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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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