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Topic 13 Classwork Assignment
Problem 1: Classify the following frieze patterns in “Hop-Step-Jump” notation.
(a)
Hop-Step-Jump Notation:
(b)
Hop-Step-Jump Notation:
(c)
Hop-Step-Jump Notation:
(d)
Hop-Step-Jump Notation:
(e)
Hop-Step-Jump Notation:
(f)
Hop-Step-Jump Notation:
(g)
Hop-Step-Jump Notation:
Problem 2: Classify the following frieze patterns in both ICN and “Hop-Step-Jump” notation. Label
enough lines of reflection and centers of rotation to support you answer.
(a)
ICN Notation:
Hop-Step-Jump Notation:
(b)
ICN Notation:
Hop-Step-Jump Notation:
(c)
ICN Notation:
Hop-Step-Jump Notation:
(d)
ICN Notation:
Hop-Step-Jump Notation:
(e)
ICN Notation:
Hop-Step-Jump Notation:
(f)
ICN Notation:
Hop-Step-Jump Notation:
(g)
ICN Notation:
Hop-Step-Jump Notation:
Topic 13
Symmetry in Geometry:
Frieze Patterns
Frieze Patterns
Frieze patterns are formed by an infinite number of repetitions of a motif along a
line. Each repetition of the basic motif must be exactly the same in size and shape as
the original figure. It can be shown that there are exactly seven frieze patterns.
2
Frieze Patterns
All frieze patterns are generated by a basic motif which is acted on by the four
transformations: translations, rotations, reflections, and glide reflections.
center
line
? translations in the direction of the strip
? reflections across a line for lines running along the center of the strip or lines
perpendicular to the strip.
? glide reflection must combine a reflection across the centerline of the strip and a
translation in the direction of the strip
? rotation about a point on the center line by 180°
Any other transformations, other angles of rotation, or other lines of reflection, would
make the copies of the original motif migrate off the strip.
3
Classification of Frieze Patterns
The seven frieze patterns may be classified with the informal notation called “HopStep-Jump” notation, as follows:
? hop
? step
? jump
? sidle or sidestep
? spinning hop
? spinning jump
? spinning sidle or spinning sidestep
4
Classification of Frieze Patterns
Frieze Pattern
Hop-Step-Jump Notation
Hop
Step
Jump
Sidle or Sidestep
Spinning Hop
Spinning Jump
Spinning Sidle or Spinning Sidestep
5
Classification of Frieze Patterns
Classify the following frieze patterns in “Hop-Step-Jump” notation.
(a) Hop-Step-Jump Notation: sidle or sidestep
(b) Hop-Step-Jump Notation: spinning jump
(c) Hop-Step-Jump Notation: hop
(d) Hop-Step-Jump Notation: spinning sidle or spinning sidestep
6
Classification of Frieze Patterns
(e) Hop-Step-Jump Notation: step
(f) Hop-Step-Jump Notation: spinning hop
(g) Hop-Step-Jump Notation: jump
7
Classification of Frieze Patterns
ICN notation is a more formal way to classify frieze patterns. ICN notation consists of
four characters, each character determined based on the transformations in the pattern,
as follows:
? The first character is always p.
? The second character is m if there is a reflection perpendicular to the center line and
l if there is no reflection perpendicular to the center line.
? The third character is m if there is a reflection parallel to the center line, g if there is
not a reflection but there is a glide reflection parallel to the center line, and l if there is
no reflection or glide reflection parallel to the center line.
? The fourth character is 2 if there is a 180° rotation and l if there is no 180° rotation.
(*) International Crystallographic Notation
(**) In ICN notation, p stands for primitive cell, m stands for mirror , and g stands for
8
glide reflection.
Classification of Frieze Patterns
ICN Notation: p111
Hop-Step-Jump Notation: hop, step, jump, sidle or sidestep, spinning hop, spinning jump,
spinning sidle or spinning sidestep
? The first character is always p.
? The second character is m if there is a reflection perpendicular to the center line and
l if there is no reflection perpendicular to the center line.
m 1
? The third character is m if there is a reflection parallel to the center line, g if there is
not a reflection but there is a glide reflection parallel to the center line, and l if there is
no reflection or glide reflection parallel to the center line.
m g l
? The fourth character is 2 if there is a 180° rotation and l if there is no 180° rotation.
9
2 1
Classification of Frieze Patterns
ICN Notation: p1g1
Hop-Step-Jump Notation: hop, step, jump, sidle or sidestep, spinning hop, spinning jump,
spinning sidle or spinning sidestep
? The first character is always p.
