Question

Bestimmen sie die Ordnungen folgender geometrischer Abbildungen der affinen Ebenen über \(\mathbf{R}\) in sich: Spiegelung, Drehung um den Winkel \(360^{\circ} / n\), Punktspiegelung.

Short answer

Reflection: Order 2, Rotation by \(360^{\circ}/n\): Order \(n\), Point Reflection: Order 2.

Step-by-step solution

Understand the concept of order of a transformation

The order of a transformation is the smallest positive integer \(k\) such that the \(k\)-th application of the transformation returns the space to its original form. In simpler terms, it's how many times you need to perform the transformation until you end up with the identity transformation.

Determine the order of a reflection

A reflection across a line in a plane is an involution, meaning performing the transformation twice returns the original position. Thus, the order of a reflection is 2, since two reflections across the same line bring you back to the starting configuration.

Determine the order of a rotation

Consider a rotation by \(360^{\circ} / n\). Applying the rotation \(n\) times results in \(n \times (360^{\circ}/n) = 360^{\circ}\), which brings points around a circle back to their original positions. Therefore, the order of this rotation is \(n\).

Determine the order of a point reflection

A point reflection in geometry, often called a half-turn, maps a point \((x, y)\) to \((-x, -y)\). Doing this reflection twice returns each point to its original location because \((-(-x), -(-y)) = (x, y)\). Hence, the order of a point reflection is also 2.

Key concepts

Order of a Transformation

The term "order of a transformation" in affine geometry refers to the smallest number of times you must apply a transformation to bring a geometrical figure back to its original form. This concept helps us understand the cyclic nature of transformations. Here's a simple breakdown:

• The order tells us how many iterations are needed for the original configuration to reappear.
• If a transformation has an order of 1, the transformation is the identity transformation, which means applying it doesn't change anything.
• Transformations with higher orders go through a sequence before returning to their starting point.

Understanding the order is crucial in studying the symmetry and periodic properties of geometric shapes and transformations.

Reflection

Reflection in geometry is a transformation where every point of a geometric figure or shape is mirrored across a line, which acts like a mirror. Here’s an easy explanation:

• Reflection involves flipping a shape over a line, known as the line of reflection.
• The original shape and its reflection have the same size and shape but are reversed.
• When you reflect a shape across the same line again, it returns to its original position, which shows why the order of reflection is 2.

Think of reflection like folding a piece of paper along a line, where the shape on one side appears over the line— unchanged except for orientation.

Rotation

Rotation is another fundamental transformation found in geometry. It involves turning a shape around a fixed point which is known as the center of rotation. Let’s dive a little deeper:

• A rotation is characterized by the angle of rotation, for example, rotating by \(360^{\circ}/n\) means dividing a circle into \(n\) equal pieces.
• The order of a rotation by \(360^{\circ}/n\) is \(n\) because applying the rotation \(n\) times: \(n \times (360^{\circ}/n) = 360^{\circ}\) brings the figure back to its starting orientation.
• Each time you perform a rotation, the shape retains its size and shape, merely altering the position.

Understanding rotation helps in visualizing how objects move and return to their initial orientation in structured steps.

Point Reflection

Point reflection, also known as a half-turn, is a unique type of transformation in geometry. It involves reflecting each point of a shape through a specific point.

• For point reflection, every point \(x, y\) is mapped to \(-x, -y\).
• Performing a point reflection twice returns each point to its original location since \(-(-x), -(-y)) = (x, y)\).
• This is why the order of a point reflection is 2.

Point reflection is like flipping an object 180 degrees around a point, ensuring every element or point returns to where it began when repeated.