lesson 6-9 frieze patterns

Frieze Patterns 363

Lesson

6-9Frieze Patterns

Lesson 6-9

Vocabulary

translation symmetry

frieze pattern

BIG IDEA The patterns found on friezes on buildings and in designs that repeat along a line can be classifi ed by their symmetries.

At this point you have studied two basic types of symmetry: refl ection symmetry and rotation symmetry. Refl ections and rotations are only two of the four isometries. What about translations and glide refl ections? Is there such thing as translation symmetry? We explore this by translating an image.

A fi gure F has translation symmetry if and only if there is a translation T with nonzero magnitude such that T(F) = F. To create a design that has translation symmetry, we need more than a fi gure and its image, as shown below. This pair of shapes does not have translation symmetry, because every time it is translated, the fi gure extends in the same direction.

v

P P

�

However, if we let this pattern continue forever in both directions and translate the whole infi nitely long strip by the vector

�v , the image of

the strip is indistinguishable from the original. Thus, this infi nitely long strip has translation symmetry!

v

P PP PPP P… P PPP P …

�

What Is a Frieze Pattern?

The pattern shown above is an example of what is known as a frieze pattern. A frieze pattern is any pattern in which a single fundamental fi gure is repeated to form a pattern that has a fi xed height but infi nite length in two opposite directions. In the pattern above, the fundamental fi gure is the because it is the smallest portion of the pattern that could be repeated to create the entire strip. (Imagine a cookie cutter that can be fl ipped, slid, and rotated.)

Multiple Choice E is the midpoint of

___ MT . Choose

the justifi cation that allows you to conclude that ___

ME � __

ET .

A Segment Congruence Theorem

B CPCF Theorem

C defi nition of congruence

D defi nition of midpoint

Mental Math

READING MATH

The name “frieze” is an architectural term that comes from the latin frisium, which means “embroidered border” or the “decorative strip at the top of the wall.”

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364 Polygons and Symmetry

Chapter 6

If you look at the top edge of older buildings, you may see many examples of frieze patterns just by walking down the street. Some other real examples include sidewalks, fences, the carpet in a hallway, a wallpaper border in a room, the border of a web page—any pattern that is in the shape of a strip in which a single pattern is used to create the strip.

The Seven Symmetries

of Frieze Patterns

All frieze patterns possess translation symmetry. Depending upon the frieze pattern, however, there may be other types of symmetries.

Example 1

Describe all of the symmetries of the frieze pattern

at the right.

Solution This frieze pattern has 180º

rotation symmetry about any point halfway

up and halfway between an M and a W. (We’ve drawn two such points at the right). It also has refl ection symmetry about a vertical

line that can be drawn through the middle of the M or the W. Finally, this pattern

has glide-refl ection symmetry because you can translate the pattern (so that

the M is on top of a W) and then refl ect the pattern over the horizontal line.

Two frieze patterns are considered to be equivalent if they have the same symmetries. For instance, the pattern is equivalent to because these two patterns have only translation symmetry. Likewise, is equivalent to

Remarkably, when we classify friezes by their symmetries, there are only seven different frieze patterns! An example of each type is shown at the top of the next page, along with the symmetries that describe it (we leave translation symmetry out, since all frieze patterns have translation symmetry). To simplify things, we use the terms “vertical” and “horizontal.” This assumes that, as you look at it, the pattern extends forever to the left and right.

Why are there only 7 frieze patterns? Why not more? After all, when you look again at the frieze pattern chart you should notice that there are four possible symmetries and each is either a yes or a no. Because there are 2 options for each, there are 2 · 2 · 2 · 2 = 16 possible combinations! Yet, there are only 7 patterns. Where did the other 9 go? We consider two of the other 9 in Example 2. You will see others in the questions.

frieze in fl ooring

railroad tracks

frieze from outside of building

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Frieze Patterns 365

Lesson 6-9

Pattern 180º rotation symmetry

Refl ection symmetry over horizontal line

Refl ection symmetry over a vertical line

Glide refl ection symmetry

1. no no no no

2. yes no no no

3. no yes no yes

4. no no yes no

5. yes yes yes yes

6. yes no yes yes

7. no no no yes

QY1

Example 2

Why is there not a frieze pattern on the chart that is marked

with no-yes-yes-yes?