? The second character is m if there is a reflection perpendicular to the center line and
l if there is no reflection perpendicular to the center line.
m 1
? The third character is m if there is a reflection parallel to the center line, g if there is
not a reflection but there is a glide reflection parallel to the center line, and l if there is
no reflection or glide reflection parallel to the center line.
m g l
? The fourth character is 2 if there is a 180° rotation and l if there is no 180° rotation.
10
2 1
Classification of Frieze Patterns
ICN Notation: p1m1
Hop-Step-Jump Notation: hop, step, jump, sidle or sidestep, spinning hop, spinning jump,
spinning sidle or spinning sidestep
? The first character is always p.
? The second character is m if there is a reflection perpendicular to the center line and
l if there is no reflection perpendicular to the center line.
m 1
? The third character is m if there is a reflection parallel to the center line, g if there is
not a reflection but there is a glide reflection parallel to the center line, and l if there is
no reflection or glide reflection parallel to the center line.
m g l
? The fourth character is 2 if there is a 180° rotation and l if there is no 180° rotation.
11
2 1
Classification of Frieze Patterns
ICN Notation: pm11
Hop-Step-Jump Notation: hop, step, jump, sidle or sidestep, spinning hop, spinning jump,
spinning sidle or spinning sidestep
? The first character is always p.
? The second character is m if there is a reflection perpendicular to the center line and
l if there is no reflection perpendicular to the center line.
m 1
? The third character is m if there is a reflection parallel to the center line, g if there is
not a reflection but there is a glide reflection parallel to the center line, and l if there is
no reflection or glide reflection parallel to the center line.
m g l
? The fourth character is 2 if there is a 180° rotation and l if there is no 180° rotation.
12
2 1
Classification of Frieze Patterns
ICN Notation: p112
Hop-Step-Jump Notation: hop, step, jump, sidle or sidestep, spinning hop, spinning jump,
spinning sidle or spinning sidestep
°
? The first character is always p.
? The second character is m if there is a reflection perpendicular to the center line and
l if there is no reflection perpendicular to the center line.
m 1
? The third character is m if there is a reflection parallel to the center line, g if there is
not a reflection but there is a glide reflection parallel to the center line, and l if there is
no reflection or glide reflection parallel to the center line.
m g l
? The fourth character is 2 if there is a 180° rotation and l if there is no 180° rotation.
13
2 1
Classification of Frieze Patterns
ICN Notation: pmm2
Hop-Step-Jump Notation: hop, step, jump, sidle or sidestep, spinning hop, spinning jump,
spinning sidle or spinning sidestep
°
? The first character is always p.
? The second character is m if there is a reflection perpendicular to the center line and
l if there is no reflection perpendicular to the center line.
m 1
? The third character is m if there is a reflection parallel to the center line, g if there is
not a reflection but there is a glide reflection parallel to the center line, and l if there is
no reflection or glide reflection parallel to the center line.
m g l
? The fourth character is 2 if there is a 180° rotation and l if there is no 180° rotation.
14
2 1
Classification of Frieze Patterns
ICN Notation: pmg2
Hop-Step-Jump Notation: hop, step, jump, sidle or sidestep, spinning hop, spinning jump,
spinning sidle or spinning sidestep
°
? The first character is always p.
? The second character is m if there is a reflection perpendicular to the center line and
l if there is no reflection perpendicular to the center line.
m 1
? The third character is m if there is a reflection parallel to the center line, g if there is
not a reflection but there is a glide reflection parallel to the center line, and l if there is
no reflection or glide reflection parallel to the center line.
m g l
? The fourth character is 2 if there is a 180° rotation and l if there is no 180° rotation.
15
2 1
Classification of Frieze Patterns
Frieze Pattern
Hop-Step-Jump Notation
ICN Notation
Hop
p111
Step
p1g1
Jump
p1m1
Sidle or Sidestep
pm11
Spinning Hop
p112
Spinning Jump
pmm2
Spinning Sidle or Spinning Sidestep
pmg2
16
Classification
of
Frieze
Patterns
Classify the following frieze patterns in both ICN and “Hop-Step-Jump” notation.
Label enough lines of reflection and centers of rotation to support you answer.
Solution:
(a) ICN Notation: p1m1
? The first character is always p.
Hop-Step-Jump Notation: jump
? The second character is m if there is a reflection
DDDDDDDDDDDDD perpendicular to the center line and l if there is no
(b) ICN Notation: p111
reflection perpendicular to the center line.
