# ILS 4430 BU Mathematics Hop Step Jump Notation Questions

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ILS 4430
Exam 2 MU
NAME: ____________________________
The following exam is worth 75 points.
While you are completing this exam, you may refer to any of the following:

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You may also discuss problems that are similar to the exam problems with others, including myself.
In reference to completing the specific eight problems on this exam, you may not collaborate with or
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If I suspect the work you submit for this exam is not your own, you will need to defend your work in a
(T12) Problem 1: Classify each pattern as either 0° ,60°, 90°, 120°, or 180°, according to the smallest
(6 points)
angle of rotation in the pattern.
(a)
(b)
Angle:
Angle:
(T13) Problem 2: Classify the following frieze patterns in “Hop-Step-Jump” notation.
(12 points)
(a)
(b)
Hop-Step-Jump Notation:
(c)
Hop-Step-Jump Notation:
(d)
Hop-Step-Jump Notation:
Hop-Step-Jump Notation:
1
(T14) Problem 3: Classify each wallpaper pattern as either p111, p1m1, p1g1, or c1m1.
(6 points)
(a)
(b)
Pattern:
Pattern:
(T14) Problem 4: Classify each wallpaper pattern as either p211, p2mm, p2mg, p2gg, or c2mm.
(6 points)
(a)
(b)
Pattern:
Pattern:
(T15) Problem 5: Classify each wallpaper pattern as either p411, p4gm, or p4mm.
(6 points)
(a)
(b)
Pattern:
Pattern:
2
(T15) Problem 6: Classify each wallpaper pattern as either p311, p31m, or p3m1.
(6 points)
(a)
(b)
Pattern:
Pattern:
(T15) Problem 7: Classify each wallpaper pattern as either p611 or p6mm.
(6 points)
(a)
(b)
Pattern:
Pattern:
3
(T16) Problem 8: Cross out any vertex configurations that violate the given rule. You may circle more
than one vertex configuration for each rule.
(12 points)
Rule 1: In a semiregular tiling of the plane, the sum of the vertex angles of the polygons meeting at each
vertex must be exactly 360°.
(a) 3.3.5.6
(b) 3.4.4.6
(c) 4.5.5
(d) 3.3.4.12
Rule 3: No semiregular tiling can have four or more different polygons meeting at a vertex. Thus, if a
semiregular tiling has four or more polygons meeting at a vertex, there must be some duplicates.
(a) 8.8.9.9
(b) 4.5.8.10.11
(c) 6.8.9.10.10
(d) 4.6.10
Rule 4: No semiregular tiling with exactly three polygons meeting at each vertex can have the vertex
configuration k.n.m where k is odd unless n = m.
(a) 4.7.12
(b) 5.6.8.9
(c) 11.12.12
(d) 5.6.10
(T16) Problem 9: Construct the dual tiling of the tiling.
(7 points)
4
(T17) Problem 10: Circle any of the following tiling modifications that have been used in the Escher-type
tiling. You may circle more than one tiling modification
(8 points)
(a) translation
(b) glide reflection
(c) midpoint rotation
(d) side rotation
5
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
∪∩∪∩∪∩∪∩∪∩∪∩∪∩

