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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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Explanation & Answer:

3 Problems

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mathematics

Math Problems

lines of reflection

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