SYMMETRIC PATTERNS The Nature of Mathematics and

Mathematics in Nature

SYMMETRIC PATTERNS

A figure has symmetry if there is a non-trivial transformation that maps the figure onto itself.

TYPES OF SYMMETRIC PATTERNS

A design is a figure with at least one non-trivial symmetry. A pattern is a design that has a translation symmetry. A plane pattern has symmetry if there is an isometry of the plane that preserves it.

TYPES OF SYMMETRIC PATTERNS

ROSETTE PATTERN

FRIEZE PATTERN

WALLPAPER PATTERN

ROSETTE PATTERNThis pattern consist of taking a motif or an element and rotating and/or reflecting that element.

Types of Rosette pattern: Cyclic –has n fold rotational symmetry and no reflectional symmetry and Dihedral – has n fold rotational symmetry and reflectional symmetry

ROSETTE PATTERN

EXAMPLESIdentify what type of Rosette Pattern each figure has.

A B C

EXAMPLESIdentify what type of Rosette Pattern each figure has.

D E F

FRIEZE PATTERNThis pattern is an infinitely long strip imprinted with a design given by a repeating pattern motif.

7 POSSIBLE FRIEZE PATTERNS

Frieze pattern that has only the translation symmetry.

7 POSSIBLE FRIEZE PATTERNS

Frieze pattern that has only the translation and horizontal symmetry.

7 POSSIBLE FRIEZE PATTERNS

Frieze pattern that has only the translation symmetry and glide reflection.

7 POSSIBLE FRIEZE PATTERNS

Frieze pattern that has only the translation symmetry and rotation.

7 POSSIBLE FRIEZE PATTERNS

Frieze pattern that has only the translation , horizontal symmetry and glide reflection.

7 POSSIBLE FRIEZE PATTERNS

Frieze pattern that has only the translation , vertical symmetry, rotation and glide reflection.

7 POSSIBLE FRIEZE PATTERNS

Frieze pattern that has only the translation , vertical , horizontal symmetry, rotation and glide reflection.

FIRMING UPIdentify the FRIEZE PATTERN used in each design.

WALLPAPER PATTERN

Whereas a frieze pattern can be mapped onto itself by a horizontal translation, a wallpaper pattern covers the plan and can be mapped onto itself by translation in more than one direction

WALLPAPER PATTERN

Whereas a frieze pattern can be mapped onto itself by a horizontal translation, a wallpaper pattern covers the plan and can be mapped onto itself by translation in more than one direction

(Insert both pictures found on page 298)

EXAMPLE1. Using the motif

2. Reflect the motif vertically to obtain

3. Using the resulting figure rotate 120° and 240° to produce

4. Translate the resulting figure to obtain

5. Glide the resulting pattern downward to produce.

EXAMPLES

EXAMPLES

EXAMPLES

DEEPENING

1. Explain why a frieze patern can have only 1800 rotation.

2. Use the motif below to draw seven frieze patterns, one with each symmetry group.