● 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
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
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
Topic 16
Symmetry in Geometry
Tilings – Part I
Polygons
A polygon is a closed plane figure with straight edges. The edges are called sides, and
the points where two edges meet are called vertices or corners. The interior of a
polygon is sometimes called its body.
Examples:
An n-gon is a polygon with n sides; for example, a triangle is a 3-gon.
2
Polygons
Polygons are commonly named by prefixes from Greek numbers.
Number of Sides
3
4
5
6
7
8
9
10
11
12
Prefix
tri
penta
hexa
hepta
octa
nona
deca
hendeca
dodeca
Polygon
triangle
pentagon
hexagon
heptagon
octagon
nonagon
decagon
hendecagon
dodecagon
3
Polygons
A polygon is a convex if given any two points, A and B, in the polygon, the line
segment AB lies in the polygon. A polygon that is not convex is called a non-convex
polygon.
Examples:
Convex Polygon
Non-Convex Polygon
4
Regular Polygons
A polygon is regular if all of its sides and all of its angles are equal; i.e., if it is be both
equilateral and equiangular.
Examples:
equilateral
triangle
square
regular
pentagon
regular
hexagon
regular
octagon
regular
nonagon
regular
decagon
regular
hendecagon
A polygon that is not regular is irregular.
regular
heptagon
regular
dodecagon
5
Regular Polygons
The vertex angle, θ, in degrees, of an n-sided regular polygon measures θ =
Example:
n−2 180°
.
n
The vertex angle of an equilateral triangle is θ = 60°.
60°
60°
60°
θ=
n−2 180° n = 3 3−2 180°
=
n
3
=
1 180°
3
= 60°
equilateral
triangle
n=3
6
Example: Regular Polygons
Calculate the vertex angle, θ, of a square.
Solution:
90°
90°
90°
90°
square
n=4
θ=
n−2 180° n = 4 4−2 180°
=
n
4
The vertex angle of a square is θ = 90°.
=
2 180°
4
= 90°
7
Vertex Angles of Some Regular Polygons
Regular Polygon
triangle
square
pentagon
hexagon
heptagon
octagon
nonagon
decagon
hendecagon
dodecagon
Number of Sides
3
4
5
6
7
8
9
10
11
12
Vertex Angle
60°
90°
108°
120°
128.57°
135°
140°
144°
147.27°
150°
8
Tilings
A tesselation or tiling of the plane is a pattern of repeated copies of figures covering
the plane so that the copies do not overlap and leave no gaps uncovered. The figures
are called the tiles.
9
Regular Tilings
A tiling is regular if it consists of repeated copies of a single regular polygon, meeting
edge to edge so that at every vertex the same number of polygons meet.
10
Regular Tilings
Squares, equilateral triangles and hexagons are the only three regular polygons that
may be positioned to tile the plane in a regular pattern.
Therefore, there are exactly three regular tilings.
Regular Tiling
of Squares
Regular Tiling of
Equilateral Triangles
Regular Tiling of
Hexagons
11
Regular Tilings: Vertex Configuration
We denote a regular tiling by describing the number of sides of the polygons meeting at
a vertex.
Example:
For a regular tiling of squares, we have:
four sides four sides

four sides four sides
Vertex Configuration: 4.4.4.4
12
Example: Regular Tilings: Vertex Configuration
What is the vertex configuration for a regular tiling of equilateral triangles?
Solution:
3 3
3 ● 3
3 3
Vertex Configuration: 3.3.3.3.3.3
13
Regular Tilings: Vertex Configuration
Regular
Polygon
triangle
Regular
Tiling
Vertex
Configuration
3.3.3.3.3.3
square
4.4.4.4
hexagon
6.6.6
14
Rule for Regular Tilings
Rule: In a tiling of the plane, the sum of the vertex angles of the regular polygons
meeting at each vertex must be exactly 360°. Therefore, for k regular polygons meeting
at a vertex, each with a vertex angle θ, kθ = 360°.
Example:
For a regular tiling of squares, we have:
k = 4 (Four squares meet at a vertex.)

θ = 90° (the vertex angle of a squares is 90°.)
kθ = 4 90° = 360°
15
Example: Regular Tilings
Show that kθ = 360° for a regular tiling of equilateral triangles, where k is the number
of regular polygons meeting at a vertex, and θ is the vertex angle.
Solution:

k = 6 (Six equilateral triangles meet at a vertex.)
θ = 60° (The vertex angle of an equilateral triangle is 60°.)
kθ = 6 60° = 360°
16
Semiregular Tilings
A semiregular or Archimedean tiling is a tiling in which each tile is a regular
polygon and each vertex is identical.
There are exactly eight semiregular tilings.
17
Semiregular Tilings
Semiregular Tiling of
Hexagons and Triangles
Semiregular Tiling of
Triangles, Squares, and Hexagons
Semiregular Tiling of
Octagons and Squares
Semiregular Tiling of
Squares, Hexagons, and Dodecagons
18
Semiregular Tilings
Semiregular Tiling of
Triangles and Squares
Semiregular Tiling of
Triangles and Hexagons
Semiregular Tiling of
Triangles and Squares
Semiregular Tiling of
Triangles and Dodecagons
19
Semiregular Tilings: Vertex Configuration
As with regular tilings, we denote semiregular tilings by describing the number of sides
of the polygons meeting at a vertex. For semiregular tilings, we begin with the number
of sides of the smallest polygon and list the number of sides of the remaining polygons
in either clockwise or counterclockwise order.
Example:
For a semiregular tiling hexagons and triangles, we have:
six sides
three sides

three sides six sides
Vertex Configuration: 3.6.3.6
20
Example: Semiregular Tilings: Vertex Configuration
What are the vertex configurations for the two semiregular tilings of triangles and
squares?
Solution:
4
4

