ILS 4430 Mathematics Lines of Reflection Problems

Question Description

I’m trying to study for my Mathematics course and I need some help to understand this question.

2 attachmentsSlide 1 of 2attachment_1attachment_1attachment_2attachment_2

Unformatted Attachment Preview

Topic 15 Classwork Assignment
Problem 1: Classify each wallpaper pattern as either p411, p4gm, or p4mm. Label enough lines of
reflection and centers of rotation in the patterns to support you answer. Use a black dotted line to
designate the lines of reflection of reflections. Use a “×” to designate a 90° rotation.
(a)
(b)
Pattern:
(c)
Pattern:
Pattern:
Problem 2: Classify each wallpaper pattern as either p311, p31m, or p3m1. Label enough lines of
reflection and centers of rotation in the patterns to support you answer. Use a black dotted line to
designate the lines of reflection of reflections. Use a “∆” to designate a 120° rotation.
(a)
(b)
Pattern:
Pattern:
(c)
Pattern:
Problem 3: Classify each wallpaper pattern as either p611 or p6mm. Label enough lines of reflection and
centers of rotation in the patterns to support you answer. Use a black dotted line to designate the lines
of reflection of reflections. Use a “*” to designate a 60° rotation.
(a)
(b)
Pattern:
Pattern:
Topic 15
Symmetry in Geometry:
Wallpaper Patterns Part II
Wallpaper Patterns
Wallpaper patterns are formed by repetitions of a motif in such a way as to cover a
plane. It can be shown that there are exactly seventeen wallpaper patterns.
All wallpaper patterns are generated by a basic motif which is acted on by the four
transformations: translations, rotations, reflections, and glide reflections.
Since the wallpaper is considered to cover the entire plane, there must be translations
in two different direction.
2
Classification of Wallpaper Patterns
The ICN notation used to classify wallpaper patterns consists of four characters, each
character determined based on the transformations in the pattern, as follows:
● The first character is a p or a c. There are only two wallpaper patterns of type c.
● The second character is n for the highest order of rotation: 1 for 0° rotation, 2 for
180° rotation, 3 for 120° rotation, 4 for 90°, or 6 for 60° rotation.
● The third and fourth characters are m if there are lines of reflection, g if there is no
line of reflection but there is a glide reflection, and 1 otherwise. The angle at which
these lines of symmetry meet also determines the pattern.
(*) International Crystallographic Notation
(**) In ICN notation, p stands for primitive cell, c stands for centered cell, m stands for
3
mirror , and g stands for glide reflection.
Classification of Wallpaper Patterns
Today We will consider wall paper patterns with 60°, 90°, and 120° rotations.
60°
90°
120°
4
Classification of Wallpaper Patterns
ICN Notation
p411
p4gm
p4mm
p311
p31m
p3m1
p611
p6mm
Short Form
p4
p4g
p4m
p3
p31m
p3m1
p6
p6m
5
Classification of Wallpaper Patterns
The following is a flow chart for wallpaper patterns with a smallest angle of
rotation equal to 90°.
45° Reflection
6
Classification of Wallpaper Patterns
The following is a flow chart for wallpaper patterns with a smallest angle of
rotation equal to 120°.
7
Classification of Wallpaper Patterns
The following is a flow chart for wallpaper patterns with a smallest angle of
rotation equal to 60°.
8
Wallpaper Patterns – p411 (p4)
×
Smallest Rotation Angle: 90°
Reflection: No
9
Wallpaper Patterns – p4gm (p4g)
×
Smallest Rotation Angle: 90°
Reflection: Yes
45° Reflection: No
45° Reflection
10
Wallpaper Patterns – p4mm (p4m)
×
Smallest Rotation Angle: 90°
Reflection: Yes
45° Reflection: Yes
45° Reflection
11
Wallpaper Patterns – p311 (p3)
∆
Smallest Rotation Angle: 120°
Reflection: No
12
Wallpaper Patterns – p31m (p31m)
∆
∆
Smallest Rotation Angle: 120°
Reflection: Yes
All Rotation Centers on Reflection Lines: No
13
Wallpaper Patterns – p3m1 (p3m1)
∆
Smallest Rotation Angle: 120°
Reflection: Yes
All Rotation Centers on Reflection Lines: Yes
14
Wallpaper Patterns – p611 (p6)
*
Smallest Rotation Angle: 60°
Reflection: No
15
Wallpaper Patterns – p6mm (p6m)
*
Smallest Rotation Angle: 60°
Reflection: Yes
16

Purchase answer to see full
attachment

Explanation & Answer:
3 Problems

Tags:
mathematics

Math Problems

lines of reflection

Student has agreed that all tutoring, explanations, and answers provided by the tutor will be used to help in the learning process and in accordance with Studypool’s honor code & terms of service.