Boston University Symmetry in Geometry Mathematics Worksheet

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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
Topic 16 Classwork Assignment
Problem 1: Label each polygon as either convex or non-convex.
(a)
(b)
Convex or Non-Convex:
Convex or Non-Convex:
Problem 2: Complete the table for the following regular pentagons.
Regular Polygon
Number of Sides
pentakaidecagon
octakaidecagon
icosagon
tetrakaicosagon
15
18
20
24
Vertex Angle
Problem 3: What is the vertex configuration for a regular tiling of hexagons?
Vertex Configuration:
Problem 4: Show kθ = 360° for a regular tiling of hexagons, where k is the number of regular polygons
meeting at a vertex, and θ is the vertex angle.
Problem 5: What is the vertex configuration for the following semiregular tiling? List the polygons in the
semiregular tiling. Explain how the semiregular tiling conforms to all five rules for semiregular tilings.
Vertex Configuration:
Polygons:
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 meeting 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.
Problem 6: What is the vertex configuration for the following semiregular tiling? List the polygons in the
semiregular tiling. Explain how the semiregular tiling conforms to all five rules for semiregular tilings.
Vertex Configuration:
Polygons:
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 meeting 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.
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
quad
penta
hexa
hepta
octa
nona
deca
hendeca
dodeca
Polygon
triangle
quadrilateral
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

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