Frieze Pattern

Mathematics asbyaR.Liberal MATH 122 Lecture notes Hammack Arts

Frieze Symmetries

There are five basic symmetry operations that can be applied to a frieze pattern: Translation (T), Glide reflection (G), Rotation (R), Vertical reflection (V) and Horizontal reflection (R). These are illustrated below.

We say that a frieze pattern has a particular symmetry if that symmetry operation can be applied to the pattern without changing the appearance of the pattern.

T

(Translation)

G

(Glide reflection)

Translation – moves every point a constant distance in a speci3ied direction

(Rotation)

Glide-re3lection – the combination of a re3lection in a line and a translation along that line

R

Rotation – a circular movement about center of rotation

(Vertical reflection)

V

Re3lection – a transformation in which the direction of the one axis is reversed

(Horizontal reflection)

H

Any given frieze pattern may have some of these symmetries but not others. However, a frieze pattern can have only one of seven possible combinations of symmetries, illustrated below.

1.

T

2.

TG

3.

TGH

4.

TV

5.

TR

6.

TGRV

7.

TGRVH

Further examples are shown on the next page.

1

The Seven Frieze Patterns

1.

T

2.

TG

3.

TR

4.

TV

5.

TGRV

6.

TGH

7.

TGRVH

KEY: T = Translation R = Rotation by 180 V = Vertical Reflection

G = Glide Reflection H = Horizontal Reflection

2

Frieze patterns can of be compared to periodic functions. Considergive the Q1. FindQ2.which (if any) the above transformations trigonometric functions.

Use graphing calculators to graph each pattern (3ill in the chart).

Put a check mark on the the tan(x) function. Determine if this is a frieze pattern. column for each symmetry that were executed on the pattern.

Data T

R G V

H

type

a

Typical Answers

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Determine the Frieze pattern of the following