Hop-Step-Jump Notation: hop
GGGGGGGGGGGGG
(c) ICN Notation: pm11
Hop-Step-Jump Notation: sidle
YYYYYYYYYYYYY
(d) ICN Notation: p1g1
Hop-Step-Jump Notation: step
bpbpbpbpbpbp
? The third character is m if there is a reflection
parallel to the center line, g if there is not a
reflection but there is a glide reflection parallel
to the center line, and l if there is no reflection
or glide reflection parallel to the center line.
? The fourth character is 2 if there is a 180°
rotation and l if there is no 180° rotation.
17
Classification of Frieze Patterns
Solution(continued):
(e) ICN Notation: pmg2
? The first character is always p.
Hop-Step-Jump Notation: spinning sidle
?n?n?n?n?n?n?n
°
? The second character is m if there is a reflection
perpendicular to the center line and l if there is no
reflection perpendicular to the center line.
°
XXXXXXXXXXXXX
? The fourth character is 2 if there is a 180°
rotation and l if there is no 180° rotation.
(f) ICN Notation: p112
Hop-Step-Jump Notation: spinning hop
? The third character is m if there is a reflection
parallel to the center line, g if there is not a
°
ZZZZZZZZZZZZZ
reflection but there is a glide reflection parallel
to the center line, and l if there is no reflection
(g) ICN Notation: pmm2
Hop-Step-Jump Notation: spinning jump or glide reflection parallel to the center line.
18
Topic 14 Classwork Assignment
Problem 1: Classify each wallpaper pattern as either p111, p1m1, p1g1, or c1m1. Label enough lines of
reflection in the patterns to support you answer. Use a black dotted line to designate the lines of
reflection of reflections and a red dotted line to designate the lines of reflection of glide reflections.
(a)
(b)
Pattern:
(c)
Pattern:
(d)
Pattern:
Pattern:
Problem 2: Classify each wallpaper pattern as either p211, p2mm, p2mg, p2gg, or c2mm. 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 and a red dotted line to designate the lines of reflection
of glide reflections. Use a “°” to designate a 180° rotation.
(a)
(b)
Pattern:
(c)
Pattern:
(d)
Pattern:
(e)
Pattern:
Pattern:
Topic 14
Symmetry in Geometry:
Wallpaper Patterns – Part I
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
We will first consider wallpaper patterns with a smallest angle of rotation of 0°
and a smallest angle of rotation of 180°.

180°
4
Classification of Wallpaper Patterns
ICN Notation
Short Form
p111
p1g1
p1m1
c1m1
p211
p2gg
p2mg
p2mm
c2mm
p1
pg
pm
cm
p2
pgg
pmg
pmm
cmm
5
Classification of Wallpaper Patterns
The following is a flow chart for wallpaper patterns with a smallest angle of
rotation equal to 0°.
6
Classification of Wallpaper Patterns
The following is a flow chart for wallpaper patterns with a smallest angle of
rotation equal to 180°.
7
Wallpaper Patterns – p111 (p1)
Smallest Rotation Angle: 0°
Reflection: No
Glide Reflection: No
8
Wallpaper Patterns – p1g1 (pg)
Smallest Rotation Angle: 0°
Reflection: No
Glide Reflection: Yes
9
Wallpaper Patterns – p1m1 (pm)
Smallest Rotation Angle: 0°
Reflection: Yes
Glide Reflection Parallel to the Reflection: No
10
Wallpaper Patterns – c1m1 (cm)
Smallest Rotation Angle: 0°
Reflection: Yes
Glide Reflection Parallel to the Reflection: Yes
11
Wallpaper Patterns – p211 (p2)
°
Smallest Rotation Angle: 180°
Reflection: No
Glide Reflection: No
12
Wallpaper Patterns – p2gg (pgg)
°
Smallest Rotation Angle: 180°
Reflection: No
Glide Reflection: Yes
13
Wallpaper Patterns – p2mg (pmg)
°
Smallest Rotation Angle: 180°
Reflection: Yes
Two Reflections: No
14
Wallpaper Patterns – p2mm (pmm)
° spinning jump
Smallest Rotation Angle: 180°
Reflection: Yes
Two Reflections: Yes
Rotation Centers on Reflection Lines: Yes
15
Wallpaper Patterns – c2mm (cmm)
°
Smallest Rotation Angle: 180°
Reflection: Yes
Two Reflections: Yes
Rotation Centers on Reflection Lines: No
16

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