by the reference:
Rotational symmetry is the number of times a shape can “fit into itself” when it is rotated 360 degrees about its centre.
This means that during a full 360-degree turn, the shape or pattern aligns perfectly with its original position multiple times before returning to the start. The number of times it aligns is called the order of rotational symmetry.
Example from the Reference: The Rectangle
The provided reference uses a simple shape to illustrate this concept:
E.g. A rectangle has a rotational symmetry of order 2 shown below where one vertex is highlighted with a circle and the centre of the shape is indicated with an 'x'.
Let's break down what this means for a rectangle:
- Order 2: A rectangle has rotational symmetry of order 2.
- Rotation: When you rotate a rectangle around its exact center, it will look identical to its original position twice within a full 360-degree rotation.
- Positions: These two positions are the starting (0 degrees) position and the position after a 180-degree rotation.
Imagine the rectangle with one corner marked. After a 180-degree turn, the marked corner is now where the opposite corner used to be, but the rectangle as a whole looks exactly the same. You need to rotate it another 180 degrees (total 360) to get the marked corner back to its original spot.
Connecting Shapes to Patterns
While the reference uses a single shape (a rectangle) as an example of something having rotational symmetry, this property is central to creating and understanding patterns that exhibit rotational symmetry. A pattern can have rotational symmetry in a couple of ways:
- Individual Elements: The repeating shapes or motifs within a pattern might individually possess rotational symmetry (like repeating stars, pinwheels, or the rectangles from the example).
- Overall Arrangement: The arrangement of the elements across the entire pattern or a section of it can exhibit rotational symmetry around a central point.
For instance, if you arranged four rectangles from the example around a central point, you could create a larger pattern that might also have rotational symmetry, possibly of a higher order depending on the arrangement.
Examples of Rotational Symmetry in Patterns and Shapes
Many common shapes and patterns display rotational symmetry. Here are a few examples:
- Shapes:
- Squares (Order 4)
- Equilateral Triangles (Order 3)
- Regular Hexagons (Order 6)
- Circles (Infinite Order)
- Patterns/Objects:
- Snowflakes (often Order 6)
- Pinwheels (Order depends on the number of blades, e.g., Order 4)
- Many types of Mandalas (often high orders)
- Certain tiled designs when viewed around a central point
- Fan blades or propellers
Shapes like the rectangle mentioned in the reference, which have rotational symmetry, are the building blocks or fundamental examples that help us understand more complex patterns exhibiting this property.
| Shape/Pattern Example | Rotational Symmetry Order |
|---|---|
| Rectangle | 2 |
| Square | 4 |
| Equilateral Triangle | 3 |
| Snowflake | Typically 6 |
| Pinwheel | Varies (e.g., 4 or 5) |
Understanding the rotational symmetry of individual components, like the rectangle described in the reference, is key to recognizing and creating patterns with this beautiful geometric property.