Solution Suppose a frieze pattern F had a vertical symmetry

line v and a horizontal symmetry line h. In Lesson 5-7, we proved

that if a fi gure has two intersecting symmetry lines, it must also

have rotation symmetry. Thus, no-yes-yes-yes is not possible.

QY2

Questions

COVERING THE IDEAS

1. What is a frieze pattern?

2. Why is it impossible to have translation or glide refl ection symmetry in a shape that has fi nite length?

In 3−9, a frieze pattern is given. Refer to the table at the top of

this page. Give the number of the frieze pattern that has the same

symmetry as this one.

3. . . . . . . 4. . . . . . .

5. . . . . . . 6. . . . . . .

7. . . . . . . 8. . . . . . .

9. . . . . . .

QY1

Why is 180º the only magnitude for a rotation symmetry that a frieze pattern can have?

QY2

The explanation given in the Solution for Example 2 shows that another combination of symmetries is not possible. Which one?


366 Polygons and Symmetry

Chapter 6

APPLYING THE MATHEMATICS

In 10−16, repeat the directions for Questions 3−9.

10. 11.

12. 13.

14. 15.

16.

In 17 and 18, refer to the table on page 365. Which frieze pattern

best matches the graph of the function, ignoring the axes? (You

will study these functions in later mathematics courses.)

17. 18.

19. a. Recall that all frieze patterns have translation symmetry. Use this to explain why if a frieze pattern has a horizontal symmetry line, then it also must have glide refl ection symmetry.

b. What arrangements of yes’s and no’s (from the chart) does your answer from Part a allow you to eliminate as possible descriptions of frieze patterns?

20. Create one new example of each type of frieze pattern. Number your examples 1−7 to match the patterns in the chart.

y

y = sin (x)

x5 10

24

�2�4

�5�10

y

x2 4

2

4

�2

�4

�2�4

y = tan (x)


Frieze Patterns 367

Lesson 6-9

In 21−23, recall Lesson 4-8 on transformations in music. Consider

the frieze pattern found in the oval note heads only. Assume notes

are repeated forever (even though in the music they are not). Give

the number of the equivalent frieze pattern.

21. This is from Ludwig van Beethoven’s Piano Sonata No. 14 in C-sharp minor, Op. 27, No. 2, the “Moonlight Sonata.”

22. This is from Igor Stravinsky’s ballet Petrushka.

23. This is from Wolfgang Amadeus Mozart’s Requiem Mass.

REVIEW

24. Suppose � and m are symmetry lines for a regular hexagon. What is the smallest possible value for the measure of the acute angle between � and m? (Lesson 6-8)

25. The points A = (–1, 0), B = (1, 0) and E = (–1, 2 √ � 3 ) are three

of the six vertices of a regular hexagon. (Lesson 6-8)

a. Find the center of the circle on which A, B, and E lie. b. Is this the center of the hexagon? Explain.

26. Draw an example of a fi gure with 5-fold rotation symmetry that has no lines of symmetry. (Lesson 6-7)

27. Determine whether the following statement is always, sometimes but not always, or never true: A rhombus has exactly two lines of symmetry. (Lessons 6-4, 6-1)

28. Explain what the refl exive, symmetric, and transitive properties of congruence say. (Lesson 5-1)

EXPLORATION

29. Find one real-life example of at least four of the different types of frieze patterns. For each pattern, provide an image (a sketch or a picture), a description of what pattern it matches, and where you found it. We ask you to fi nd only four because fi nding examples of all seven is often diffi cult. Can you fi nd more than four?

QY ANSWERS

1. Any other magnitude would cause the pattern to be at a tilt, and thus not at a fi xed height.

2. no-yes-yes-no


lesson 6-9 frieze patterns

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