3
3
3
Vertex Configuration: 3.3.3.4.4
4
3
3 ●
3 4
Vertex Configuration: 3.3.4.3.4
21
Semiregular Regular Tilings: Vertex Configuration
Regular
Polygons
Hexagons,
Triangles
Semiregular
Vertex
Tiling
Configuration
3.6.3.6
Regular
Polygons
Semiregular
Vertex
Tiling
Configuration
Triangles,
Squares
3.3.3.4.4
Octagons,
Squares
4.8.8
Triangles,
Squares
3.3.4.3.4
Triangles,
Squares,
Hexagons
Squares,
Hexagons,
Dodecagons
3.4.6.4
Triangles,
Hexagons
3.3.3.3.6
4.6.12
Triangles,
Dodecagons
3.12.12
22
Rules for Semiregular Tilings
Rule 1: In a semiregular tiling of the plane, the sum of the vertex angles of the
polygons meeting at each vertex must be exactly 360°.
Rule 2: A semiregular tiling must have at least three and no more than five polygons
meeting at each vertex.
Rule 3: No semiregular tiling can have four or more different polygons meeting at a
vertex. Thus, if a semiregular tiling has four or more polygons at a vertex, there must be
some duplicates.
Rule 4: No semiregular tiling with exactly three polygons meeting at each vertex can
have the vertex configuration k.n.m where k is odd unless n = m.
Rule 5: No semiregular tiling with exactly four polygons meeting at each vertex can
have the vertex configuration 3.k.n.m unless k = m.
23
Example: Semiregular Tilings
Explain how the semiregular tiling conforms to all five
3 6
rules for semiregular tilings.

Solution:
6 3
Vertex Configuration: 3.6.3.6
Polygons: triangle hexagon triangle hexagon
Rule 1: In a semiregular tiling of the plane, the sum of the vertex angles of the
polygons meeting at each vertex must be exactly 360°.
triangle hexagon triangle hexagon
60° + 120° + 60° + 120° = 360°
Rule 2: A semiregular tiling must have at least three and no more than five polygons
meeting at each vertex.
# of polygons meeting at each vertex = 4
Rule 3: No semiregular tiling can have four or more different polygons meeting at a
vertex. Thus, if a semiregular tiling has four or more polygons meeting at a vertex, there
must be some duplicates.
24
four polygons with two duplicates.
triangle hexagon triangle hexagon
Example: Semiregular Tilings
Solution (continued):
Vertex Configuration: 3.6.3.6
Polygons: triangle hexagon triangle hexagon
3 6

6 3
Rule 4: No semiregular tiling with exactly three polygons meeting at each vertex can
have the vertex configuration k.n.m where k is odd unless n = m.
The semiregular tiling 3.6.3.6 has exactly four polygons meeting at each vertex.
Therefore Rule 4 does not apply.
Rule 5: No semiregular tiling with exactly four polygons meeting at each vertex can
have the vertex configuration 3.k.n.m unless k = m.
3.6.3.6
3.k.n.m
k=6
n=3
m=6
k=m
25
Example: Semiregular Tilings
Which of the following semiregular tiling patterns violates Rule 4 for semiregular tiling
configurations?
7

Solution:
3
4
42

8
8
Vertex Configuration: 4.8.8
Vertex Configuration: 3.7.42
Rule 4: No semiregular tiling with exactly three polygons meeting at each vertex can
have the vertex configuration k.n.m where k is odd unless n = m.
k = 4 (even)
4.8.8
Since k is even, the semiregular tiling configuration 4.8.8
n=8
k.n.m
does not violate Rule 4.
m=8
k = 3 (odd) Since k is odd and n ≠ m, the semiregular tiling configuration
3.7.42
n=7
3.7.42 violates Rule 4.
k.n.m
26
m = 42
Example: Semiregular Tilings
Which of the following semiregular tiling patterns violates Rule 5 for semiregular tiling
configurations?
Solution:
4
3
4 3

