# ILS 4430 Mathematics Lines of Reflection Problems

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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

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Math Problems

lines of reflection

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