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Chapter 4: Problem 6

Find the order of every element in the symmetry group of the square, \(D_{4}\).

Short Answer

Expert verified
Answer: In the symmetry group \(D_{4}\), the orders of the elements are as follows: the order of the identity element (\(e\)) is 1, the order of the 90-degree rotation element (\(r\)) is 4, the order of the 180-degree rotation element (\(r^2\)) is 2, the order of the 270-degree rotation element (\(r^3\)) is 4, and the order of the reflection elements (\(h\), \(v\), \(d_1\), and \(d_2\)) is 2.

Step by step solution

01

List all the elements of \(D_{4}\)

There are 4 rotation elements: Identity (\(e\)), 90-degree rotation (\(r\)), 180-degree rotation (\(r^2\)), and 270-degree rotation (\(r^3\)). There are also 4 reflection elements: horizontal reflection (\(h\)), vertical reflection (\(v\)), and two diagonal reflections (\(d_1\) and \(d_2\)).
02

Find the order of rotation elements

1. Identity (\(e\)): Since applying the identity operation once results in no change, the order of \(e\) is 1. 2. 90-degree rotation (\(r\)): Applying \(r\) four times results in a full rotation (360 degrees) to the original position. Therefore, the order of \(r\) is 4. 3. 180-degree rotation (\(r^2\)): Applying \(r^2\) two times results in a full rotation (360 degrees) to the original position. Therefore, the order of \(r^2\) is 2. 4. 270-degree rotation (\(r^3\)): Applying \(r^3\) four times results in a full rotation (360 degrees) to the original position. Therefore, the order of \(r^3\) is 4.
03

Find the order of reflection elements

1. Horizontal reflection (\(h\)): Applying the horizontal reflection twice results in the original position. Therefore, the order of \(h\) is 2. 2. Vertical reflection (\(v\)): Applying the vertical reflection twice results in the original position. Therefore, the order of \(v\) is 2. 3. Diagonal reflection (\(d_1\)): Applying the diagonal reflection (over one of the diagonals) twice results in the original position. Therefore, the order of \(d_1\) is 2. 4. Diagonal reflection (\(d_2\)): Applying the diagonal reflection (over the other diagonal) twice results in the original position. Therefore, the order of \(d_2\) is 2. Summary: - Order of \(e\) is 1 - Order of \(r\) is 4 - Order of \(r^2\) is 2 - Order of \(r^3\) is 4 - Order of \(h\) is 2 - Order of \(v\) is 2 - Order of \(d_1\) is 2 - Order of \(d_2\) is 2

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotation Elements
In geometry, rotation elements are transformation operations that rotate a shape around a fixed point, like the center of a square. For the symmetry group of the square, denoted as \(D_4\), there are four essential rotation elements:
  • Identity \(e\): This rotation doesn't change the position of the square. It is the equivalent of doing nothing at all and is called the identity element.
  • 90-degree rotation \(r\): Rotates the square in a clockwise direction by 90 degrees.
  • 180-degree rotation \(r^2\): Continues the rotation further to 180 degrees, which is equivalent to flipping the square upside down.
  • 270-degree rotation \(r^3\): Completes the full turn for 270 degrees when the square is finally rotated back to its original position at 360 degrees (since 270 + 90 = 360).
The rotation operations have different 'orders', which describe the number of times you need to apply the operation for the square to return to its original configuration. For instance, the order of the 90-degree rotation is 4 because you must apply \(r\) four times to rotate a full 360 degrees.
Reflection Elements
Reflection elements involve flipping a shape over a line, which acts as a mirror reflection, changing its orientation. In the context of the symmetry group of the square, there are four reflection elements:
  • Horizontal reflection \(h\): This flips the square over a horizontal axis, meaning the bottom swaps with the top.
  • Vertical reflection \(v\): This reflects the square over a vertical axis, swapping the left side with the right side.
  • Diagonal reflection \(d_1\): Reflects the square over one of the diagonals.
  • Diagonal reflection \(d_2\): Reflects the square over the other diagonal.
For reflection elements, the order is almost always 2. This means applying the reflection operation twice returns the square to its original state, because performing two identical symmetrical flips brings the object back to its starting position.
Order of Elements
The 'order' of an element in a symmetry group is a fundamental concept that tells us how many applications of an element (like a rotation or reflection) are needed to bring a figure back to its initial position. In the symmetry group of the square, \(D_4\), the order of elements varies:
  • The identity element \(e\): Has an order of 1, as it requires no application to revert to the initial state.
  • The rotation \(r\) of 90 degrees and \(r^3\) 270 degrees: Each have an order of 4, because only after applying them four times does the square return to its original position.
  • The rotation \(r^2\) of 180 degrees: Has an order of 2, meaning applying it twice will complete a full cycle (360 degrees).
  • Each of the reflection elements \(h\), \(v\), \(d_1\), and \(d_2\): Share an order of 2, because performing each reflection two times will bring the square back to its original configuration.
Understanding the order of elements is crucial for comprehending how different symmetry operations interact and combine in various transformations."

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