6
4
6
4
Vertex Configuration: 3.4.6.4
Vertex Configuration: 3.4.4.6
Rule 5: No semiregular tiling with exactly four polygons meeting at a vertex can have
vertex configuration 3.k.n.m unless k = m.
k = 4 Since k ≠ m, the semiregular tiling configuration 3.4.4.6
3.4.4.6
n = 4 violates Rule 5.
3.k.n.m
m=6
3.4.6.4
3.k.n.m
k=4
n=6
m=4
Since k = m, the semiregular tiling configuration 3.4.6.4 does
not violate Rule 5.
27
Topic 17
Symmetry in Geometry
Tilings – Part II
Constructing Dual Tilings
To each polygonal tiling we can associate a dual tiling.
A dual tiling is a type of tiling that is superimposed a polygon tiling.
A dual tiling may or may not be regular or semiregular.
2
Constructing Dual Tilings
Example:
The dual tiling of the semiregular tiling 4.8.8 (grey) is a tiling of 45°- 45°- 90° right
triangles (blue).
3
Constructing Dual Tilings
To construct the dual tiling of a polygon tiling:
Step 1: Place a vertex at the center of each polygon in the original tiling.
Step 2: Whenever two polygons share an edge in the original tiling, draw a dual
edge connecting the new vertices at the centers of those polygons.
4
Example: Dual Tilings
Construct the dual tiling of the semiregular tiling 3.4.6.4.
Solution:
5
Irregular Tilings
An irregular tiling is a tiling of polygons that are not regular. Any quadrilateral can
tile the plane if the copies are put together right.
6
Irregular Tilings
Any convex quadrilateral can tile the plane.
7
Irregular Tilings
Similarly, any non-convex quadrilateral can tile the plane.
8
Example: Irregular Tilings
Show that any parallelogram can tile the plane.
Solution:
9
Reptiles
A reptile (short for repeating tile) is a tile that can be arranged to form a larger copy
of itself. The larger copy must be an exact scaled replica of the original.
Example:
10
Example: Reptiles
Show that the following shape is a reptile.
Solution:
11
Escher Tilings
Maurits Cornelis Escher was a mathematical artist.
Escher developed a own system of cataloging patterns, not only in terms of 17
wallpaper patterns, but also in terms of the relationship between motif and underlying
grid, and in terms of coloring.
12
Escher Tilings
To construct an Escher tiling,
Step 1: Choose a grid.

The grid my be a regular or semiregular tiling, or a tiling by irregular polygons.
Step 2: Modify a tile in the grid using one or more or the following transformations:

Translation
Glide Reflection
Midpoint Rotation
Side Rotation
Step 3: Use the modified tile to tile the plane.
13
Escher Tilings: Modifying a Tile Using Translation
To modify a tile using translation, the original tile must have at least one pair of
parallel sides of equal length.
Grid: squares, rectangles, parallelograms, hexagons.
Tile Modification: Cut out a section of the tile that leaves the vertices of the tile intact.
Translate this section to the parallel side of the tile. The modification may be repeated
for the other pair of parallel sides of the tile.
Example:
14
Escher Tilings: Modifying a Tile Using Translation
Escher Example:
Grid: squares
Modification: translation
15
Escher Tilings: Modifying a Tile Using Translation
Escher Example:
16
Escher Tilings: Modifying a Tiling Using Glide Reflection
To modify a tile using glide reflection, the original tile must have at least one pair of
parallel sides of equal length.
Grid: squares, rectangles, parallelograms, hexagons
Tile Modification: Take the original tile and cut something off of one of the parallel
sides. Flip the cut piece over and move it to the parallel side.
17
Escher Tilings: Modifying a Tiling Using Glide Reflection
Escher Like Example:
Grid: parallelograms
Modification: glide reflection and translation
18
Escher Tilings: Modifying a Tiling Using Glide Reflection
Escher Like Example:
Grid: parallelograms
Modification: glide reflection and translation
19
Escher Tilings: Modifying a Tiling Using Glide Reflection
Escher Like Example:
20
Escher Tilings: Modifying a Tiling Using Midpoint Rotation
To modify a tile using midpoint rotation, any original tiling may be used.
Grid: any tiling
Tile Modification: Cut out a section of the tile that begins at one endpoint and ends at
the midpoint of the same side. Rotate the section by 180° about the midpoint.
O
21
Escher Tilings: Modifying a Tile Using Midpoint Rotation
Escher Like Example:
Grid: parallelograms
O
Modification: midpoint rotation
O
22
Escher Tilings: Modifying a Tile Using Midpoint Rotation
Escher Like Example:
23
Escher Tilings: Modifying a Tiling Using Side Rotation
To modify a tile using side rotation, the original tile must have a pair of adjacent sides of
equal length
Grid: squares, equilateral triangles, rhombi, or regular hexagons.
Tile Modification: Cut out a section of the tile that that begins at one vertex of one of
the equal edges of the tile and ends at the other vertex. Rotate the section about the
vertex connecting the two equal sides and fix it to the adjacent edge. The modification
may be repeated for the other sides of the tile.
O
24
Escher Tilings: Modifying a Tile Using Side Rotation
Escher Example:
Modification: side rotation
Modification: side rotation
O
O
25
Escher Tilings: Modifying a Tile Using Side Rotation
Escher Example:
26
ILS 4430
Exam 2
NAME: ____________________________
The following exam is worth 75 points.
While you are completing this exam, you may refer to any of the following:

