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Chapter 11: Problem 57

Consider a border pattern in a horizontal direction with a repeating motif that has horizontal reflection symmetry. If the motif has symmetry type \(D_{4},\) what is the symmetry type of the border pattern?

Short Answer

Expert verified
The symmetry type of the border pattern with a motif of \(D_{4}\) symmetry in horizontal direction is \(D_{2}\).

Step by step solution

01

Understanding Symmetry

A motif with symmetry type \(D_{4}\) means it can rotate 0掳, 90掳, 180掳, and 270掳, and also reflect along the vertical, horizontal, and both diagonal axes.
02

Apply the Symmetry to a Border Pattern

When we construct a border pattern in a horizontal or vertical direction with this motif and it has horizontal reflection symmetry, the symmetry of the border motif will lose some of its original \(D_{4}\) symmetries. Because we place the motifs in a single line, it would retain its horizontal reflection symmetry but lose its vertical and diagonal reflections, and it would also lose its 90掳 and 270掳 rotations.
03

Resulting Symmetry Type of Border Pattern

The symmetry group left is the one containing two rotations (0掳 and 180掳) and one reflection (in the horizontal axis). This group is denoted \(D_{2}\), which is the result.

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Key Concepts

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

Reflection Symmetry
Reflection symmetry is like looking in a mirror. It means that one half of an object or design is a mirror image of the other half. Think of a butterfly whose left wing mirrors the right. This symmetry can be seen in various forms:
  • Horizontal reflection: Mirror line runs sideways.
  • Vertical reflection: Mirror line runs up and down.
  • Diagonal reflection: Mirror line runs slantwise.
To understand how this relates to a border pattern, imagine a pattern that can flip over a horizontal line without changing its appearance. In such patterns, horizontal reflection symmetry helps create balance and harmony. As seen in the original exercise, even though the initial motif may have multiple types of reflection symmetries, when it's laid out in a line, some of these symmetries might be lost.
Rotation Symmetry
Rotation symmetry occurs when a shape or design looks the same even after a certain amount of rotation. Picture a clock that can turn a quarter, half, or three-quarters around its center and still look the same. The degree of rotation in a rotation symmetry can vary, and common degrees are:
  • 90掳 鈥� a quarter turn
  • 180掳 鈥� a half turn
  • 270掳 鈥� three-quarters turn
  • 360掳 鈥� a full turn, returning to the starting position
For motifs with symmetry type \( D_4 \), rotations include 0掳, 90掳, 180掳, and 270掳. This means the motif can appear identical when rotated by these angles. However, in the context of a border pattern, only the rotations that align with the pattern's layout remain; usually, just 0掳 and 180掳.
Symmetry Groups
Symmetry groups are a way of classifying the symmetries of objects or designs. These groups help us understand the mathematical structure behind symmetrical patterns. Each symmetry group is defined by:
  • The types of transformations it includes, such as reflections and rotations.
  • How these transformations can combine.
The designation \( D_n \) is typically used for symmetry groups where an object has n rotational and reflection symmetries. For instance, \( D_4 \) indicates multiple rotation and reflection symmetries, while \( D_2 \) is less complex, exhibiting fewer symmetrical transformations. Understanding these groups is crucial for identifying and predicting changes in symmetrical designs, like when a motif is turned into a border pattern.
D4 Symmetry
The \( D_4 \) symmetry group is robust, encompassing a selection of swaps and rotational symmetries. Specifically, it refers to figures that can:
  • Rotate by 0掳, 90掳, 180掳, and 270掳
  • Reflect across horizontal, vertical, and both diagonal axes
This group is often associated with squares or objects that have similar symmetry capabilities. When these motifs are organized into a linear border pattern, the layout affects their symmetry. As the original exercise explains, a linear horizontal arrangement retains the horizontal reflections and full or half turns but loses other complex symmetries. Hence, the resulting symmetry group becomes \( D_2 \), which is a simpler version with reduced symmetry.
Border Patterns
Border patterns illustrate how simple repetitions of symmetrical motifs form attractive designs. They repeat a basic motif along a straight line, either horizontally or vertically. Factors crucial in border patterns include:
  • The basic symmetry of the motif
  • The spacing and alignment of repeated motifs
Depending on these factors, the overall pattern might gain or lose some of its original symmetry. For instance, the exercise highlighted how horizontal reflection symmetry is preserved in a motif with \( D_4 \) symmetry when it becomes a border pattern along one direction, resulting in a \( D_2 \) symmetry type. It's a fascinating way to see how symmetry changes during the transformation from a single motif to an extended border.

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Most popular questions from this chapter

The minute hand of a clock is pointing at the number 9 , and it is then wound clockwise 7080 degrees. (a) How many full hours has the hour hand moved? (b) At what number on the clock does the minute hand point at the end?

In each case, state whether the rigid motion \(M\) is proper or improper. (a) \(M\) is the product of a proper rigid motion and an improper rigid motion. (b) \(M\) is the product of an improper rigid motion and an improper rigid motion. (c) \(A\) is the product of a reflection and a rotation. (d) \(M\) is the product of two reflections.

Suppose that a rigid motion \(M\) is the product of a reflection with axis \(l_{1}\) and a reflection with axis \(l_{2},\) where \(l_{1}\) and \(l_{2}\) in tersect at a point \(C .\) Explain why \(M\) must be a rotation with center \(C\)

In each case, give an answer between \(0^{\circ}\) and \(360^{\circ}\). (a) A clockwise rotation by an angle of \(710^{\circ}\) is equivalent to a counterclockwise rotation by an angle of_____ (b) A clockwise rotation by an angle of \(710^{\circ}\) is equivalent to a clockwise rotation by an angle of ___ (c) A counterclockwise rotation by an angle of \(7100^{\circ}\) is equivalent to a clockwise rotation by an angle of ____ (d) A clockwise rotation by an angle of \(71,000^{\circ}\) is equivalent to a clockwise rotation by an angle of___

(a) Explain why a border pattern cannot have a reflection symmetry along an axis forming a \(45^{\circ}\) angle with the direction of the pattern. (b) Explain why a border pattern can have only horizontal and/or vertical reflection symmetry.
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