Materials posted to Ulearn for ILS 4430.
Any textbook, including the ILS 4430 textbooks.
Any website that is purely informational.
You-Tube, Kahn Academy, or similar videos.
You may also discuss problems that are similar to the exam problems with others, including myself.
In reference to completing the specific eight problems on this exam, you may not collaborate with or
obtain help from others in any way. This includes but is not limited to fellow students, friends, family
members, and tutors. Question and answer websites are also prohibited.
If I suspect the work you submit for this exam is not your own, you will need to defend your work in a
1
(T12) Problem 1: Classify each pattern as either 0° ,60°, 90°, 120°, or 180°, according to the smallest
(6 points)
angle of rotation in the pattern.
(a)
(b)
Angle: 0 degree.
Angle: 60 degrees.
(T13) Problem 2: Classify the following frieze patterns in “Hop-Step-Jump” notation.
(12 points)
(a)
(b)
Hop-Step-Jump Notation: tg
Hop-Step-Jump Notation: mt
(c)
(d)
Hop-Step-Jump Notation: t2
Hop-Step-Jump Notation: t2mm
2
(T14) Problem 3: Classify each wallpaper pattern as either p111, p1m1, p1g1, or c1m1.
(6 points)
(a)
(b)
Pattern: p1g1
Pattern: c1m1
(T14) Problem 4: Classify each wallpaper pattern as either p211, p2mm, p2mg, p2gg, or c2mm.
(6 points)
(a)
(b)
Pattern: p2mg
Pattern: p2gg
(T15) Problem 5: Classify each wallpaper pattern as either p411, p4gm, or p4mm.
(6 points)
(a)
(b)
Pattern: p411
Pattern: p4mm
3
(T15) Problem 6: Classify each wallpaper pattern as either p311, p31m, or p3m1.
(6 points)
(a)
(b)
Pattern: p3m1
Pattern: p311
(T15) Problem 7: Classify each wallpaper pattern as either p611 or p6mm.
(6 points)
(a)
(b)
Pattern: p6mm
Pattern: p611
4
(T16) Problem 8: Cross out any vertex configurations that violate the given rule. You may cross out
more than one vertex configuration for each rule.
(12 points)
Rule 1: In a semiregular tiling of the plane, the sum of the vertex angles of the polygons meeting at each
vertex must be exactly 360°.
(a) 3.3.6.6
(b) 4.8.4
(c) 3.4.9
(d) 7.7.8.12
Rule 3: No semiregular tiling can have four or more different polygons meeting at a vertex. Thus, if a
semiregular tiling has four or more polygons meeting at a vertex, there must be some duplicates.
(a) 3.4.9
(b) 8.9.10.11
(c) 5.6.6.11
(d) 4.7.8.9.11
Rule 4: No semiregular tiling with exactly three polygons meeting at each vertex can have the vertex
configuration k.n.m where k is odd unless n = m.
(a) 5.7.7
(b) 3.7.8.12
(c) 6.9.11
(d) 3.3.8
(T17) Problem 9: Construct the dual tiling of the tiling.
(7 points)
5
(T17) Problem 10: Circle any of the following tiling modifications that have been used in the Escher-type
tiling. You may circle more than one tiling modification.
(8 points)
(a) translation
(b) glide reflection
(c) midpoint rotation
(d) side rotation
6

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HopStepJump Notation

PATTERNS CLASSIFICATION

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